people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · models hydrod. 2nd class before question...
TRANSCRIPT
![Page 1: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/1.jpg)
Models Hydrod. 2nd class Before Question BCRW
Second class particles can perform simplerandom walks (in some cases)
Joint withGyorgy Farkas, Peter Kovacs and Attila Rakos
Marton Balazs
University of Bristol
Warwick Probability SeminarOctober 23, 2013.
![Page 2: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/2.jpg)
Models Hydrod. 2nd class Before Question BCRW
The modelsAsymmetric simple exclusion processZero range processGeneralized ZRPBricklayers processStationary distributions
Hydrodynamics
The second class particle
Earlier results
The question
Branching coalescing random walk
![Page 3: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/3.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
![Page 4: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/4.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 5: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/5.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 6: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/6.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 7: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/7.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 8: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/8.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 9: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/9.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 10: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/10.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 11: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/11.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 12: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/12.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 13: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/13.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 14: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/14.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 15: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/15.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 16: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/16.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 17: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/17.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 18: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/18.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 19: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/19.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 20: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/20.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 21: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/21.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 22: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/22.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 23: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/23.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 24: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/24.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 25: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/25.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 26: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/26.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 27: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/27.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 28: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/28.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 29: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/29.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 30: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/30.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 31: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/31.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 32: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/32.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 33: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/33.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 34: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/34.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 35: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/35.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 36: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/36.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 37: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/37.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 38: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/38.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 39: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/39.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 40: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/40.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 41: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/41.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 42: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/42.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 43: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/43.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 44: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/44.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 45: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/45.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 46: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/46.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 47: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/47.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 48: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/48.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 49: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/49.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 50: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/50.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 51: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/51.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 52: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/52.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 53: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/53.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 54: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/54.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Asymmetric simple exclusion
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •
Particles try to jump
to the right with rate p,to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied byanother particle.
![Page 55: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/55.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •••
•
![Page 56: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/56.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •••
•
}
ω−3=2
![Page 57: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/57.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •••
•
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 58: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/58.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •••
•
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 59: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/59.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •••
•
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 60: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/60.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• ••• •
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 61: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/61.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• • •
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 62: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/62.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• • •
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 63: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/63.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• ••
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 64: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/64.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• ••
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 65: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/65.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• ••
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 66: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/66.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• ••
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 67: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/67.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• ••
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 68: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/68.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• ••
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 69: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/69.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• ••
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 70: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/70.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• • •
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 71: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/71.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• • •
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 72: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/72.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• • •
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 73: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/73.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• • •
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 74: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/74.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• • •
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 75: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/75.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• • •
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 76: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/76.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric zero range process
-3 -2 -1 0 1 2 3 4 i
•• •• • •
}
ω−3=2
Particles jump to the right from site i with rate r(ωi)
(r non-decreasing).
![Page 77: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/77.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
![Page 78: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/78.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 79: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/79.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 80: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/80.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 81: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/81.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 82: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/82.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 83: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/83.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 84: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/84.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 85: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/85.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 86: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/86.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 87: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/87.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 88: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/88.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 89: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/89.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 90: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/90.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 91: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/91.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 92: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/92.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 93: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/93.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 94: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/94.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 95: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/95.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 96: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/96.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 97: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/97.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 98: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/98.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 99: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/99.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 100: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/100.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 101: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/101.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 102: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/102.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 103: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/103.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 104: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/104.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 105: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/105.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 106: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/106.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 107: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/107.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 108: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/108.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 109: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/109.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 110: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/110.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 111: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/111.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 112: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/112.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 113: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/113.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 114: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/114.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 115: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/115.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
A generalized totally asymmetric zero range process:ωi ∈ Z
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi).
(r non-decreasing).
![Page 116: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/116.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
![Page 117: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/117.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 118: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/118.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 119: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/119.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 120: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/120.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 121: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/121.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 122: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/122.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 123: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/123.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 124: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/124.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 125: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/125.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 126: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/126.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 127: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/127.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 128: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/128.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 129: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/129.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 130: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/130.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 131: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/131.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
}
ωi+1=−1
(
ωi
ωi+1
)
||(
1−1
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 132: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/132.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 133: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/133.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 134: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/134.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 135: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/135.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 136: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/136.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 137: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/137.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 138: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/138.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 139: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/139.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 140: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/140.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 141: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/141.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 142: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/142.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 143: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/143.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 144: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/144.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 145: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/145.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 146: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/146.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 147: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/147.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 148: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/148.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 149: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/149.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 150: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/150.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 151: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/151.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 152: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/152.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 153: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/153.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 154: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/154.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 155: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/155.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 156: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/156.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 157: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/157.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 158: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/158.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 159: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/159.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 160: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/160.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
![Page 161: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/161.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
The totally asymmetric bricklayers process
i i + 1
(
ωi
ωi+1
)
→
(
ωi − 1ωi+1 + 1
)
|| ||(
1−1
)
→
(
00
)
a brick is added with rate r(ωi) + r(−ωi+1).
A mirror-symmetrized version of the extended zero range. Leftand right jumps of the dynamics cooperate, if(r(ω) · r(1 − ω) = 1; r non-decreasing).
![Page 162: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/162.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Stationary product distributions
For the ASEP: the Bernoulli() distribution is time-stationary forany (0 ≤ ≤ 1).
![Page 163: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/163.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Stationary product distributions
For the ASEP: the Bernoulli() distribution is time-stationary forany (0 ≤ ≤ 1).
For zero range, bricklayers: the product of marginals
µθ(ωi) =eθωi
r(ωi)!·
1Z (θ)
is stationary for any θ ∈ R that makes Z (θ) finite.
![Page 164: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/164.jpg)
Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.
Stationary product distributions
For the ASEP: the Bernoulli() distribution is time-stationary forany (0 ≤ ≤ 1).
For zero range, bricklayers: the product of marginals
µθ(ωi) =eθωi
r(ωi)!·
1Z (θ)
is stationary for any θ ∈ R that makes Z (θ) finite.
Here r(0)! : = 1, and r(z + 1)! = r(z)! · r(z + 1) for all z ∈ Z.
![Page 165: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/165.jpg)
Models Hydrod. 2nd class Before Question BCRW
Hydrodynamics (very briefly)
The density = (θ) : = Eθ(ω) and the hydrodynamic fluxH = Hθ : = Eθ[growth rate] both depend on a parameter or θof the stationary distribution.
![Page 166: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/166.jpg)
Models Hydrod. 2nd class Before Question BCRW
Hydrodynamics (very briefly)
The density = (θ) : = Eθ(ω) and the hydrodynamic fluxH = Hθ : = Eθ[growth rate] both depend on a parameter or θof the stationary distribution.
◮ H() = Hθ() is the hydrodynamic flux function.
![Page 167: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/167.jpg)
Models Hydrod. 2nd class Before Question BCRW
Hydrodynamics (very briefly)
The density = (θ) : = Eθ(ω) and the hydrodynamic fluxH = Hθ : = Eθ[growth rate] both depend on a parameter or θof the stationary distribution.
◮ H() = Hθ() is the hydrodynamic flux function.◮ If the process is locally in equilibrium, but changes over
some large scale (variables X = εi and T = εt), then
∂T(T , X ) + ∂X H((T , X )) = 0 (conservation law).
![Page 168: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/168.jpg)
Models Hydrod. 2nd class Before Question BCRW
Hydrodynamics (very briefly)
The density = (θ) : = Eθ(ω) and the hydrodynamic fluxH = Hθ : = Eθ[growth rate] both depend on a parameter or θof the stationary distribution.
◮ H() = Hθ() is the hydrodynamic flux function.◮ If the process is locally in equilibrium, but changes over
some large scale (variables X = εi and T = εt), then
∂T(T , X ) + ∂X H((T , X )) = 0 (conservation law).
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 169: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/169.jpg)
Models Hydrod. 2nd class Before Question BCRW
Hydrodynamics (very briefly)
For the ASEP, H() = (p − q) · (1 − ), concave.
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 170: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/170.jpg)
Models Hydrod. 2nd class Before Question BCRW
Hydrodynamics (very briefly)
For the ASEP, H() = (p − q) · (1 − ), concave.
For the zero range and bricklayers, H() is convex/concave ifthe rate function r is convex/concave.
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 171: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/171.jpg)
Models Hydrod. 2nd class Before Question BCRW
Hydrodynamics (very briefly)
For the ASEP, H() = (p − q) · (1 − ), concave.
For the zero range and bricklayers, H() is convex/concave ifthe rate function r is convex/concave.
Special case: r(ω) = const. · eβω; H() is convex.
will be interested in TAGEZRP, TAEBLP.
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 172: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/172.jpg)
Models Hydrod. 2nd class Before Question BCRW
Hydrodynamics (very briefly)
For the ASEP, H() = (p − q) · (1 − ), concave.
For the zero range and bricklayers, H() is convex/concave ifthe rate function r is convex/concave.
Special case: r(ω) = const. · eβω; H() is convex.
will be interested in TAGEZRP, TAEBLP.
Either convex or concave, discontinuous shock solutionsexist.
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 173: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/173.jpg)
Models Hydrod. 2nd class Before Question BCRW
Rarefaction wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
left
right
Traffic jam
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 174: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/174.jpg)
Models Hydrod. 2nd class Before Question BCRW
Rarefaction wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
left
right
Traffic jam
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
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right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 253: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/253.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 254: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/254.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 255: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/255.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 256: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/256.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 257: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/257.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 258: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/258.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 259: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/259.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 260: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/260.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 261: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/261.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 262: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/262.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 263: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/263.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 264: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/264.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 265: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/265.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
∂T+ ∂X H() = 0 = E(ω), H() = Eθ()[growth rate]
![Page 266: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/266.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
Discontinuous shock appears. Its velocity is given by theRankine-Hugoniot speed for densities left and right
R =H(left)− H(right)
left − right.
![Page 267: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/267.jpg)
Models Hydrod. 2nd class Before Question BCRW
Shock wave
X
(T , X)
0.2
0.4
0.6
0.8
1.0
Traffic jam
left
right
Discontinuous shock appears. Its velocity is given by theRankine-Hugoniot speed for densities left and right
R =H(left)− H(right)
left − right.
Let’s look for the corresponding microscopic structure.
![Page 268: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/268.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• •• ••
![Page 269: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/269.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• •• ••
![Page 270: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/270.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• •• ••
Growth on the right:rate≤rate
![Page 271: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/271.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• •• ••
Growth on the right:rate≤ratewith rate:
![Page 272: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/272.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•••• •• ••
•••
Growth on the right:rate≤ratewith rate:
![Page 273: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/273.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•••• •• ••
•••
Growth on the right:rate≤ratewith rate:
![Page 274: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/274.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•••• •• ••
•••
Growth on the right:rate≤ratewith rate:
![Page 275: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/275.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•••• • • ••
•••
Growth on the right:rate≤ratewith rate:
![Page 276: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/276.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•••• • • ••
•••
Growth on the right:rate≤ratewith rate:
![Page 277: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/277.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•••• • • •
•
•••
Growth on the right:rate≤ratewith rate:
![Page 278: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/278.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•••• • ••
•
•••
Growth on the right:rate≤ratewith rate:
![Page 279: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/279.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•••• • ••
•
•••
Growth on the right:rate≤ratewith rate:
![Page 280: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/280.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•••• • ••
•
•••
Growth on the right:rate≤ratewith rate:
![Page 281: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/281.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•••• • ••
•
•••
Growth on the right:rate≤ratewith rate:
![Page 282: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/282.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• •• ••
Growth on the right:rate≤ratewith rate-rate:
![Page 283: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/283.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• •• ••
Growth on the right:rate≤ratewith rate-rate:
![Page 284: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/284.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• •• ••
Growth on the right:rate≤ratewith rate-rate:
![Page 285: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/285.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• •• ••
Growth on the right:rate≤ratewith rate-rate:
![Page 286: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/286.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• • • ••
Growth on the right:rate≤ratewith rate-rate:
![Page 287: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/287.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• • • ••
Growth on the right:rate≤ratewith rate-rate:
![Page 288: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/288.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• • • •
•
Growth on the right:rate≤ratewith rate-rate:
![Page 289: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/289.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• • ••
•
Growth on the right:rate≤ratewith rate-rate:
![Page 290: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/290.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• • ••
•
Growth on the right:rate≤ratewith rate-rate:
![Page 291: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/291.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• • ••
•
Growth on the right:rate≤ratewith rate-rate:
![Page 292: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/292.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
i
•• • •••• • ••
•
Growth on the right:rate≤ratewith rate-rate:
![Page 293: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/293.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥rate
i
•• • •••• •• ••
![Page 294: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/294.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
•• • •••• •• ••
![Page 295: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/295.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
• ••• ••• ••
• •
![Page 296: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/296.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
• ••• ••• ••
• •
![Page 297: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/297.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
• ••• ••• ••
• •
![Page 298: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/298.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
• ••• ••• ••
• •
![Page 299: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/299.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
• ••• ••• ••
• •
![Page 300: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/300.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
• ••• ••• •
•
• •
![Page 301: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/301.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
• ••• ••••
•
••
![Page 302: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/302.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
• ••• ••••
•
••
![Page 303: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/303.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
• ••• ••••
•
••
![Page 304: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/304.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate:
i
• ••• ••••
•
••
![Page 305: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/305.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
•• • •••• •• ••
![Page 306: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/306.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
• •
![Page 307: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/307.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
• •
![Page 308: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/308.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
• •
![Page 309: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/309.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
• •
![Page 310: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/310.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
• •
![Page 311: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/311.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
• •
![Page 312: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/312.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
••
![Page 313: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/313.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
••
![Page 314: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/314.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
••
![Page 315: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/315.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
••
![Page 316: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/316.jpg)
Models Hydrod. 2nd class Before Question BCRW
The second class particle
States ω and ω only differ at one site.
Growth on the left:rate≥ratewith rate-rate:
i
• •••• •• ••
••
A single discrepancy , the second class particle, is conserved.
![Page 317: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/317.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: as seen by the second class particle
From now on: ASEP, TAGEZRP, TAEBLP only; “E”=exponential.
Theorem (Derrida, Lebowitz, Speer ’97)For the ASEP, the Bernoulli product distribution with densities
-3 -2 -1 0 1 2 3
left 0
right
i
i
is stationary for the process, as seen from the second class particle, if
right · (1 − left)
left · (1 − right)=
pq.
![Page 318: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/318.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: as seen by the second class particle
From now on: ASEP, TAGEZRP, TAEBLP only; “E”=exponential.
Theorem (Derrida, Lebowitz, Speer ’97)For the ASEP, the Bernoulli product distribution with densities
-3 -2 -1 0 1 2 3
left 0
1right
i
i
is stationary for the process, as seen from the second class particle, if
right · (1 − left)
left · (1 − right)=
pq.
![Page 319: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/319.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Theorem (Belitsky and Schutz ’02)For the ASEP with the very same parameters, the Bernoulliproduct distribution µ0 with densities
-3 -2 -1 0 1 2 3
left
right
i
i
evolves according to
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0].
![Page 320: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/320.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Theorem (Belitsky and Schutz ’02)For the ASEP with the very same parameters, the Bernoulliproduct distribution µ0 with densities
-3 -2 -1 0 1 2 3
left
right
i
i
evolves according to
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0].
Multiple shocks and their interactions are also handled.
![Page 321: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/321.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · leftright
:
-3 -2 -1 0 1 2 3
left
right
i
i
X (t) = 0µ0
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 322: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/322.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · leftright
:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 323: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/323.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · leftright
:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 324: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/324.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · leftright
:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 325: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/325.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · leftright
:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 326: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/326.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · leftright
:
-3 -2 -1 0 1 2 3
left
right
i
i
X (t) = −1µ−1
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 327: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/327.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate q ·rightleft
:
-3 -2 -1 0 1 2 3
left
right
i
i
X (t) = 0µ0
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 328: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/328.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate q ·rightleft
:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 329: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/329.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate q ·rightleft
:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 330: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/330.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate q ·rightleft
:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 331: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/331.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate q ·rightleft
:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 332: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/332.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate q ·rightleft
:
-3 -2 -1 0 1 2 3
left
right
i
i
X (t) = 1µ1
ddtµ0 = p ·
left
right· [µ
−1 − µ0] + q ·right
left· [µ1 − µ0]
![Page 333: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/333.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: as seen by the second class particle
Theorem (B. ’01)For the TAEBLP, the product distribution of marginals µi withdensities
-3 -2 -1 0 1 2 3
left
right
i
i
is stationary for the process, as seen from the second classparticle, if
left − right = 1.
![Page 334: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/334.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: as seen by the second class particle
Theorem (B. ’01)For the TAEBLP, the product distribution of marginals µi withdensities
-3 -2 -1 0 1 2 3
left
right
i
i
is stationary for the process, as seen from the second classparticle, if
left − right = 1.
![Page 335: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/335.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Theorem (B. ’04)For the very same parameters, the product distribution µ0 withdensities
-3 -2 -1 0 1 2 3
left
right
i
i
evolves according to
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 336: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/336.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Theorem (B. ’04)For the very same parameters, the product distribution µ0 withdensities
-3 -2 -1 0 1 2 3
left
right
i
i
evolves according to
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
Multiple shocks and their interactions are also handled.
![Page 337: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/337.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cleft:
-3 -2 -1 0 1 2 3
left
right
i
i
X (t) = 0µ0
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 338: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/338.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cleft:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 339: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/339.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cleft:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 340: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/340.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cleft:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 341: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/341.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cleft:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 342: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/342.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cleft:
-3 -2 -1 0 1 2 3
left
right
i
i
X (t) = −1µ−1
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 343: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/343.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cright:
-3 -2 -1 0 1 2 3
left
right
i
i
X (t) = 0µ0
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 344: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/344.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cright:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 345: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/345.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cright:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 346: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/346.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cright:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 347: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/347.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cright:
-3 -2 -1 0 1 2 3
left
right
i
i
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 348: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/348.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate Cright:
-3 -2 -1 0 1 2 3
left
right
i
i
X (t) = 1µ1
ddtµ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].
![Page 349: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/349.jpg)
Models Hydrod. 2nd class Before Question BCRW
The question
Of course, the drift of the walk X (t) is the same as theexpected drift of the second class particle in its stationaryshock distribution,
![Page 350: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/350.jpg)
Models Hydrod. 2nd class Before Question BCRW
The question
Of course, the drift of the walk X (t) is the same as theexpected drift of the second class particle in its stationaryshock distribution,
and also agrees with the Rankine Hugoniot formula for thespeed of shocks.
![Page 351: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/351.jpg)
Models Hydrod. 2nd class Before Question BCRW
The question
Of course, the drift of the walk X (t) is the same as theexpected drift of the second class particle in its stationaryshock distribution,
and also agrees with the Rankine Hugoniot formula for thespeed of shocks.
Is it the second class particle that performs the simple randomwalk in the middle of a shock?
![Page 352: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/352.jpg)
Models Hydrod. 2nd class Before Question BCRW
The question
Of course, the drift of the walk X (t) is the same as theexpected drift of the second class particle in its stationaryshock distribution,
and also agrees with the Rankine Hugoniot formula for thespeed of shocks.
Is it the second class particle that performs the simple randomwalk in the middle of a shock?
In what sense? Annealed w.r.t. the initial shock distribution...But what does this mean?
![Page 353: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/353.jpg)
Models Hydrod. 2nd class Before Question BCRW
Here is the question:
For the ASEP, let ν0 be the Bernoulli product distribution
ν0 =(
⊗
i<0
µleft
)
⊗(
δ)
⊗(
⊗
i>0
µright
)
,
where
µ(ω = ω) =
{
, if ω = 1,
1 − , if ω = 0;δ(0, 1) = 1.
-3 -2 -1 0 1 2 3
left 0
1right
i
i
![Page 354: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/354.jpg)
Models Hydrod. 2nd class Before Question BCRW
Here is the question:
For the ASEP, let ν0 be the Bernoulli product distribution
-3 -2 -1 0 1 2 3
left 0
1right
i
i
Does it satisfy
ddtν0 = p ·
left
right· [ν
−1 − ν0] + q ·right
left· [ν1 − ν0]
whenright · (1 − left)
left · (1 − right)=
pq
?
![Page 355: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/355.jpg)
Models Hydrod. 2nd class Before Question BCRW
Here is the question:
For the TAEBLP, let ν0 be the product distribution
ν0 =(
⊗
i<0
µleft
)
⊗(
δright
)
⊗(
⊗
i>0
µright
)
,
where
µ(ω = ω) =eθ()·ω
r(ω)!·
1Z (θ())
;
δ(ω, ω + 1) =eθ()·ω
r(ω)!·
1Z (θ())
.
-3 -2 -1 0 1 2 3
left
right
i
i
![Page 356: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/356.jpg)
Models Hydrod. 2nd class Before Question BCRW
Here is the question:For the TAEBLP, let ν0 be the product distribution
ν0 =(
⊗
i<0
µleft
)
⊗(
δright
)
⊗(
⊗
i>0
µright
)
,
-3 -2 -1 0 1 2 3
left
right
i
i
Does it satisfy
ddtν0 = Cleft · [ν−1 − ν0] + Cright · [ν1 − ν0].
whenleft − right = 1 ?
![Page 357: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/357.jpg)
Models Hydrod. 2nd class Before Question BCRW
Here is the answer:
Theorem (Gy. Farkas, P. Kovacs, A. Rakos, B. ’10)Yes, and yes. Even more, the thing also works for theTAGEZRP.
![Page 358: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/358.jpg)
Models Hydrod. 2nd class Before Question BCRW
Here is the answer:
Theorem (Gy. Farkas, P. Kovacs, A. Rakos, B. ’10)Yes, and yes. Even more, the thing also works for theTAGEZRP.
The second class particle, annealed w.r.t. the initial shockproduct distribution, does perform a drifted simple random walkin these cases.
![Page 359: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/359.jpg)
Models Hydrod. 2nd class Before Question BCRW
Here is the answer:
Theorem (Gy. Farkas, P. Kovacs, A. Rakos, B. ’10)Yes, and yes. Even more, the thing also works for theTAGEZRP.
The second class particle, annealed w.r.t. the initial shockproduct distribution, does perform a drifted simple random walkin these cases.
This explains both types of the previous results.
![Page 360: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/360.jpg)
Models Hydrod. 2nd class Before Question BCRW
Here is the answer:
Theorem (Gy. Farkas, P. Kovacs, A. Rakos, B. ’10)Yes, and yes. Even more, the thing also works for theTAGEZRP.
The second class particle, annealed w.r.t. the initial shockproduct distribution, does perform a drifted simple random walkin these cases.
This explains both types of the previous results.
The presence of a second class particle in the measuresignificantly simplifies the computations. This is how wediscovered the TAGEZRP.
![Page 361: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/361.jpg)
Models Hydrod. 2nd class Before Question BCRW
Nice, since
This might open up the path for applying methods physicistslike.
![Page 362: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/362.jpg)
Models Hydrod. 2nd class Before Question BCRW
Nice, since
This might open up the path for applying methods physicistslike.
It also gives a rough tail bound for the second class particle in aflat initial distribution; essential in the t2/3 proofs for theexponential models.
![Page 363: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/363.jpg)
Models Hydrod. 2nd class Before Question BCRW
Interactions:
We also see that shocks+second class particles◮ locally interact by exclusion in ASEP, and don’t locally
interact in TAGEZRP, TAEBLP, but
![Page 364: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/364.jpg)
Models Hydrod. 2nd class Before Question BCRW
Interactions:
We also see that shocks+second class particles◮ locally interact by exclusion in ASEP, and don’t locally
interact in TAGEZRP, TAEBLP, but
◮ their jump rates depend on their position, making themattract each other such that
![Page 365: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/365.jpg)
Models Hydrod. 2nd class Before Question BCRW
Interactions:
We also see that shocks+second class particles◮ locally interact by exclusion in ASEP, and don’t locally
interact in TAGEZRP, TAEBLP, but
◮ their jump rates depend on their position, making themattract each other such that
◮ their center of mass has the Rankine-Hugoniot velocity forthe large shock they jointly represent.
![Page 366: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/366.jpg)
Models Hydrod. 2nd class Before Question BCRW
Interactions:
We also see that shocks+second class particles◮ locally interact by exclusion in ASEP, and don’t locally
interact in TAGEZRP, TAEBLP, but
◮ their jump rates depend on their position, making themattract each other such that
◮ their center of mass has the Rankine-Hugoniot velocity forthe large shock they jointly represent.
Macroscopically it’s one shock after all.
![Page 367: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/367.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
![Page 368: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/368.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate p: jump to the right
![Page 369: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/369.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate p: jump to the right
![Page 370: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/370.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate p: jump to the right
![Page 371: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/371.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate p: jump to the right
![Page 372: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/372.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate p: jump to the right
![Page 373: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/373.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate p: jump to the right
![Page 374: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/374.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate p: jump to the right
![Page 375: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/375.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate p: jump to the right
![Page 376: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/376.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate p: jump to the right
![Page 377: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/377.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate p: jump to the right
![Page 378: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/378.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate q: jump to the left
![Page 379: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/379.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate q: jump to the left
![Page 380: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/380.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate q: jump to the left
![Page 381: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/381.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate q: jump to the left
![Page 382: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/382.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate q: jump to the left
![Page 383: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/383.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate q: jump to the left
![Page 384: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/384.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate q: jump to the left
![Page 385: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/385.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate q: jump to the left
![Page 386: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/386.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate q: jump to the left
![Page 387: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/387.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate q: jump to the left
![Page 388: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/388.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cr : coalescence to the right
![Page 389: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/389.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cr : coalescence to the right
![Page 390: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/390.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cr : coalescence to the right
![Page 391: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/391.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cr : coalescence to the right
![Page 392: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/392.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cr : coalescence to the right
![Page 393: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/393.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cr : coalescence to the right
![Page 394: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/394.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cr : coalescence to the right
![Page 395: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/395.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cr : coalescence to the right
![Page 396: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/396.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cr : coalescence to the right
![Page 397: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/397.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cr : coalescence to the right
![Page 398: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/398.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate bl : branching to the left
![Page 399: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/399.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate bl : branching to the left
![Page 400: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/400.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate bl : branching to the left
![Page 401: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/401.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate bl : branching to the left
![Page 402: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/402.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate bl : branching to the left
![Page 403: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/403.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate bl : branching to the left
![Page 404: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/404.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate bl : branching to the left
![Page 405: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/405.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate bl : branching to the left
![Page 406: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/406.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate bl : branching to the left
![Page 407: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/407.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate bl : branching to the left
![Page 408: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/408.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cl : coalescence to the left
![Page 409: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/409.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cl : coalescence to the left
![Page 410: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/410.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cl : coalescence to the left
![Page 411: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/411.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cl : coalescence to the left
![Page 412: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/412.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cl : coalescence to the left
![Page 413: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/413.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cl : coalescence to the left
![Page 414: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/414.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cl : coalescence to the left
![Page 415: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/415.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cl : coalescence to the left
![Page 416: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/416.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cl : coalescence to the left
![Page 417: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/417.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate cl : coalescence to the left
![Page 418: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/418.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 419: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/419.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 420: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/420.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 421: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/421.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 422: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/422.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 423: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/423.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 424: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/424.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 425: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/425.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 426: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/426.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 427: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/427.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 428: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/428.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
![Page 429: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/429.jpg)
Models Hydrod. 2nd class Before Question BCRW
A similar result: branching coalescing random walk
-3 -2 -1 0 1 2 3 4 i
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••
With rate br : branching to the right
The Bernoulli(∗) distribution is stationary for
∗ =bl + br
bl + br + cl + cr.
![Page 430: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/430.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: as seen by the rightmost particle
TheoremFor the BCRW, the Bernoulli product distribution with densities
-3 -2 -1 0 1 2 3
∗
= 0
i
i
is stationary for the process, as seen from the rightmostparticle.
![Page 431: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/431.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Theorem (Krebs, Jafarpour and Schutz ’03)For the BCRW with the very same parameters, the Bernoulliproduct distribution µ0 with densities
-3 -2 -1 0 1 2 3
∗
= 0
i
i
evolves according to
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 432: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/432.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate cl ·(bl+br )+q·(cl+cr )bl+br+cl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
X (t) = 0µ0
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 433: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/433.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate cl ·(bl+br )+q·(cl+cr )bl+br+cl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 434: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/434.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate cl ·(bl+br )+q·(cl+cr )bl+br+cl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 435: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/435.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate cl ·(bl+br )+q·(cl+cr )bl+br+cl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 436: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/436.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate cl ·(bl+br )+q·(cl+cr )bl+br+cl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 437: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/437.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate cl ·(bl+br )+q·(cl+cr )bl+br+cl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
X (t) = −1µ−1
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 438: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/438.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · bl+br+cl+crcl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
X (t) = 0µ0
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 439: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/439.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · bl+br+cl+crcl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 440: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/440.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · bl+br+cl+crcl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 441: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/441.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · bl+br+cl+crcl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 442: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/442.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · bl+br+cl+crcl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 443: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/443.jpg)
Models Hydrod. 2nd class Before Question BCRW
Earlier results: random walking shocks
Interpretation: random walking shock µ(t) = µX(t):
with rate p · bl+br+cl+crcl+cr
:
-3 -2 -1 0 1 2 3
∗
= 0
i
i
X (t) = 1µ1
ddtµ0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [µ
−1 − µ0]
+ p ·bl + br + cl + cr
cl + cr· [µ1 − µ0].
![Page 444: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/444.jpg)
Models Hydrod. 2nd class Before Question BCRW
The question:
Is it the rightmost particle that performs the random walk?
![Page 445: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/445.jpg)
Models Hydrod. 2nd class Before Question BCRW
Here is the question:For the BCRW, let ν0 be the Bernoulli product distribution
ν0 =(
⊗
i<0
µ∗)
⊗(
δ)
⊗(
⊗
i>0
µ0)
,
where δ(0) = 1.
-3 -2 -1 0 1 2 3
∗
= 0
i
i
1
Does it satisfy
ddtν0 =
cl · (bl + br ) + q · (cl + cr )
bl + br + cl + cr· [ν
−1 − ν0]
+ p ·bl + br + cl + cr
cl + cr· [ν1 − ν0]?
![Page 446: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/446.jpg)
Models Hydrod. 2nd class Before Question BCRW
The answer
◮ ... is, of course, yes again. [Gy. Farkas, P. Kovacs, A.Rakos, B. ’10]
![Page 447: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/447.jpg)
Models Hydrod. 2nd class Before Question BCRW
The answer
◮ ... is, of course, yes again. [Gy. Farkas, P. Kovacs, A.Rakos, B. ’10]
◮ Fronts of the other direction: 0 − 1 − ∗ can also behandled.
![Page 448: people.maths.bris.ac.ukmb13434/balazs_warwick13.pdf · Models Hydrod. 2nd class Before Question BCRW The question Of course, the drift of the walk X(t)is the same as the expected](https://reader036.vdocuments.us/reader036/viewer/2022081521/5b3b65d07f8b9a26728c78d8/html5/thumbnails/448.jpg)
Models Hydrod. 2nd class Before Question BCRW
The answer
◮ ... is, of course, yes again. [Gy. Farkas, P. Kovacs, A.Rakos, B. ’10]
◮ Fronts of the other direction: 0 − 1 − ∗ can also behandled.
Thank you.