latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/latentvotertalk.pdf ·...
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![Page 1: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/1.jpg)
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Latent voter model on random regular graphs
Shirshendu Chatterjee
Cornell University (visiting Duke U.)
Work in progress with Rick Durrett
April 25, 2011
S. Chatterjee Latent voter model
![Page 2: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/2.jpg)
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Outline
Definition of voter model and duality with coalescing random walks
Voter models on “Complex Networks" (Facebook)
Definition of Latent voter model (iPad) and mean field equations
Approximate duality with branching coalescing random walk
Limiting behavior
S. Chatterjee Latent voter model
![Page 3: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/3.jpg)
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Voter model on a graph
ξt(x) ∈ {0, 1} is the opinion of x at time t.E.g., 1 = Democrat, 0 = Republican
At times T xn , n ≥ 1 of a rate 1 Poisson process, voter x decides tochange her opinion independent of other voters.At time t = T xn picks a neighbor yxn at random and ξt(x) = ξt(yxn).For a convenient construction of the process we draw an arrowfrom (x, T xn ) to (yxn, T
xn ).
Arrows point to the opposite direction of the flow of information.But this choice will be useful in defining a dual process.
S. Chatterjee Latent voter model
![Page 4: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/4.jpg)
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Voter model on a graph
ξt(x) ∈ {0, 1} is the opinion of x at time t.E.g., 1 = Democrat, 0 = RepublicanAt times T xn , n ≥ 1 of a rate 1 Poisson process, voter x decides tochange her opinion independent of other voters.At time t = T xn picks a neighbor yxn at random and ξt(x) = ξt(yxn).
For a convenient construction of the process we draw an arrowfrom (x, T xn ) to (yxn, T
xn ).
Arrows point to the opposite direction of the flow of information.But this choice will be useful in defining a dual process.
S. Chatterjee Latent voter model
![Page 5: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/5.jpg)
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Voter model on a graph
ξt(x) ∈ {0, 1} is the opinion of x at time t.E.g., 1 = Democrat, 0 = RepublicanAt times T xn , n ≥ 1 of a rate 1 Poisson process, voter x decides tochange her opinion independent of other voters.At time t = T xn picks a neighbor yxn at random and ξt(x) = ξt(yxn).For a convenient construction of the process we draw an arrowfrom (x, T xn ) to (yxn, T
xn ).
Arrows point to the opposite direction of the flow of information.But this choice will be useful in defining a dual process.
S. Chatterjee Latent voter model
![Page 6: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/6.jpg)
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Graphical Representation
time
6
vertices of the graph
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S. Chatterjee Latent voter model
![Page 7: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/7.jpg)
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Graphical Representation
time
6
vertices of the graph
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S. Chatterjee Latent voter model
![Page 8: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/8.jpg)
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Graphical Representation
time
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vertices of the graph
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S. Chatterjee Latent voter model
![Page 9: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/9.jpg)
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What is x at time t?
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S. Chatterjee Latent voter model
![Page 10: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/10.jpg)
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What is x at time t?
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S. Chatterjee Latent voter model
![Page 11: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/11.jpg)
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What is x at time t?
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S. Chatterjee Latent voter model
![Page 12: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/12.jpg)
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What is x at time t?
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S. Chatterjee Latent voter model
![Page 13: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/13.jpg)
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What is x at time t?
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S. Chatterjee Latent voter model
![Page 14: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/14.jpg)
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What is x at time t?
To compute the state of x at time t we work backwards in time todefine ζx,ts , s ≤ t.
ζx,tr = x for r < s, the first time so that t− s = T xn for some n.
Then set ζx,ts = yxn, and repeat.
For all 0 ≤ s ≤ t, ξt(x) = ξt−s(ζx,ts ).
Follow the arrow. Jump to the neighbor you imitated.
ζx,ts is a random walk that jumps x→ y at rate 1/d(x)if y is a neighbor of x.
S. Chatterjee Latent voter model
![Page 15: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/15.jpg)
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Coalescing random walks
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S. Chatterjee Latent voter model
![Page 16: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/16.jpg)
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Duality
ζB,ts = {ζx,ts : x ∈ B} for s ≤ t. (Dual rw’s)Let ηAt = {x : ξt(x) = 1} when A = {x : ξ0 = 1}.Voter model written as a set-valued process.
{ηAt ∩B 6= ∅} = {ζB,tt ∩A 6= ∅}
Define ζBs , s ≥ 0 so that ζBs =d ζB,ts for s ≤ t
P (ηAt ∩B 6= ∅) = P (ζBt ∩A 6= ∅)
ζBt is a coalescing random walk: particles move independent untilthey hit, and coalesce to one when they hit.
S. Chatterjee Latent voter model
![Page 17: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/17.jpg)
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Finite graphs G
The voter model on a finite graph is a finite Markov chain with twoabsorbing states (all 0’s and all 1’s) and so eventually it reachescomplete consensus.
Q. How long does it take?
Let X1t and X2
t be independent random walks on the graph.
Let A = {(x, x) : x is a vertex of the graph},TA = inf{t : X1
t = X2t }
How big is TA when X10 and X2
0 are randomly chosen?
This is a lower bound on the time to consensus and is the rightorder of magnitude.
S. Chatterjee Latent voter model
![Page 18: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/18.jpg)
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Write Pπ for the law of (X1t , X
2t ), t ≥ 0 when X1
0 and X20 are
independent with distribution π.
Proposition 23 in Chapter 3 Aldous and Fill’s book on ReversibleMarkov Chains.
supt|Pπ(TA > t)− exp(−t/EπTA)| ≤ τ2
EπTA
where τ2 is the relaxation time, i.e., 1/spectral gap
Why? Make the stronger assumption that the mixing timetn � EπTA. In the limit we have the lack of memory property.
P (TA > (t+ s)EπTA|TA > sEπTA) ≈ P (TA > tEπTA)
S. Chatterjee Latent voter model
![Page 19: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/19.jpg)
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Write Pπ for the law of (X1t , X
2t ), t ≥ 0 when X1
0 and X20 are
independent with distribution π.
Proposition 23 in Chapter 3 Aldous and Fill’s book on ReversibleMarkov Chains.
supt|Pπ(TA > t)− exp(−t/EπTA)| ≤ τ2
EπTA
where τ2 is the relaxation time, i.e., 1/spectral gap
Why? Make the stronger assumption that the mixing timetn � EπTA. In the limit we have the lack of memory property.
P (TA > (t+ s)EπTA|TA > sEπTA) ≈ P (TA > tEπTA)
S. Chatterjee Latent voter model
![Page 20: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/20.jpg)
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Computing Eπ(TA)
For the underlying discrete chain, where at each step one particleis chosen at random and allowed to jump,
1/π(A) = EA(TA)= o(n) + EA(TA|TA � tn)PA(TA � tn)= o(n) + Eπ(TA)PA(TA � tn),
which implies
Eπ(TA) ≈ 1π(A)
· 1PA(TA � tn)
.
Naive guess for waiting time must be corrected by multiplying bythe expected clump size. Have a geometric number of quickreturns with “success probability" PA(TA � tn).
×× ××× × ×××clump
S. Chatterjee Latent voter model
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Random Regular Graphs
Sood and Redner (2005) have considered graphs with apower-law degree distribution. Other locally tree-like graphs havesame qualitative behavior..
Rancom regular graph is locally a tree with degree r.
Distance between two random walks, when positive, increases by1 with probability (r − 1)/r and decreases by 1 with prob. 1/r
PA(TA � tn) = 1− 1/(r − 1) = (r − 2)/(r − 1) for r ≥ 3.
If each particle jump at rate 1, then
Eπ(TA) ≈ 12
1π(A)
· 1PA(TA � tn)
=n(r − 1)r − 2
· 12
S. Chatterjee Latent voter model
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Summary for voter model on finite graphs
Eventually it reaches complete consensus.
The consensus time for locally-tree like random graphs on nvertices having degree distribution with finite second moment isO(n).
Sood and Redner (2005) have considered graphs with apower-law degree distribution. In all cases, the consensus time isat most linear in the number of vertices.
Starting from product measure, the quasi-stationary density ofstate 0 is the same as the initial proportion.
S. Chatterjee Latent voter model
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Latent Voter Model: Motivation
Lambiottei, Saramaki, and Blondel (2009) Phys. Rev. E. 79, paper046107
Consider the states of the voter model to be 0 = IBM laptop and1 = iPad. or Blu-Ray versus HD-DVD.
“It is likely that choice of a customer is influenced by hisacquaintances. However it is unlikely that the customer willreplace his equipment immediately after a purchase."
To reflect this, after an opinion change the voter enters an inactivestate that lasts for an exponentially distributed amount of time withmean 1/λ.
S. Chatterjee Latent voter model
![Page 24: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/24.jpg)
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Mean field equations.
We write + and − instead of 1 and 0 ( or ↑ and ↓ in LSB):
dρ+
dt= ρa−ρ+ − ρa+(1− ρ+)
dρa+dt
= −ρa+(1− ρ+) + λ(ρ+ − ρa+)
dρa−dt
= −ρa−ρ+ + λ(1− ρ+ − ρa−)
From the second equation we see that in equilibrium
0 = −ρa+ + ρa+ρ+ + λρ+ − λρa+
Using these in first equation we find three roots ρ+ = 0, 1/2 or 1
0 =λρ+(1− ρ+)(1− 2ρ+)(ρ+ + λ)(1− ρ+ + λ)
S. Chatterjee Latent voter model
![Page 25: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/25.jpg)
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Completing the solution, we see thatIf ρ+ = 1 then ρa+ = 1.If ρ+ = 0 then ρa− = 1.If ρ+ = 1/2 then ρa+/ρ+ = 2λ/(1 + 2λ).
Straightforward calculations show that the roots with ρ+ = 0 or 1are unstable while the one with ρ+ = 1/2 is locally attracting.
Three dimensional ODE: (ρ+, ρa+, ρa−).
S. Chatterjee Latent voter model
![Page 26: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/26.jpg)
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Voter model perturbations
We want to use some techniques similar to those in Cox, Durrett,and Perkins to study the latent voter model when λ is large.At each site there is a rate λ Poisson process of “wake-up dots."For each neighbor y, x consults y at rate 1/d(x) where d(x) is thedegree of x.
Number of consulting times for x between two wake-up dots isgeometric with success probability λ/(λ+ 1). The distribution is:
0 :λ
1 + λ1 :
λ
(1 + λ)22 :
λ
(1 + λ)3≥ 3 :
1(1 + λ)3
Scale time at rate λ and in the limit we can ignore intervals withthree or more arrows.
S. Chatterjee Latent voter model
![Page 27: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/27.jpg)
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Voter model perturbations
We want to use some techniques similar to those in Cox, Durrett,and Perkins to study the latent voter model when λ is large.At each site there is a rate λ Poisson process of “wake-up dots."For each neighbor y, x consults y at rate 1/d(x) where d(x) is thedegree of x.
Number of consulting times for x between two wake-up dots isgeometric with success probability λ/(λ+ 1). The distribution is:
0 :λ
1 + λ1 :
λ
(1 + λ)22 :
λ
(1 + λ)3≥ 3 :
1(1 + λ)3
Scale time at rate λ and in the limit we can ignore intervals withthree or more arrows.
S. Chatterjee Latent voter model
![Page 28: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/28.jpg)
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Graphical representation for rescaled latent votermodel
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate λ
0
t
time
6
vertices of the graph
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S. Chatterjee Latent voter model
![Page 29: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/29.jpg)
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Graphical representation for rescaled latent votermodel
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate λ
0
t
time
6
vertices of the graph
•
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S. Chatterjee Latent voter model
![Page 30: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/30.jpg)
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Graphical representation for rescaled latent votermodel
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate λ
0
t
time
6
vertices of the graph
•
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S. Chatterjee Latent voter model
![Page 31: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/31.jpg)
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Graphical representation for rescaled latent votermodel
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate λ
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![Page 32: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/32.jpg)
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Graphical representation for rescaled latent votermodel
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate λ
0
t
time
6
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S. Chatterjee Latent voter model
![Page 33: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/33.jpg)
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Changes in one interval
If there is only one voting time in an interval between two wake-updots then this is a voter event.
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y x
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y x z
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Considering the four cases we see that x will become 1 if and onlyat least one of x and y is a 1. (Redner’s vacillating voter model.)If y is 1 then x flips to 1 but is inactive and ignores z. If y is 0 . . .
S. Chatterjee Latent voter model
![Page 34: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/34.jpg)
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Changes in one interval
If there is only one voting time in an interval between two wake-updots then this is a voter event.
�
0
y x
•
•
�
-
0
y x z
•
•
Considering the four cases we see that x will become 1 if and onlyat least one of x and y is a 1. (Redner’s vacillating voter model.)If y is 1 then x flips to 1 but is inactive and ignores z. If y is 0 . . .
S. Chatterjee Latent voter model
![Page 35: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/35.jpg)
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Dual process
To determine the state of x at time λt we work backwards.
In an interval with one arrow x→ y the particle follows the arrowand jumps, since x imitates y at that time.
In an interval with two arrows, x→ y and x→ z, x stays in thedual and we add y and z, since we need to know the state of x, yand z to see what will happen.
S. Chatterjee Latent voter model
![Page 36: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/36.jpg)
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Dual process in the graphical representation
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate ≈ λ
0
t
time
6
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S. Chatterjee Latent voter model
![Page 37: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/37.jpg)
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Dual process in the graphical representation
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate ≈ λ
0
t
time
6
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S. Chatterjee Latent voter model
![Page 38: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/38.jpg)
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Dual process in the graphical representation
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate ≈ λ
0
t
time
6
vertices of the graph
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S. Chatterjee Latent voter model
![Page 39: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/39.jpg)
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Dual process in the graphical representation
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate ≈ λ
0
t
time
6
vertices of the graph
•
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S. Chatterjee Latent voter model
![Page 40: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/40.jpg)
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Dual process in the graphical representation
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate ≈ λ
0
t
time
6
vertices of the graph
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S. Chatterjee Latent voter model
![Page 41: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/41.jpg)
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Dual process in the graphical representation
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate ≈ λ
0
t
time
6
vertices of the graph
•
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S. Chatterjee Latent voter model
![Page 42: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/42.jpg)
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Dual process in the graphical representation
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate ≈ λ
0
t
time
6
vertices of the graph
•
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S. Chatterjee Latent voter model
![Page 43: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/43.jpg)
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Dual process in the graphical representation
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate ≈ λ
0
t
time
6
vertices of the graph
•
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S. Chatterjee Latent voter model
![Page 44: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/44.jpg)
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Dual process in the graphical representation
• wake up dotsat rate λ2
× voting timesat rate λ
• branching wake updots at rate ≈ 1
• single wake updots at rate ≈ λ
0
t
time
6
vertices of the graph
•
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S. Chatterjee Latent voter model
![Page 45: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/45.jpg)
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Schematic diagram of the dual
time↓
Space
S. Chatterjee Latent voter model
![Page 46: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/46.jpg)
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Schematic diagram of the dual
time↓
Space
S. Chatterjee Latent voter model
![Page 47: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/47.jpg)
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Schematic diagram of the dual
Space
S. Chatterjee Latent voter model
![Page 48: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/48.jpg)
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Schematic diagram of the dual
Space
S. Chatterjee Latent voter model
![Page 49: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/49.jpg)
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Schematic diagram of the dual
Space
S. Chatterjee Latent voter model
![Page 50: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/50.jpg)
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Schematic diagram of the dual
Space
S. Chatterjee Latent voter model
![Page 51: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/51.jpg)
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Schematic diagram of the dual
Space
S. Chatterjee Latent voter model
![Page 52: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/52.jpg)
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Schematic diagram of the dual
Space
S. Chatterjee Latent voter model
![Page 53: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/53.jpg)
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Schematic diagram of the dual
Space
S. Chatterjee Latent voter model
![Page 54: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/54.jpg)
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Schematic diagram of the dual
Space
S. Chatterjee Latent voter model
![Page 55: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/55.jpg)
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Schematic diagram of the dual
time↓
Space
S. Chatterjee Latent voter model
![Page 56: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/56.jpg)
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Schematic diagram of the dual
time↓
Space
S. Chatterjee Latent voter model
![Page 57: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/57.jpg)
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Limit theorem
For concreteness consider random regular graphs.
TheoremIf log n� λ� n/ log n, where n is the number of vertices of therandom regular graph, then
P (x has state 0 at timeλt)→ 12
as n→∞.
Similar result should be true for other locally tree-like random graphs.
S. Chatterjee Latent voter model
![Page 58: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/58.jpg)
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Ingredients for the proof of limit theorem
If the random walks from x, y, z all coalesce then we can ignorethe event since nothing can happen in the branching random walk.If xy|z or xz|y i.e., two coalesce but avoid the other until time λtthen the situation reduces to an ordinary voter arrow. This is alsotrue if x|yz.
Let
r1n := P ( all three members of a family coalesce before ε log n jumps),
r2n := P (none of the pairs in the family coalesce before ε log n jumps).
r1n → r1 and r2n → r2 as n→∞, e.g. the limit of r2 is theprobability of no coalescence among three random walks on theinfinite tree with degree r starting from neighboring sites.
S. Chatterjee Latent voter model
![Page 59: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/59.jpg)
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Ingredients for the proof of limit theorem
If the random walks from x, y, z all coalesce then we can ignorethe event since nothing can happen in the branching random walk.If xy|z or xz|y i.e., two coalesce but avoid the other until time λtthen the situation reduces to an ordinary voter arrow. This is alsotrue if x|yz.
Let
r1n := P ( all three members of a family coalesce before ε log n jumps),
r2n := P (none of the pairs in the family coalesce before ε log n jumps).
r1n → r1 and r2n → r2 as n→∞, e.g. the limit of r2 is theprobability of no coalescence among three random walks on theinfinite tree with degree r starting from neighboring sites.
S. Chatterjee Latent voter model
![Page 60: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/60.jpg)
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Sketch of the proof of the limit theorem
The integral equation for vt = P (x has state 0 at time λt) is
vt =∫ t
0(1− r1)e−(1−r1)s
[r2
1− r1{v3t−s + (1− vt−s)(1− (1− vt−s)2)}
+1− r1 − r2
1− r1vt−s
]ds+ v0e
−(1−r1)t.
A little calculation shows that
v′t = Kvt(1− vt)(1− 2vt) for some constant K = K(r1, r2)
S. Chatterjee Latent voter model
![Page 61: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/61.jpg)
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Consensus time
TheoremConsensus time is at least nb for any b <∞.
Reason
The dual processes starting from distant vertices areasymptotically independent.
Using Markov inequality for higher moments of the number ofvoters at time λt in state 0, it stays close to its mean withprobability ≥ 1− cn−b for any b <∞.
S. Chatterjee Latent voter model
![Page 62: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/62.jpg)
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Why is the condition on λ required in theargument?
The conditions on λ guarantees that with high probabilitythe particles become uniformly distributed on the graph betweensuccessive branching events,only those particles which are involved in the same branchingcoalesce,dual starting from distant particles are asymptotically independent.
Future question
Q. What happens if λ is large but O(1)?
S. Chatterjee Latent voter model
![Page 63: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/63.jpg)
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Why is the condition on λ required in theargument?
The conditions on λ guarantees that with high probabilitythe particles become uniformly distributed on the graph betweensuccessive branching events,only those particles which are involved in the same branchingcoalesce,dual starting from distant particles are asymptotically independent.
Future question
Q. What happens if λ is large but O(1)?
S. Chatterjee Latent voter model
![Page 64: Latent voter model on random regular graphsshirshendu.ccny.cuny.edu/s/LatentVoterTalk.pdf · university-logo Voter model on a graph ˘ t(x) 2f0;1gis the opinion of xat time t. E.g.,](https://reader033.vdocuments.us/reader033/viewer/2022052103/603e3c85a259ff55d84f4129/html5/thumbnails/64.jpg)
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Thank you
S. Chatterjee Latent voter model