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Activated Random Walks on Z d Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar – July 2020

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Page 1: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Activated Random Walks on Zd

Lectures 9 and 10

Leonardo T. Rolla

THU-PKU-BNU joint probability webinar – July 2020

Page 2: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tentative plan

Page 3: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tentative plan

Recall of previous lectures

Particle-wise construction is well-defined [§11.3]

Uniqueness of the critical density [§8]

Weak and strong stabilization [§7]

Page 4: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Recall

Page 5: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Recall

Dynamics and phase space

Odometer and toppling procedures

Counting arguments

Exploring instructions in advance

Coarse-grained flow between blocks

The particle-wise construction and applications

Page 6: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Phase space [§1.5]

slow

d = 1 directed d = 1 biased d = 1 unbiased

d = 2 unbiasedd ≥ 3 unbiasedd ≥ 2 biased

fast

slow

scaling limit

ζ

λ

1

ζ

λ

1

ζ

λ

1

ζ

λ

1

ζ

λ

1

ζ

λ

1

Page 7: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Odometer and Abelian property [§2.2]

All deterministic: finite sequences of topplings α, β

mα(x) := #times x appears in α

mV,η := supβ⊆V legal

mβ.

mV,η 6 mα if α stabilizes η in V

mV,η ↑ mη

Now make it random

Page 8: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Counting arguments [§3]

10−1−L

N0=

0

N1=

2

N2=

1

N3=

2

N4=

2

N5=

1

N6=

0

NL

· · · · · ·

−L+ 1 −L+ 2 · · ·

(folklore)

Page 9: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Counting arguments [§3]

v

(Taggi; R, Tournier)

Page 10: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Exploring instructions in advance [§4]

(R, Sidoravicius)

Page 11: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Coarse-grained flow between blocks [§5]

Analyze the odometer m1, . . . ,mn at the buffers

Mass Balance Equations and Single-Block Dynamics

(Basu, Ganguly, Hoffman)

Page 12: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

The particle-wise construction [§10.1]

Labeled particles

Constructed from η0, CTRW, Clocks

Defined through a limit (see below)

Fixation equivalence: Sites fixate ⇔ Particles fixate

Conservation: If fixate, E[start at 0] = E[settle at 0]

Corollary: ζc 6 1

Page 13: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Averaged condition for activity [§10.2]

lim supnEMn

|Vn|> 0 =⇒ Activity

n

m (R, Tournier)

Page 14: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Fixation equivalence [§10.4]

Theorem 10.7. Sites fixate ⇔ Particles fixate

Idea: extra randomness so as to spread out the effect

of non-fixating particles and control variance.

Implementation: for each particle that stays active, tag

at random one of the n first sites visited after time t.

(Amir, Gurel-Gurevich)

Page 15: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Resampling [§10.3]

Theorem 10.4. For i.i.d. random initial configurations

with average ζ = 1, the system a.s. stays active.

Take λ =∞, change rates to particle-hole model.

Suppose finitely many particles visit 0.

Resample η0 on the sites where they may start, so

wpp site 0 is never visited. Conclude that ζ < 1.

(Cabezas, R, Sidoravicius)

Page 16: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

§11.3

The particle-wise construction

is well-defined

Page 17: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Construction [§10.1]

For a triple (η0,X,P), we we say that

(η0,X,P) 7→ (η0,Y ,γ) is well-defined if:

(i) for each x, y ∈ Zd, j ∈ N and t > 0, both

(Y x,js )s∈[0,t] and (γx,js )s∈[0,t] are the same in the

systems (η0 · 1Byn,X,P) for all but finitely many n;

(ii) the limit (η0,Y ,γ) does not depend on y.

Page 18: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Statement

Theorem 10.6. If supx E|η0(x)| <∞, then the above

particle-wise construction is a.s. well-defined.

(R, Tournier)

Page 19: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Overview of the proof

` For arbitrary fixed Vn ↑ Zd there is an a.s. limit.

Add particles one by one, updating the whole evolution

` Life of each particle is well-defined through some limit

Main step: ∀x, T , the number of particle additions that

affect site x by time T has finite expectation

Page 20: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Page 21: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Page 22: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Page 23: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Page 24: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Page 25: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Page 26: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Page 27: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Page 28: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Page 29: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Page 30: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Tracking differences

Dominate by a supercritical branching process.

E[green] = e(2+λ)t

Re-index sums etc, Borel-Cantelli...

Page 31: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

§8Uniqueness of the critical density

Page 32: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Uniqueness of the critical density

Theorem 2.13. Given the dimension d, sleep rate λ,

and jump distribution p(·), there is a number ζc such

that, for every translation-ergodic distribution ν

supported on (N0)Zd

with average density ζ, the ARW

dynamics satisfies

Pν(system stays active) =

0, ζ < ζc,

1, ζ > ζc.

(R, Sidoravicius, Zindy)

Page 33: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Equivalent statement

Theorem 8.1. Let d, λ and p(·) be given. Let ν1 and

ν2 be two spatially ergodic distributions on (N0)Zd

,

with respective densities ζ1 < ζ2. If the ARW system is

a.s. fixating with initial state ν2, then it is also a.s.

fixating with initial state ν1.

Page 34: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

The more general version

Open Problem. Suppose ν is a translation-ergodic

active state (active means ν is supported on

(Ns)Zd \ {0, s}Zd

) with density ζ > ζc. Show that the

ARW with initial state ν a.s. stays active.

Page 35: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Idea of the proof

- Embedding the initial configuration into another one

with higher density (decoupling)

- Stabilization of the embedded configuration

- Stabilization of the original configuration

Remark. Not a sequential procedure like previous ones

Page 36: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Decoupling

Sample η0 ∼ ν1, ξ0 ∼ ν2 and I independently → ω.

Assume wlog ν1 or ν2 mixing, hence ω ergodic.

` A doubly-infinite procedure which is a factor of ω.

Let A0 = {x : η0(x) > ξ0(x)}. Topple every site in A0.

Result η1 is insensitive to the order, hence a factor.

Repeat for η0, η1, η2, . . . . Limit η∞ = η′0 exists.

Each site is toppled finitely often, otherwise ζ1 > ζ2.

Page 37: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Stabilization of the larger configuration

Delete the instructions used in the previous stage.

Zero out odometer.

Conditioning on the outcome of the first step, the

remaining instructions are again i.i.d. with the correct

distribution.

Stabilize ξ0. Odometer mξ0(x) < +∞ by assumption.

Page 38: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Stabilization of the original configuration

Since η′0 6 ξ0, we also have mη′0(x) < +∞.

Two stages: embedding and then stabilizing.

Some topplings in the first stage were not legal

(because we made forced sleepy particles to wake up

and jump).

Hence, the sum of the (locally finite) odometers

obtained in these two stages is an upper bound for the

odometer of η0.

Page 39: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

§7Weak and strong stabilization

Page 40: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Results

Theorem 7.1. For any jump distribution in any

dimension, ζc > λ1+λ .

Theorem 7.2. If d > 2, then ζc < 1 for every λ <∞and ζc → 0 as λ→ 0.

(Stauffer, Taggi)

Open Problem. Prove a similar statement for unbiased

walks on Z2.

Page 41: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Weak and strong stabilization

We say that 0 is w-stable if η(0) 6 1, and we say that

0 is s-stable if η(0) = 0. Otherwise we say that 0 is

w-unstable or s-unstable. For y 6= 0 we say that y is

stable, w-stable, and s-stable if η(y) 6 s.

Comparison:

mwV,η 6 mV,η 6 ms

V,η.

ηwV and η′V : configuration after (weakly) stabilizing

Page 42: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Proof of Theorem 7.1

Using Abelian Property, one way to stabilize η0 on Bn

is to first weakly stabilize it and then stabilize it.

` mV,η0(0) > 1 =⇒ ηwV (0) = 1

` P(η′V (0) = s

)> λ

1+λ P(ηwV (0) = 1

)Hence, P

(η′V (0) = s

)> λ

1+λ P(mV,η0(0) > 1

)Using monotonicity and amenability... non-fixation

implies ζ > λ1+λ .

Page 43: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Jump odometer and extra particles

Define the “jump odometer” m̄V,η by counting only the

number of jump instructions performed at each site

when η is stabilized in V .

Define m̄sV,η and m̄w

V,η similarly.

Let η+ = η + δ0 denote the result of adding an active

particle at 0 to a configuration η.

Page 44: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Strong − weak = extra particle

Lemma 7.5. We have m̄sV,η = m̄w

V,η+ .

Proof. A sequence of topplings β is w-legal for η+ if

and only if it is s-legal for η.

Page 45: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Getting rid of the extra particle

Lemma 7.6. We have E[m̄wV,η+(0)

]6 G+E

[m̄wV,η(0)

].

Sketch. Force the particle to move.

Corollary 7.7. E[m̄sV,η(0)− m̄w

V,η(0)]6 G.

Page 46: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Successive weak stabilizations

Particle at 0?

Move on to thenext round

strongstabilizationachieved

stabilizationachieved

First round

Topple 0

Perform weakstabilization in V

Yes

No

Yes

No

Jumpinstruction?

illegal but acceptable

there is a single particle at 0

w-stable: η(0) 6 1

stable: η(0) 6 s

s-stable: η(0) = 0

Page 47: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Successive weak stabilizations (cont)

Let TV and T sV count the number of rounds needed for

stabilization and strong stabilization to be achieved,

respectively (weak stabilization is always achieved in the

first round). From this definition we have

TV = 1 ⇐⇒ ηwV (0) = 0 ⇐⇒ T sV = 1.

Page 48: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Successive weak stabilizations (cont)

Lemma 7.8. m̄sV,η(0) > m̄w

V,η(0) + T sV − 1.

Corollary 7.9. ETV 6 ET sV 6 1 +G.

Page 49: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020

Proof of the main theorems (overview)

P(η′V (0) = s

)=

∞∑n=2

P(η′(0) = s, TV = n

)

P(η′V (0) = s, TV = n) 6 λ1+λ

(1

1+λ

)n−2

P(η′V (0) = s, TV = n

∣∣TV > n)

= λ1+λ

Page 50: Lectures 9 and 10 - Beijing Normal Universitymath0.bnu.edu.cn/~hehui/webinarsRolla05.pdf · Lectures 9 and 10 Leonardo T. Rolla THU-PKU-BNU joint probability webinar { July 2020