Activated Random Walks on Zd

Lectures 9 and 10

Leonardo T. Rolla

THU-PKU-BNU joint probability webinar – July 2020

Tentative plan

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]

Recall

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

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

∞

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

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)

Counting arguments [§3]

v

(Taggi; R, Tournier)

Exploring instructions in advance [§4]

(R, Sidoravicius)

Coarse-grained flow between blocks [§5]

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

Mass Balance Equations and Single-Block Dynamics

(Basu, Ganguly, Hoffman)

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

Averaged condition for activity [§10.2]lim supn

EMn|Vn|

> 0 =⇒ Activity

n

m (R, Tournier)

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)

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)

§11.3The particle-wise construction

is well-defined

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 (γ

x,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.

Statement

Theorem 10.6. If supx E|η0(x)|

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 thataffect site x by time T has finite expectation

Tracking differences

Tracking differences

Tracking differences

Tracking differences

Tracking differences

Tracking differences

Tracking differences

Tracking differences

Tracking differences

Tracking differences

Tracking differences

Dominate by a supercritical branching process.

E[green] = e(2+λ)t

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

§8Uniqueness of the critical density

Uniqueness of the critical density

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

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

Equivalent statement

Theorem 8.1. Let d, λ and p(·) be given. Let ν1 andν2 be two spatially ergodic distributions on (N0)Z

d,

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.

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.

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

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.

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.

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.

§7Weak and strong stabilization

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

Weak and strong stabilization

We say that 0 is w-stable if η(0) 6 1, and we say that0 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 isstable, w-stable, and s-stable if η(y) 6 s.

Comparison:

mwV,η 6 mV,η 6 msV,η.

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

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

)> λ1+λ P

(ηwV (0) = 1

)Hence, P

(η′V (0) = s

)> λ1+λ P

(mV,η0(0) > 1

)Using monotonicity and amenability... non-fixation

implies ζ > λ1+λ .

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 m̄wV,η similarly.

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

particle at 0 to a configuration η.

Strong − weak = extra particle

Lemma 7.5. We have m̄sV,η = m̄wV,η+ .

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

and only if it is s-legal for η.

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)− m̄wV,η(0)

]6 G.

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 1stable: η(0) 6 s

s-stable: η(0) = 0

Successive weak stabilizations (cont)

Let TV and TsV 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.

Successive weak stabilizations (cont)

Lemma 7.8. m̄sV,η(0) > m̄wV,η(0) + T sV − 1.

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

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(

11+λ

)n−2

P(η′V (0) = s, TV = n

∣∣TV > n) = λ1+λ