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Activated Random Walks on Zd
Lectures 9 and 10
Leonardo T. Rolla
THUPKUBNU joint probability webinar – July 2020

Tentative plan

Tentative plan
Recall of previous lectures
Particlewise construction is welldefined [§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
Coarsegrained flow between blocks
The particlewise 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)

Coarsegrained flow between blocks [§5]
Analyze the odometer m1, . . . ,mn at the buffers
Mass Balance Equations and SingleBlock Dynamics
(Basu, Ganguly, Hoffman)

The particlewise 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
EMnVn
> 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 nonfixating 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, GurelGurevich)

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 particlehole 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 particlewise construction
is welldefined

Construction [§10.1]
For a triple (η0,X,P), we we say that(η0,X,P) 7→ (η0,Y ,γ) is welldefined 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 Eη0(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 welldefined 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
Reindex sums etc, BorelCantelli...

§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 translationergodic 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 translationergodic
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 doublyinfinite 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 wstable if η(0) 6 1, and we say that0 is sstable if η(0) = 0. Otherwise we say that 0 is
wunstable or sunstable. For y 6= 0 we say that y isstable, wstable, and sstable 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... nonfixation
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 wlegal for η+ if
and only if it is slegal 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
wstable: η(0) 6 1stable: η(0) 6 s
sstable: 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