# Universality Conjectures for Activated Random Walk

Lionel Levine  Department of Mathematics, Cornell University, Ithaca, NY
14853<EMAIL_ADDRESS>and  Vittoria Silvestri  Department of
Mathematics, University of Rome La Sapienza, 00185 Rome, Italy..
<EMAIL_ADDRESS>
(Date: June 14, 2023)

###### Abstract.

Activated Random Walk is a particle system displaying Self-Organized
Criticality, in that the dynamics spontaneously drive the system to a critical
state. How universal is this critical state? We state many interlocking
conjectures aimed at different aspects of this question: scaling limits,
microscopic limits, temporal and spatial mixing, incompressibility, and
hyperuniformity.

###### Key words and phrases:

abelian network, activated random walk, hyperuniformity, incompressibility,
microscopic limit, quadrature inequality, scaling limit, spatial mixing,
stationary distribution, temporal mixing

###### 2020 Mathematics Subject Classification:

60K35, 82C22, 82C26, 82B26, 82B27,

##  1\. Introduction

Many complex systems in nature are driven by a steady source of energy which
builds up slowly and is released in intermittent bursts. An example is the
steady accumulation of pressure between continental plates, which is released
in sudden bursts in the form of earthquakes. Wildfires, landslides,
avalanches, and financial crises all have this character.

In a famous 1987 paper [4], Bak, Tang and Wiesenfeld proposed both a general
mechanism for how such systems arise, and a prototypical example. Their term
for the mechanism was self-organized criticality (SOC), and their example was
a pile of sand resting on the surface of a table. Given the current slope,
$\zeta$, of the pile, we can sprinkle more sand on top and measure two things:

  * •

How much sand falls off the table?

  * •

Does $\zeta$ tend to increase or decrease?

In experiments, one finds that the system drives itself to a critical slope
$\zeta_{c}$: If the pile is too flat ($\zeta<\zeta_{c}$), then very little
sand falls off the table and $\zeta$ increases as sand is added. If the pile
is too steep ($\zeta>\zeta_{c}$), then the sprinkling causes a lot of sand to
fall off the table, so that $\zeta$ _decreases_ as sand is added. No matter
the initial sand profile, the effect of adding more sand is therefore to drive
the pile toward its critical slope $\zeta_{c}$.

This new idea of self-organization to criticality led initially to a lot of
excitement (exemplified by Per Bak’s ambitiously titled book _How Nature
Works_), but its success in making specific testable predictions has been
modest so far. We do not know of a “universal model of SOC” in the way that,
for instance, Brownian motion is a “universal model of diffusion.”

###  1.1. In search of a universal model of SOC

The most intensively studied model of SOC is called the Abelian Sandpile. In
this model, the pile of sand is a collection of indistinguishable particles on
the vertices of a fixed graph, for example the $d$-dimensional cubic lattice
$\mathbb{Z}^{d}$. When a vertex has at least as many particles as the number
of neighbors in the graph ($2d$, in the case of $\mathbb{Z}^{d}$), it
_topples_ by sending one particle to each neighboring vertex. As a result,
some of those neighboring vertices may now have enough particles to topple,
enabling some of their neighbors in turn to topple, and so on: an avalanche.
Dhar [9] discovered a beautiful algebraic structure underlying this model.

_Abelian networks_ [3] form a larger class of SOC models. Among these, the
Stochastic Sandpile [10], the Oslo model [17], and Activated Random Walk [19,
37] (but not the original Abelian Sandpile!) seem to have some “universality”
in the sense that when the system size is large, its behavior does not depend
much on details like the initial condition, the boundary conditions, or the
underlying graph. However, the meaning of “universality” is rarely spelled
out. The purpose of this survey is to state precisely several senses in which
one of these models, Activated Random Walk, seems to be “universal”.

Activated Random Walk (ARW) is a particle system with two species, active
particles (${\tt{a}}$) and sleeping particles (${\tt{s}}$) that become active
if an active particle encounters them (${\tt{a}}+{\tt{s}}\rightarrow
2{\tt{a}}$). Active particles perform random walk at rate 1. When an active
particle is alone, it falls asleep (${\tt{a}}\rightarrow{\tt{s}}$) at rate
$\lambda$. A sleeping particle stays asleep until an active particle steps to
its location. The parameter $\lambda>0$ is called the _sleep rate_. We denote
this dynamics by ARW($\mathbb{Z}^{d},\lambda)$.

To draw out the analogy between ARW and sandpiles: The sleeping particles play
the role of sand grains, the movement of the active particles plays the role
of toppling, and the awakening of sleeping particles by active particles can
trigger an avalanche in which many particles wake up. The density of particles
in the system plays the role of the slope of the sandpile. Just as a pile of
high slope can easily be destabilized by adding a single sand grain, an ARW
configuration with a high density of sleeping particles can easily be
destabilized by adding a single active particle.

###  1.2. Plan of the paper

We discuss ARW dynamics in six settings, differing in the initial condition,
underlying graph, or boundary conditions. Our conjectures focus on the
approach to criticality, and on shared properties of the corresponding
critical states. Sections 2 and 3 consider finite particle configurations in
infinite volume. Our conjectures touch on the existence of scaling limits
(Conjectures 1, 2, 6), microscopic limits (Conjectures 3, 5, 7), and extra
symmetry acquired in the limit.

In Section 4 we discuss infinite (stationary ergodic) particle configurations
on $\mathbb{Z}^{d}$. We conjecture existence of a microscopic limit as the
threshold density is approached from below (Conjecture 9).

In Sections 5, 6, 7 we consider three different Markov chains on ARW stable
configurations on a finite graph. The main themes here are temporal mixing
(the system quickly forgets its initial condition: Conjectures 19,24), spatial
mixing (the boundary condition does not affect observables in the bulk:
Question 14, Conjecture 15), and a slow-to-fast phase transition (Question 22,
Conjecture 25).

Sections 8 and 9 discuss statistical properties of these ARW systems:
hyperuniformity (Conjectures 26, 27, Question 28) and site correlations
(Tables 1,2,3).

Several conjectures on shared properties of critical or stationary states in
the different settings are offered throughout the article (see Conjectures
12,17,21, Question 14 and Proposition 18).

We conclude in Section 10 by contrasting the conjectured behavior of ARW with
what is known about the Abelian Sandpile model.

##  2\. Point Source

|  |
---|---|---
$\lambda=4$ | $\lambda=1$ | $\lambda=0.25$
$\zeta_{\mathrm{a}}\approx 0.91$ | $\zeta_{\mathrm{a}}\approx 0.68$ | $\zeta_{\mathrm{a}}\approx 0.34$
Figure 1. ARW aggregates formed by stabilizing a point source of $10000$
active particles at the origin in the square lattice $\mathbb{Z}^{2}$, at
three different sleep rates $\lambda$. Each pixel represents a site of
$\mathbb{Z}^{2}$: sites with a sleeping particle are colored blue, and empty
sites are colored white. Particles spread farther when the sleep rate is
lower, so the aggregate density $\zeta_{\mathrm{a}}$ is an increasing function
of $\lambda$.

###  2.1. Spherical limit shape

Consider $\mathrm{ARW}(\mathbb{Z}^{d},\lambda)$ with initial configuration
$n\delta_{0}$, consisting of $n$ active particles at the origin and all other
sites empty. After a dynamical phase in which each particle performs random
walk, and may fall asleep and be awakened many times, activity will die out
when all particles fall asleep at distinct sites. We refer to the final
configuration of $n$ sleeping particles as the _ARW aggregate_ (Figure 1).

######  Conjecture 1.

(Aggregate density $\zeta_{\mathrm{a}}$) Let $A_{n}$ denote the random set of
sites visited by at least one walker during the dynamical phase of ARW started
from $n$ particles at the origin in $\mathbb{Z}^{d}$.

There exists a positive constant
$\zeta_{\mathrm{a}}=\zeta_{\mathrm{a}}(\mathbb{Z}^{d},\lambda)$ such that for
any $\epsilon>0$, with probability tending to $1$ as $n\to\infty$, the random
set $A_{n}$ contains all sites of $\mathbb{Z}^{d}$ that belong to the origin-
centered Euclidean ball of volume $(1-\epsilon)n/\zeta_{\mathrm{a}}$; and
$A_{n}$ is contained in the origin-centered Euclidean ball of volume
$(1+\epsilon)n/\zeta_{\mathrm{a}}$.

A weak form of Conjecture 1 in dimension $1$ is proved in [33]. As the sleep
rate $\lambda\uparrow\infty$, Activated Random Walk degenerates to Internal
DLA, whose limit shape is proved to be a Euclidean ball [26]. The main barrier
to applying Internal DLA methods is proving that sleeping particles are spread
uniformly, which is the topic of our next conjecture.

###  2.2. Macroscopic structure of the aggregate

An _ARW configuration_ in $\mathbb{Z}^{d}$ is a map

$\eta:\mathbb{Z}^{d}\to\mathbb{N}\cup\\{{\tt{s}}\\}$

where $\eta(x)={\tt{s}}$ indicates that there is a sleeping particle at
$x\in\mathbb{Z}^{d}$, and $\eta(x)=k$ indicates that there are $k$ active
particles at $x$. We write

$\mathcal{S}:(\mathbb{N}\cup\\{{\tt{s}}\\})^{\mathbb{Z}^{d}}\to\\{0,{\tt{s}}\\}^{\mathbb{Z}^{d}}$

for the operation of _stabilizing_ an ARW configuration $\eta$: running ARW
dynamics until all particles fall asleep. 111$\mathcal{S}(\eta)$ is always
defined if $\eta$ has finitely many particles, but it may be undefined in
general. The situation of infinite $\eta$ is discussed in Section 4.

Consider $\mathcal{S}(n\delta_{0})$, the ARW aggregate formed by stabilizing
$n$ particles at the origin in $\mathbb{Z}^{d}$. We will rescale the aggregate
and take a limit as $n\to\infty$. For $x\in\mathbb{R}^{d}$ let

$a_{n}(x)=1_{\big{\\{}\mathcal{S}(n\delta_{0})(\left\lfloor{n^{1/d}x}\right\rfloor)={\tt{s}}\big{\\}}}.$

Write $f_{n}\stackrel{{\scriptstyle*}}{{\rightarrow}}f$ for weak-$*$
convergence: $\int f_{n}\phi\,\mathrm{d}x\to\int f\phi\,\mathrm{d}x$ for all
bounded continuous test functions $\phi$ on $\mathbb{R}^{d}$, where
$\,\mathrm{d}x$ is Lebesgue measure on $\mathbb{R}^{d}$.

######  Conjecture 2.

(Uniformity of the aggregate) The rescaled ARW aggregates $a_{n}$ satisfy

$a_{n}\stackrel{{\scriptstyle*}}{{\rightarrow}}\zeta_{a}\mathbf{1}_{B}$

with probability one, where $B$ is the origin-centered ball of volume
$1/\zeta_{a}$ in $\mathbb{R}^{d}$.

In other words, in the weak-$*$ scaling limit, the random locations of the
sleeping particles in the aggregate blur out to a constant density $\zeta_{a}$
everywhere in the ball.

###  2.3. Microscopic structure of the aggregate

Figure 2. An ARW aggregate of $100000$ particles in $\mathbb{Z}^{2}$ at sleep
rate $\lambda=0.25$, with a zoom-in of the microscopic structure deep inside.

The next conjecture zooms in to the fine scale random structure of the
aggregate near the origin (Figure 2).

Write $\alpha_{n}$ for the law of the aggregate $\mathcal{S}(n\delta_{0})$.
This is a probability measure on $\\{0,{\tt{s}}\\}^{\mathbb{Z}^{d}}$. We
examine its marginals 222For a probability measure $\mu$ on
$\\{0,{\tt{s}}\\}^{\mathbb{Z}^{d}}$ and a finite set
$V\subseteq\mathbb{Z}^{d}$, we write $\mu|_{V}$ for the marginal distribution
on $\\{0,{\tt{s}}\\}^{V}$, that is
$\mu|_{V}(\xi):=\mu(\\{\eta\in\\{0,{\tt{s}}\\}^{\mathbb{Z}^{d}}:\eta(v)=\xi(v)\;\forall
v\in V\\})$, for $\xi\in\\{0,{\tt{s}}\\}^{V}$.  on a finite subset of
$\mathbb{Z}^{d}$, as $n\to\infty$.

######  Conjecture 3.

(Microscopic limit of the aggregate) For all finite $V\subset\mathbb{Z}^{d}$
and all $\xi\in\\{0,{\tt{s}}\\}^{V}$, the sequence $\alpha_{n}|_{V}(\xi)$
converges as $n\to\infty$.

This conjecture would imply, by Kolmogorov’s extension theorem, the existence
of the infinite-volume limit

$\alpha:=\lim_{V\uparrow\mathbb{Z}^{d}}\lim_{n\to\infty}\alpha_{n}|_{V},$

which is a probability measure on the set of infinite stable configurations
$\\{0,{\tt{s}}\\}^{\mathbb{Z}^{d}}$. The outer limit is over an exhaustion of
$\mathbb{Z}^{d}$, that is, a sequence of finite sets $V_{1}\subset
V_{2}\subset\cdots$ such that $\bigcup_{n\geq 1}V_{n}=\mathbb{Z}^{d}$. To
spell the limit out: For any finite $V\subset\mathbb{Z}^{d}$ and any
configuration $\xi\in\\{0,{\tt{s}}\\}^{V}$,

$\alpha_{n}|_{V}(\xi)\to\alpha|_{V}(\xi).$

Note the order of limits: we are restricted to a fixed window $V$ as the size
of the aggregate $n\to\infty$. Even though $\alpha$ is supported on
configurations with an infinite number of particles, a sample from $\alpha$ is
best imagined as tiny piece of an even larger aggregate!

######  Conjecture 4.

The limit $\alpha$ is invariant with respect to translations of
$\mathbb{Z}^{d}$.

######  Conjecture 5.

The limit $\alpha$ is supported on configurations of density
$\zeta_{\mathrm{a}}$.

##  3\. Multiple sources

|  |
---|---|---
$\downarrow\mathcal{S}_{4}$ | $\downarrow\mathcal{S}_{1}$ | $\downarrow\mathcal{S}_{\frac{1}{4}}$
|  |
$\int_{A}u\geq 0.91\int_{A^{*}}u$ | $\int_{R}u\geq 0.68\int_{R^{*}}u$ | $\int_{W}u\geq 0.34\int_{W^{*}}u$
Figure 3. Top: The capital letters $A,R,W$ viewed as subsets of
$\mathbb{R}^{2}$. Middle: The stabilization of
$1_{A\cap\epsilon\mathbb{Z}^{2}}$ at sleep rate 4, and
$1_{R\cap\epsilon\mathbb{Z}^{2}}$ at sleep rate 1, and
$1_{W\cap\epsilon\mathbb{Z}^{2}}$ at sleep rate $\frac{1}{4}$. Bottom:
According to Conjecture 6, the resulting supports $A^{*},R^{*},W^{*}$ are
characterized by quadrature inequalities for superharmonic functions. The
constant appearing in the inequality is the aggregate density
$\zeta_{\mathrm{a}}$ ($\approx 0.91,0.68,0.34$ for sleep rates
$4,1,\frac{1}{4}$).

Let $A\subset\mathbb{R}^{d}$ be a bounded open set satisfying
$\int_{\bar{A}\setminus A}\,\mathrm{d}x=0$ where $\,\mathrm{d}x$ denotes
$d$-dimensional Lebesgue measure. For $\epsilon>0$, let
$\mathcal{S}(1_{A}\cap\epsilon\mathbb{Z}^{d})$ be the configuration of
sleepers that results from starting one active particle at each point of
$A\cap\epsilon\mathbb{Z}^{d}$ and running activated random walk on
$\epsilon\mathbb{Z}^{d}$ with sleep rate $\lambda$. The following conjectured
scaling limit for ARW is inspired by Theorem 1.2 of [31], which describes the
scaling limit of internal DLA in $\mathbb{Z}^{d}$.

######  Conjecture 6.

(Quadrature inequality) As $\epsilon\to 0$,

$\mathcal{S}(1_{A}\cap\epsilon\mathbb{Z}^{d})\stackrel{{\scriptstyle*}}{{\rightarrow}}\zeta_{\mathrm{a}}1_{A^{*}}$

where $A^{*}$ is the unique (up to measure zero) open subset of
$\mathbb{R}^{d}$ satisfying

$\int_{A}u\,\mathrm{d}x\geq\zeta_{\mathrm{a}}\int_{A^{*}}u\,\mathrm{d}x$ (1)

for all integrable superharmonic functions $u$ on $A^{*}$.

The intuition behind this conjecture is that the uniform density $1$ on $A$
spreads out to uniform density $\zeta_{\mathrm{a}}<1$ on the larger set
$A^{*}$. If $u$ is a superharmonic function on $A^{*}$, then the sum of the
values of $u$ at all particle locations is approximately a supermartingale,
leading to (1) by optional stopping. In the case of multiple point sources,
$A^{*}$ is a smash sum of Euclidean balls [31, Theorem 1.4].

The next conjecture examines the microstructure of the aggregate near
$\mathbf{0}$.

######  Conjecture 7.

(Microstructure looks the same everywhere) Assume $\mathbf{0}\in A^{*}$. For
any finite $V\subset\mathbb{Z}^{d}$, the law of
$\mathcal{S}(1_{A}\cap\epsilon\mathbb{Z}^{d})|_{\epsilon V}$ has a limit as
$\epsilon\to 0$, in the sense that for any configuration
$\eta\in\\{0,{\tt{s}}\\}^{V}$

$\mathbb{P}\big{(}\mathcal{S}(1_{A}\cap\epsilon\mathbb{Z}^{d})(\epsilon
v)=\eta(v)\text{ for all }v\in V\big{)}\to\alpha|_{V}(\eta)$

The limiting probability measure $\alpha$ on
$\\{0,{\tt{s}}\\}^{\mathbb{Z}^{d}}$ is the same as in Conjecture 3. In
particular, $\alpha$ does not depend on $A$.

So far we have examined initial conditions with a finite number of particles
only. The next section examines infinite configurations.

##  4\. Stationary Ergodic

For $\mathrm{ARW}(\mathbb{Z}^{d},\lambda)$, start with a stationary ergodic
configuration $\eta:\mathbb{Z}^{d}\to\mathbb{N}$, where all particles are
initially active. Running ARW dynamics, will all particles fall asleep? If
each site of $\mathbb{Z}^{d}$ is visited only finitely often, then we say that
$\eta$ _stabilizes_. Rolla, Sidoravicius, and Zindy proved the remarkable fact
that stabilizing depends only on the mean number of particles per site

$\zeta:=\mathbb{E}(\eta(\mathbf{0})).$

######  Theorem 8.

(Universality of threshold density $\zeta_{\mathrm{c}}$, [38]) There exists a
constant $\zeta_{\mathrm{c}}=\zeta_{\mathrm{c}}(\mathbb{Z}^{d},\lambda)$ such
that if $\zeta<\zeta_{\mathrm{c}}$ then $\eta$ stabilizes with probability
$1$, and if $\zeta>\zeta_{\mathrm{c}}$ then with probability $1$, $\eta$ does
not stabilize.

###  4.1. Approaching the threshold from below

Theorem 8 ensures that the stabilization $\mathcal{S}(\eta)$ is always defined
if $\zeta<\zeta_{\mathrm{c}}$. What happens to the microstructure of
$\mathcal{S}(\eta)$ as $\zeta\uparrow\zeta_{\mathrm{c}}$? Start with a
stationary ergodic configuration
$\eta_{0}:\mathbb{Z}^{d}\to\mathbb{N}\cup\\{{\tt{s}}\\}$ with mean
$\zeta_{0}<\zeta_{\mathrm{c}}$, and sprinkle some extra active particles:
Letting $(\xi_{t}(x))_{x\in\mathbb{Z}^{d}}$ be independent Poisson random
variables with mean $t<\zeta_{\mathrm{c}}-\zeta_{0}$, the configuration
$\eta_{0}+\xi_{t}$ stabilizes with probability 1.

######  Conjecture 9.

(Universal limit of subcritical measures) Fix $\lambda>0$ and let $\mu_{t}$ be
the law of the ARW stabilization of $\eta_{0}+\xi_{t}$ with sleep rate
$\lambda$. There exists a limiting measure

$\mu:=\lim_{t\uparrow\zeta_{\mathrm{c}}-\zeta_{0}}\mu_{t}$

supported on configurations $\eta\in\\{0,{\tt{s}}\\}^{\mathbb{Z}^{d}}$ of
density $\zeta_{\mathrm{c}}$. Moreover, $\mu$ depends only on $\lambda$ and
not on the initial configuration $\eta_{0}$.

##  5\. The wired Markov chain

Fix a finite set $V\subset\mathbb{Z}^{d}$, and consider the particle system
$\mathrm{ARW}(V,\lambda)$ in which particles evolve as in ARW with sleep rate
$\lambda$, with the additional rule that when a particle exits $V$ it is
killed (i.e. removed from the system). Fix $v\in V$. The ARW _wired Markov
chain_ $(w_{k})_{k\geq 0}$ on the state space $\\{0,{\tt{s}}\\}^{V}$ has the
update rule: add one active particle at $v$ and stabilize, i.e.

$w_{k+1}=\mathcal{S}_{V}(w_{k}+\delta_{v}),$

where $\mathcal{S}_{V}$ denotes ARW stabilization with killing of any
particles that exit $V$.

###  5.1. Stationary distribution

The stationary distribution of the Markov chain $(w_{k})_{k\geq 0}$ does not
depend on the choice of the site $v$ where particles are added, as for
different $v$ the Markov transition operators commute! The next result gives
an efficient way to sample exactly from the stationary distribution of this
chain.

Start with the configuration $\mathbf{1}_{V}$, consisting of one active
particle on each site of $V$, and let the particles perform
$\mathrm{ARW}(V,\lambda)$ until no active particles remain. Some particles
exit the system, and the remaining particles fall asleep in $V$. Denote by
$\mathcal{S}_{V}(\mathbf{1}_{V})$ the resulting random configuration of
sleepers.

######  Proposition 10 (Exact sampling, [28]).

The law of $\mathcal{S}_{V}(\mathbf{1}_{V})$ is the unique stationary
distribution of the ARW wired Markov chain on $V$ with sleep rate $\lambda$.

|  |
---|---|---
$\lambda=4$ | $\lambda=1$ | $\lambda=0.25$
$\zeta_{\mathrm{s}}\approx 0.91$ | $\zeta_{\mathrm{s}}\approx 0.68$ | $\zeta_{\mathrm{s}}\approx 0.34$
Figure 4. Stationary configurations for the ARW wired chain on a $100\times
100$ box, at three different sleep rates $\lambda$. The stationary density
$\zeta_{s}$ is an increasing function of $\lambda$.

For any given set $V\subset\mathbb{Z}^{d}$ let $\partial V$ denote its
boundary and $\\#V$ denote its cardinality. Write
$|\mathcal{S}_{V}(\mathbf{1}_{V})|$ for the total number of (sleeping)
particles in $\mathcal{S}_{V}(\mathbf{1}_{V})$.

######  Conjecture 11.

(Stationary density $\zeta_{\mathrm{s}}$) Then there exists a constant
$\zeta_{\mathrm{s}}=\zeta_{\mathrm{s}}(\mathbb{Z}^{d},\lambda)$ such that for
any exhaustion $V_{1}\subset V_{2}\subset\cdots\subset\mathbb{Z}^{d}$
satisfying $\\#(\partial V_{n})/\\#V_{n}\to 0$ as $n\to\infty$,

$\lim_{n\to\infty}\frac{|\mathcal{S}_{V_{n}}(\mathbf{1}_{V_{n}})|}{\\#V_{n}}=\zeta_{\mathrm{s}}$

in probability.

######  Conjecture 12.

The critical densities from Sections 2, 4 and 5 coincide:

$\zeta_{\mathrm{a}}=\zeta_{c}=\zeta_{\mathrm{s}}.$

###  5.2. Infinite volume limit

Let $\pi_{V}$ denote the stationary distribution of the ARW wired Markov
chain, as defined above. For a subset $W\subset V$, write $\pi_{V}|_{W}$ for
the restriction of $\pi_{V}$ to $W$.

######  Conjecture 13.

For any fixed finite set $W\subset\mathbb{Z}^{d}$, the measures $\pi_{V}|_{W}$
have a limit as $V\uparrow\mathbb{Z}^{d}$, and this limit does not depend on
the exhaustion of $\mathbb{Z}^{d}$.

This conjecture would imply, by Kolmogorov’s extension theorem, the existence
of a limiting probability measure

$\pi=\lim_{W\uparrow\mathbb{Z}^{d}}\lim_{V\uparrow\mathbb{Z}^{d}}\pi_{V}|_{W}.$
(2)

on the space of infinite stable configurations
$w:\mathbb{Z}^{d}\to\\{0,{\tt{s}}\\}$. We can then ask how this limit relates
to the measures $\alpha$ and $\mu$ from Sections 2 and 4 above.

######  Question 14.

Is $\pi=\mu=\alpha$ ?

Can the wired boundary condition be felt deep inside $V$? We conjecture that
as $V\uparrow\mathbb{Z}^{d}$, the particle density deep inside $V$ coincides
with the overall density $\zeta_{s}$.

######  Conjecture 15.

For $w\sim\pi$ we have $\pi\\{w(\mathbf{0})={\tt{s}}\\}=\zeta_{s}$.

###  5.3. The hockey stick conjecture

Write $(w_{k})_{k\geq 0}$ for the ARW wired chain on the box
$V:=[1,L]^{d}\subset\mathbb{Z}^{d}$ with initial state $w_{0}=0$ (all sites
are empty) and with _uniform driving_ : instead of adding particles at a fixed
vertex, we add them at a sequence of independent vertices $v_{1},v_{2},\ldots$
with the uniform distribution on $V$:

$w_{k+1}=\mathcal{S}_{V}(w_{k}+\delta_{v_{k+1}}),\qquad k\geq 0.$

When does the wired chain begin to lose a macroscopic number of particles at
the boundary? A theorem of Rolla and Tournier partially answers this question.
Define

$\zeta_{w}:=\inf\Big{\\{}t>0\,:\,\limsup_{L}\frac{\mathbb{E}(|w_{tL^{d}}|)}{L^{d}}<t\Big{\\}}$

where, as usual, $|w_{k}|$ denotes the number of particles in $w_{k}$.

######  Theorem 16.

[36, Proposition 3] $\zeta_{w}\geq\zeta_{c}$.

We conjecture $\zeta_{w}=\zeta_{\mathrm{c}}$, and that the stabilized density
has the following simple piecewise linear form.

######  Conjecture 17 (Hockey stick).

The wired chain on $[1,L]^{d}$ with uniform driving satisfies

$\frac{|w_{tL^{d}}|}{L^{d}}\to\begin{cases}t,&t\leq\zeta_{\mathrm{c}}\\\
\zeta_{c},&t\geq\zeta_{\mathrm{c}}\end{cases}$

in probability as $L\to\infty$, where $\zeta_{\mathrm{c}}$ is the threshold
density of Theorem 8.

The name for this conjecture comes from the graph of the piecewise linear
limit, which has the shape of a hockey stick (Figure 5).

Figure 5. The hockey stick: As particles are added, the density of the ARW
wired chain increases to $\zeta_{c}$ and then flatlines. Here
$V\subset\mathbb{Z}^{2}$ is a box of side length $L=256$, the sleep rate is
$\lambda=2$, and $\zeta_{c}\approx 0.813$. “Global density” is the total
number of particles divided by $L^{2}$. “Bulk density” is the number of
particles in a central window of side length $L/2$, divided by $(L/2)^{2}$.

######  Proposition 18.

If Conjecture 17 holds, then $\zeta_{w}=\zeta_{\mathrm{c}}$.

###### Proof.

By Theorem 16, it suffices to prove that $\zeta_{w}\leq\zeta_{\mathrm{c}}$. We
argue by contradiction: suppose that $\zeta_{\mathrm{c}}<\zeta_{w}$. Then
there is a $\zeta\in(\zeta_{\mathrm{c}},\zeta_{w})$ such that both

$\limsup_{L}\frac{\mathbb{E}(|w_{\zeta L^{d}}|)}{L^{d}}\geq\zeta$ (3)

since $\zeta<\zeta_{w}$, and for any $\epsilon>0$

$\mathbb{P}\bigg{(}\bigg{|}\frac{|w_{\zeta
L^{d}}|}{L^{d}}-\zeta_{\mathrm{c}}\bigg{|}\geq\epsilon\bigg{)}\to
0\qquad\mbox{ as }L\to\infty,$ (4)

since $\zeta>\zeta_{\mathrm{c}}$ and Conjecture 17 holds true. But by (3)
there exists a diverging subsequence $(L_{k})_{k\geq 1}$ such that

$\lim_{k\to\infty}\frac{\mathbb{E}(|w_{\zeta
L_{k}^{d}}|)}{L_{k}^{d}}\geq\zeta,$

and since each term in this sequence is at most $\zeta$ by construction, it
must be that the above limit actually equals $\zeta$. It follows that for any
$\epsilon>0$

$\lim_{k\to\infty}\mathbb{P}\bigg{(}\bigg{|}\frac{|w_{\zeta
L_{k}^{d}}|}{L_{k}^{d}}-\zeta\bigg{|}\geq\epsilon\bigg{)}=\lim_{k\to\infty}\mathbb{P}\bigg{(}\zeta-\frac{|w_{\zeta
L_{k}^{d}}|}{L_{k}^{d}}\geq\epsilon\bigg{)}\leq\lim_{k\to\infty}\frac{1}{\epsilon}\bigg{(}\zeta-\frac{\mathbb{E}(|w_{\zeta
L_{k}^{d}}|)}{L_{k}^{d}}\bigg{)}=0.$

This contradicts (4) by uniqueness of the limit in probability. ∎

###  5.4. Fast mixing

Consider the ARW wired chain in a discrete Euclidean ball
$V=\\{x\in\mathbb{Z}^{d}\,:\,\sum x_{i}^{2}<L^{2}\\}$ with uniform driving. We
highlight the following conjecture from [28] to the effect that this chain
mixes immediately after reaching the stationary density $\zeta_{\mathrm{s}}$.

######  Conjecture 19.

(Cutoff, [28]) The ARW wired chain has cutoff in total variation at the time

$t_{mix}=\zeta_{\mathrm{s}}\\#V.$

In [28] it is shown that $t_{mix}\leq(1+o(1))\\#V$. The proof uses a coupling
between Activated Random Walk and Internal DLA. Bristiel and Salez [7] show
that the relaxation time is much smaller: $O(L^{d-1})$ in dimensions $d\neq 2$
and $O(L\log L)$ in dimension $2$. They also prove separation cutoff at time
$\\#V$.

###  5.5. Incompressibility

A recurring challenge in proving several of the above conjectures is to show
that “dense clumps” are unlikely. We conjecture that clumps denser than the
mean in the infinite-volume stationary state $w\sim\pi$ have exponentially
small probability. Write
$|w|_{L}:=\sum_{x\in[1,L]^{d}}1_{\\{w(x)={\tt{s}}\\}}$ for the number of
particles in the cube $[1,L]^{d}$.

######  Conjecture 20.

(Incompressibility) For each $\zeta>\zeta_{\mathrm{s}}$, there is a constant
$c=c(\zeta,\lambda)>0$ such that for $w\sim\pi$

$P(|w|_{L}\geq\zeta L^{d})\leq\exp(-cL^{d}).$

The ideas introduced in [1, 16] may be useful in proving incompressiblity for
$\zeta$ sufficiently close to 1.

##  6\. The free Markov chain

Fix a finite connected graph $V$, an initial configuration
$\phi_{0}:V\to\\{0,{\tt{s}}\\}$, and let

$\phi_{k+1}=\mathcal{S}(\phi_{k}+\delta_{v_{k+1}})$

be the configuration of sleeping particles obtained by adding one active
particle at a random vertex $v_{k}$, and then stabilizing by ARW dynamics in
$V$ with sleep rate $\lambda$. The vertices $v_{1},v_{2},\ldots$ are
independent with the uniform distribution on $V$.

Unlike the ARW wired Markov chain in Section 5, particles cannot escape $V$.
So the total number of particles is deterministic: $|\phi_{k}|=|\phi_{0}|+k$.
As long as this number does not exceed $\\#V$, stabilization happens in finite
time, but if the number of particles is large then it could take a long time
(even exponentially long, [5])! We will define the _threshold time_ as the
first time $k$ such that $\phi_{k}$ takes “too long” to stabilize.

Let $(\phi_{k})_{k\geq 0}$ denote the ARW free chain on $V$ initiated from the
empty configuration $\phi_{0}=0$. For $k\geq 0$, denote by $U_{k}$ be the
total number of random walk steps needed to stabilize
$\phi_{k}+\delta_{v_{k+1}}$. For any function $f:\mathbb{N}\to\mathbb{R}$, let

$\tau_{f}(V)=\inf\\{k\geq 0\,:\,U_{k}\geq f(\\#V)\\}.$

######  Conjecture 21.

(Concentration of the threshold time) Let $V=\mathbb{Z}_{L}^{d}$ be the
$d$-dimensional torus of side-length $L$. There exists a superlinear function
$f:\mathbb{N}\to\mathbb{R}$ such that, as $L\uparrow\infty$,

$\lim_{L\uparrow\infty}\frac{\tau_{f}(\mathbb{Z}_{L}^{d})}{L^{d}}=\zeta_{\mathrm{c}}$

in probability, where $\zeta_{\mathrm{c}}$ is the threshold density of Section
4.

A stronger formulation would posit a sharp transition from linear to
exponential time:

######  Question 22.

Is it true that

$U_{tL^{d}}=\begin{cases}O(L^{d}),&t<\zeta_{c}\\\
\exp(\Omega(L^{d})),&t>\zeta_{c}?\end{cases}$

##  7\. The Wake Markov Chain

Fix a finite connected graph $V$ and an initial configuration
$\varphi_{0}:V\to\\{0,{\tt{s}}\\}$ with $|\varphi_{0}|\leq\\#V$. Let
$\mathcal{W}$ denote the operator that acts on stable particle configurations
on $V$ by waking all particles up. The ARW wake Markov chain, supported on
stable particle configurations on $V$, is defined by

$\varphi_{k+1}=\mathcal{S}(\mathcal{W}(\varphi_{k})).$

So in one time step of the wake chain, we wake all particles up and then
stabilize. Note that stabilization is always possible, though it may take a
long time, since $|\varphi_{k}|=|\varphi_{0}|\leq\\#V$ for all $k\geq 0$.

###  7.1. Stationary measure

Take $V=\mathbb{Z}_{L}^{d}$ and let $\nu_{L,\zeta}$ denote the stationary
measure of the ARW wake chain on $V$. Denote further by
$\tilde{\nu}_{L,\zeta}$ the law of the ARW free chain $(\phi_{k})_{k\geq 0}$
at time $k=\zeta L^{d}$. Note that sleepers configurations drawn according to
$\nu_{L,\zeta}$ and $\tilde{\nu}_{L,\zeta}$ have the same density $\zeta$ (in
fact, the same number of particles). It is natural to conjecture that in the
supercritical regime these measures are close.

######  Conjecture 23.

If $\zeta>\zeta_{c}$ then
$d_{\textrm{TV}}\big{(}\nu_{L,\zeta},\tilde{\nu}_{L,\zeta}\big{)}=o(1)$ as
$L\to\infty$, where $d_{\textrm{TV}}$ denotes the Total Variation distance.

This would follow from the following, more general conjecture.

######  Conjecture 24.

(Dense stabilized configurations are hard to distinguish) Fix the dimension
$d$ and sleep rate $\lambda$, and let $\zeta_{c}$ be the threshold density of
Theorem 8. For each $\zeta>\zeta_{c}$ and $\epsilon>0$ there is an $L_{0}$
such that for all $L\geq L_{0}$ and any two configurations $\eta,\tilde{\eta}$
of active particles on the discrete torus $\mathbb{Z}_{L}^{d}$ with
$|\eta|=|\tilde{\eta}|\geq\zeta L^{d}$, their stabilizations
$\mathcal{S}(\eta),\mathcal{S}(\tilde{\eta})$ satisfy

$d_{\mathrm{TV}}(\mathcal{S}(\eta),\mathcal{S}(\tilde{\eta}))<\epsilon.$

The underlying mechanism here is that the system takes a long time to
stabilize. In particular, it is known that for small enough sleep rate,
stabilization takes exponentially many steps in $L$ with high probability:
This was proved in dimension $1$ by Basu, Ganguly, Hoffman and Richey [5], and
recently in all dimensions by Forien and Gaudillière [16, Theorem 3]. Their
result plus a coupling argument proves Conjecture 24 for $\lambda$ small
enough: One can couple the trajectories of any two particles in the ARW
systems starting from $\eta$ and $\tilde{\eta}$ so that they will meet prior
to stabilization with high probability. Provided all these couplings are
successful, the processes stabilize to the same configuration
$\mathcal{S}(\eta)=\mathcal{S}(\tilde{\eta})$.

Figure 6. Site covariance of the ARW Wake Markov Chain on the torus
$\mathbb{Z}_{L}\times\mathbb{Z}_{L}$ with $L=2501$, sleep rate $1$, and
subcritical density $0.3$ (much less than $\zeta_{c}\approx 0.68$). Left to
right: covariances after $k=1,2,10$ steps of the wake chain from a uniform
random initial condition with $\left\lfloor{0.3L^{2}}\right\rfloor$ particles.
Each non-central site is shaded according to the covariance between the events
that a sleeping particle is located at that site and at the central site.
Initial positive correlations with nearby sites become negative after a few
time steps.

###  7.2. Mixing time

How long does it take for the ARW wake chain to reach stationarity? We
conjecture a transition from slow to instantaneous mixing at the threshold
density $\zeta_{c}$.

######  Conjecture 25.

Let $\eta$ be any stable configuration on the $d$-dimensional cycle
$\mathbb{Z}_{L}^{d}$ on $L^{d}$ vertices, and denote by $\zeta=|\eta|/L^{d}$
its particle density. Then the total variation mixing time of the ARW wake
chain starting from $\eta$ is $1$ if $\zeta>\zeta_{c}$, while it is
$\Omega(L^{2})$ if $\zeta<\zeta_{c}$, where $\zeta_{c}$ is the threshold
density of Theorem 8.

The first part of this conjecture, fast mixing at high density, would follow
directly from Conjecture 24.

##  8\. Hyperuniformity

Experiments suggest that the stationary states for the Activated Random Walk
Markov chains introduced in Sections 5,6,7 above are _hyperuniform_.

###  8.1. The wired chain

For a random configuration $\eta\in\\{0,{\tt{s}}\\}^{\mathbb{Z}^{d}}$ with
$\eta\sim\pi$, write $|\eta|_{L}$ for the total number of particles in the
cube $[1,L]^{d}$.

######  Conjecture 26.

Under $\pi$, the variance of $|\eta|_{L}$ is $O(L^{\alpha})$ as $L\to\infty$,
for some $\alpha<d$.

Figure 7. The onset of hyperuniformity in the ARW wired chain on a square of
side length $L=50$ at sleep rate $\lambda=2$. The variance
$\mathbb{E}|\eta_{t}|^{2}-(\mathbb{E}|\eta_{t}|)^{2}$ of the total number of
particles in the system increases and then levels off after time
$t=\zeta_{s}L^{2}$. The variance of the number of particles in the bulk
(central square of side length $L/2$) peaks and then drops steeply as
$t/L^{2}$ approaches the stationary density $\zeta_{s}\approx 0.81$.

For comparison, observe that if the particles are placed in the box
$[1,L]^{d}$ in an i.i.d. fashion, the variance is of order $L^{d}$. Thus
hyperuniformity implies a kind of rigid repulsion among particles: in order to
make the variance of the number of particles in the box $[1,L]^{d}$ grow
sublinearly with its volume $L^{d}$, the particle counts $\eta(v)$ for
$v\in[1,L]^{d}$ must have significant negative correlations [18].

Burdzy has proved hyperuniformity in a related particle system called the
Meteor Model [8]. The challenge in adapting his proof to ARW lies in adapting
the i.i.d. driving of the meteors to the correlated driving that results from
active particles waking sleeping particles.

###  8.2. The free chain

For the free chain the number of particles increases by one at each time step,
and we expect hyperuniformity to manifest starting at the threshold density
$\zeta_{c}$. To state a hyperuniformity conjecture for the free chain, we will
count particles in a box $B\subset\mathbb{Z}_{L}^{d}$. Write
$(\phi_{k})_{k\geq 0}$ for the ARW free chain on the torus
$\mathbb{Z}_{L}^{d}$, and write $|\phi_{k}|_{B}$ for the total number of
particles in $B$ at time $k$.

######  Conjecture 27.

(Onset of hyperuniformity in the free chain) There exists $\epsilon>0$ such
that for any box
$B=[0,\ell_{1}-1]\times\cdots\times[0,\ell_{d}-1]\subset\mathbb{Z}_{L}^{d}$ we
have

$\mathrm{Var}(|\phi_{\zeta
L^{d}}|_{B})=\begin{cases}\Theta(\ell_{1}\ldots\ell_{d}),&\zeta<\zeta_{c}\\\
O((\ell_{1}\cdots\ell_{d})^{1-\epsilon}),&\zeta\geq\zeta_{c}.\end{cases}$

The implied constants depend only on $d,\lambda,\zeta$.

Figure 8.  The onset of hyperuniformity in the ARW free chain on the discrete
torus $\mathbb{Z}_{L}\times\mathbb{Z}_{L}$ with $L=50$ and sleep rate
$\lambda=2$. The initial configuration of $k$ active particles at independent
uniformly distributed sites, stabilizes to a final configuration of $k$
sleeping particles on the torus. The variance of the number of sleeping
particles in the left half $\mathbb{Z}_{L/2}\times\mathbb{Z}_{L}$ initially
increases with $k$, then peaks around $k=0.66L^{2}$, and bottoms out around
$\zeta_{c}L^{2}$ where $\zeta_{c}\approx 0.81$.

###  8.3. The wake chain

Figure 9. The ARW wake chain on the cycle $\mathbb{Z}_{N}$ starting with $N/2$
particles at uniformly random sites and run for $100N$ time steps. The
variance of the number of particles on the left half of the cycle grows
linearly with $N$ at the subcritical sleep rate $\lambda=0.2$, but sublinearly
with $N$ at the critical sleep rate $\lambda=0.15$. (Here $\zeta=1/2$ is less
than $\zeta_{c}(0.2)$ and approximately equal to $\zeta_{c}(0.15)$.)

Write $\nu_{L,\zeta}$ for the stationary distribution of the ARW wake chain of
$\left\lfloor{\zeta L^{d}}\right\rfloor$ particles on the discrete torus
$\mathbb{Z}_{L}^{d}$. For a stationary configuration $\eta\sim\nu_{L,\zeta}$,
we can once again count the number of particles in a box
$B=\prod_{i=1}^{d}[0,\ell_{i}-1]$ and ask whether the variance is linear or
sublinear in the size of $B$.

######  Question 28.

Is there $\epsilon>0$ such that the stationary state $\eta$ of the ARW wake
chain satisfies

$\mathrm{Var}_{\eta\sim\nu_{L,\zeta}}(|\eta|_{B})=\begin{cases}\Theta(\ell_{1}\ldots\ell_{d}),&\zeta<\zeta_{c}\\\
O((\ell_{1}\cdots\ell_{d})^{1-\epsilon}),&\zeta\geq\zeta_{c}?\end{cases}$

Figure 9 tests the case $d=1$ at density $\zeta=1/2$ and two different sleep
rates: $\zeta_{c}(0.2)>1/2$ and $\zeta_{c}(0.15)\approx 1/2$.

##  9\. Site correlations

To address Question 14 (Is $\pi=\mu=\alpha$?) we performed experiments
comparing the site correlations in the ARW free chain, ARW wired chain, and
ARW point source aggregate. For the free chain on the torus
$\mathbb{Z}_{L}\times\mathbb{Z}_{L}$ we are able to average with respect to
translation and reflection symmetries of the torus to increase the precision
of the numerical estimates. For the wired chain and point source, only
reflection symmetries are available, so precision is lower.

###  9.1. Free site correlations

For small $x,y$ we computed the empirical correlation coefficient

$\frac{\mathbb{E}(1_{f}(0,0)1_{f}(x,y))-\zeta^{2}}{\zeta-\zeta^{2}}$

where $1_{f}(x,y)$ is the indicator of the event that site $(x,y)$ has a
sleeping particle in the stabilization of $\zeta L^{2}$ particles started at
independent uniform random sites on the torus
$\mathbb{Z}_{L}\times\mathbb{Z}_{L}$ with side length $L=63$ at sleep rate
$\lambda=2$ and density $\zeta=0.81\approx\zeta_{c}$. We averaged over 80000
independent samples, and over translation and reflection symmetries of the
torus. The results are shown in Table 1.

###  9.2. Wired site correlations

For small $x,y$ we computed the empirical correlation coefficient

$\frac{\mathbb{E}(1_{w}(0,0)1_{w}(x,y))-\zeta^{2}}{\zeta-\zeta^{2}}$

where $1_{w}(x,y)$ is the indicator of the event that site $(x,y)$ has a
sleeping particle in the stationary state $\mathcal{S}(1_{V})$ of the ARW
wired chain on the square $V=[-31,31]^{2}\subset\mathbb{Z}^{2}$ at sleep rate
$\lambda=2$, and $\zeta=0.81\approx\zeta_{s}$. We averaged over $5\cdot
10^{7}$ independent samples, and over the $D_{8}$ symmetry of the square
lattice. The results are shown in Table 2.

###  9.3. Point source site correlations

For small $x,y$ we computed the empirical correlation coefficient

$\frac{\mathbb{E}(1_{a}(0,0)1_{a}(x,y))-\zeta^{2}}{\zeta-\zeta^{2}}$

where $1_{a}(x,y)$ is the indicator of the event that site $(x,y)$ has a
sleeping particle in the stabilization of $n=3215$ particles started at
$(0,0)$, and $\zeta=0.81\approx\zeta_{a}$. This value of $n$ was chosen to
make the total number of particles match the free chain experiment described
above. We averaged over $10^{5}$ independent samples, and over the $D_{8}$
symmetry of the square lattice. The results are shown in Table 3.

$y\backslash x$ | 0 | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
0 | 1.0000 | -0.0238 | -0.0101 | -0.0061 | -0.0045 | -0.0037
1 |  | -0.0139 | -0.0082 | -0.0056 | -0.0044 | -0.0037
2 |  |  | -0.0063 | -0.0048 | -0.0040 | -0.0035
3 |  |  |  | -0.0042 | -0.0037 | -0.0034
4 |  |  |  |  | -0.0034 | -0.0032
5 |  |  |  |  |  | -0.0031
Table 1. Short-range correlations for the ARW free Markov chain on the discrete torus $\mathbb{Z}_{63}\times\mathbb{Z}_{63}$.  $y\backslash x$ | 0 | 1 | 2 | 3
---|---|---|---|---
0 | 1.000 | -0.021 | -0.008 | -0.003
1 |  | -0.012 | -0.006 | -0.003
2 |  |  | -0.004 | -0.002
3 |  |  |  | -0.001
Table 2. Short-range correlations for the ARW wired Markov chain on a square of side length $63$ in $\mathbb{Z}^{2}$.  $y\backslash x$ | 0 | 1 | 2
---|---|---|---
0 | 1.000 | -0.021 | -0.007
1 |  | -0.010 | -0.006
2 |  |  | -0.000
Table 3. Short-range correlations for the ARW point source aggregate of $3215$
particles in $\mathbb{Z}^{2}$.

Comparing Tables 1 and 2, the spatial decay of correlations is faster in the
wired chain than in the free chain. Is this an artifact of the small system
size, or are the limiting measures $\mu$ and $\pi$ different? Comparing Tables
2 and 3, the short range correlations are consistent with the measures $\pi$
and $\alpha$ being equal, but the low precision prevents us from conjecturing
this confidently.

###  9.4. Wake chain site correlations

$\qquad\zeta=0.2\hskip 39.83368pt$ | $\hskip 28.45274pt\zeta=0.6\hskip 36.98866pt$ | $\hskip 28.45274pt\zeta=0.7\qquad\qquad$
---|---|---

Figure 10. Short-range correlations of the ARW Wake Markov Chain at three
different densities $\zeta=0.2,0.6,0.7$. Each non-central site is shaded
according to the correlation coefficient between the events that a sleeping
particle is located at that site and at the central site.

The ARW wake chain has a family of stationary distributions, one for each
density. We wondered whether the site correlations depend on the density. The
answer appears to be yes, as shown in Figure 10. For three different densities
$\zeta\in\\{0.2,0.6,0.7\\}$ we computed the empirical correlation coefficient

$\frac{\mathbb{E}(1_{z}(0,0)1_{z}(x,y))-\zeta^{2}}{\zeta-\zeta^{2}}$

where $1_{z}(x,y)$ is the indicator of the event that site $(x,y)$ has a
sleeping particle in the stationary distribution of the ARW wake chain at
density $\zeta$ on the torus $\mathbb{Z}_{15}\times\mathbb{Z}_{15}$ with sleep
rate $1$. After a burn-in period to reach stationarity, we averaged over
$100000$ subsequent time steps of the ARW wake chain and over translation
symmetries of the torus. Short-range correlations are negative at all
densities, and nothing notable seems to happen at the threshold density
$\zeta_{c}\approx 0.68$. The strongest nearest-neighbor site correlation
occurs at a density somewhat less than $\zeta_{c}$.

##  10\. Contrasts with the Abelian sandpile model

We close by comparing Activated Random Walk to the Abelian Sandpile, and in
particular highlighting which results, among the ones we believe to hold for
ARW, are known to fail for the sandpile model.

###  10.1. Point Source

Pegden and Smart [35] proved existence of a limit shape for the point source
Abelian Sandpile in $\mathbb{Z}^{d}$. The scaling limit of the Abelian
sandpile on $\mathbb{Z}^{2}$ obeys a PDE that is not rotationally symmetric
[29, 30], so its limit shape from a point source is unlikely to be a Euclidean
disk (although it has not been formally proved not to be a disk!). One symptom
of the failure of universality in the Abelian Sandpile is the existence of
“dense clumps” in the point-source sandpile. These are macroscopic regions
whose density is higher than the average density of the whole pile. By
contrast, we believe Activated Random Walkers are incompressible (Conjecture
20).

###  10.2. Stationary Ergodic

The Abelian Sandpile in $\mathbb{Z}^{d}$ for $d\geq 2$ has an interval of
threshold densities: any density between $d$ and $2d-1$ can be threshold,
depending on the law of the initial configuration [14, 34, 12, 15]). By
contrast, Activated Random Walk at a given sleep rate has a single threshold
density (Theorem 8).

Conjecture 9 fails for the Abelian Sandpile, due to slow mixing: the sandpile
stabilization of $\eta_{0}+\xi_{t}$ retains a memory of its initial state
$\eta_{0}$ even as $t\uparrow\zeta_{\mathrm{c}}-\zeta_{0}$ [27]. Terms like
“the self-organized critical state” result in lot of confusion in the physics
literature on the Abelian Sandpile because there are many such states! One of
them, the limit of the uniform recurrent state, is amenable to exact
calculations blue[40, 25, 32, 24], but slow driving from a subcritical state
will usually produce a critical state with different properties (e.g.
different density).

###  10.3. Wired Markov Chain

Fast mixing of ARW stands in contrast to the slow mixing of the Abelian
sandpile, where $t_{mix}=\Theta(L^{2}\log L)$ for the wired chain on
$(\mathbb{Z}/L\mathbb{Z})^{2}$ [20] and on $[1,L]^{2}$ [21]. This logarithmic
factor is responsible for the discrepancy between the stationary density
$\zeta_{s}=2.125000$ and the threshold density $\zeta_{c}=2.125288$ observed
in [11].

To approach Conjectures 11–13 and Question 14 it may be useful to find a
combinatorial description of the ARW stationary distribution. An important
tool available for the Abelian sandpile, which has no counterpart yet in the
ARW setting, is a bijection between recurrent states and spanning trees. This
bijection is useful because of the well-developed theory of infinite-volume
limits like (2) for trees [6]. The bijection from sandpiles to trees plays a
starring role in Athreya and Járai’s proof that the uniform recurrent sandpile
on a finite set $V\subset\mathbb{Z}^{d}$ has an infinite-volume limit [2], and
in Járai and Redig’s study of infinite-volume sandpile dynamics [23].
Hutchcroft used the bijection with spanning trees to prove universality
results for high-dimensional sandpiles [22].

###  10.4. Free Markov Chain

The Hockey Stick Conjecture 17 is believed to be false for the Abelian
Sandpile due to its slow mixing. There is, however, a weaker conjectured
relationship between the sandpile free and wired chains: the threshold time of
the free chain coincides with the first time when a macroscopic number of
particles exit the wired chain [13].

## Acknowledgments

We thank Ahmed Bou-Rabee, Hannah Cairns, Deepak Dhar, Shirshendu Ganguly,
Chris Hoffman, Feng Liang, SS Manna, Pradeep Mohanty, Leonardo Rolla, Vladas
Sidoravicius, and Lorenzo Taggi for many inspiring conversations. Thanks to
Chris Hoffman for pointing out that Conjecture 11 requires a condition on the
boundary of $V_{n}$, and that Conjecture 17 requires a condition on the
driving. This project was partly supported by the Funds for joint research
Cornell-Sapienza. LL was partly supported by the NSF grant DMS-1105960 and IAS
Von-Neumann Fellowship. We thank Cornell University, Sapienza University, IAS
and ICTS-TIFR for their hospitality.

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