# Percolation phase transition on planar spin systems

Caio Alves Email<EMAIL_ADDRESS>Alfréd Rényi Institute of
Mathematics, Budapest, 1053 Hungary. Gideon Amir Email<EMAIL_ADDRESS>Bar-Ilan University, 5290002, Ramat Gan, Israel. Rangel Baldasso Email:
<EMAIL_ADDRESS>Mathematical Institute, Leiden University, P.O.
Box 9512, 2300 RA Leiden, The Netherlands. Augusto Teixeira Email:
<EMAIL_ADDRESS>IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro,
RJ - Brazil.

###### Abstract

In this article we study the sharpness of the phase transition for percolation
models defined on top of planar spin systems. The two examples that we treat
in detail concern the Glauber dynamics for the Ising model and a Dynamic
Bootstrap process. For both of these models we prove that their phase
transition is continuous and sharp, providing also quantitative estimates on
the two point connectivity. The techniques that we develop in this work can be
applied to a variety of different dependent percolation models and we discuss
some of the problems that can be tackled in a similar fashion. In the last
section of the paper we present a long list of open problems that would
require new ideas to be attacked.

_Keywords and phrases._ Percolation, spin systems, dependence.

MSC 2010: 60K35, 82C22, 82C27.

##  1 Introduction

Since the introduction of percolation as a mathematical model in [8], the
subject has undergone an impressive development. These include a better
understanding of its subcritical and supercritical phases [33, 3, 21, 18].
Moreover, although the critical phase has remained evasive to the best of the
community’s efforts, there are two remarkable exceptions that are now well
understood: the planar model [28, 40] and the high dimensional case [22].

In parallel to the above mentioned advances, big developments were also made
in the study of dependent percolation. These works extend some of the results
that were already known in the Bernoulli case, both in arbitrary dimensions
[32, 41] as well as in the plane [7, 44, 43].

This paper continues this trend by looking at examples of dependent
percolation on the plane and establishing the continuity and the sharpness of
their phase transitions. We treat two examples in detail: Glauber dynamics of
the Ising model and dynamical bootstrap percolation.

Let us start by introducing the first model we consider. For this, fix a
density $\rho\in[0,1]$ and let us define the initial distribution of our spin
system. This will be denoted as $\sigma^{\rho}_{0}(x)$, for
$x\in\mathbb{Z}^{2}$, and will be i.i.d. taking value $1$ with probability
$\rho$ and $-1$ with probability $1-\rho$.

Having established the initial configuration, the process evolves according to
the Glauber dynamics of the Ising model at inverse temperature
$\beta\in[0,\infty]$, see Section 2 for a precise definition. This defines the
process $\sigma^{\rho}_{t}(x)$, for all $x\in\mathbb{Z}^{d}$ and $t\geq 0$.

In this article we study the percolation configuration obtained at a fixed
finite time $\tau\geq 0$ as we vary $\rho$ from zero to one. In (2.16) below
we show that, for fixed $\tau$, this process is monotone in $\rho$, meaning
that we can couple all $\sigma^{\rho}_{\tau}(x)$ in a way that if
$\rho\geq\rho^{\prime}$ then
$\sigma^{\rho}_{\tau}(x)\geq\sigma^{\rho^{\prime}}_{\tau}(x)$ for every
$x\in\mathbb{Z}^{2}$. Let $\operatorname{P}_{\rho}$ denote the distribution of
$\sigma_{\tau}^{\rho}$. We then look at
$(\sigma^{\rho}_{\tau}(x))_{x\in\mathbb{Z}^{2}}$ as a dependent percolation
model on the plane as the value of $\rho$ varies. See Figures 3 and 4 for a
simulation of the process for various values of $\tau$ and $\rho$.

Given a configuration $\sigma:\mathbb{Z}^{2}\to\\{-1,1\\}$, we write
$A\ext@arrow 9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}B$ for the
event that the sets $A,B\subseteq\mathbb{Z}^{2}$ can be connected by a
nearest-neighbor path of $+1$ spins. We can then define the critical point

$\rho_{c}(\tau)=\inf\Big{\\{}\rho\in[0,1];\operatorname{P}_{\rho}\big{[}0\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\infty\big{]}>0\Big{\\}},$
(1.1)

that depends on the time $\tau$ for which we have run the Glauber dynamics.

Inspecting the simulations in Figures 1 and 2, it becomes clear that for
certain values of $\tau$ there is a phase transition for the percolation of
$\sigma^{\rho}_{\tau}$ as we vary $\rho$, while for larger values of $\tau$
the system stays always in a single phase.

Figure 1: Simulation showing how the critical value of percolation varies as
$\tau$ increases in a low-temperature regime, with $\beta=10$. The simulations
hint at the self-duality of the process, as the critical parameters seem to
converge to $0.5$. Figure 2: Simulation showing how the critical value of
percolation varies as $\tau$ increases in a high-temperature regime, with
$\beta=0.1$. The occurrence of percolation quickly becomes impossible, as the
process resembles Bernoulli site percolation with parameter $0.5$ no matter
the initial conditions.

Our first result makes the above observation precise, by showing the existence
or absence of phase transitions in $\rho$ as we vary $\tau$ and $\beta$.

######  Theorem 1.1.

Under one of the following conditions:

  1. 1.

$\beta=\infty$,

  2. 2.

or if $\beta\in(0,\infty)$ and $\tau\leq
c_{\ref{c:t_small}}=c_{\ref{c:t_small}}(\beta)$,

the system undergoes a non-trivial phase transition, meaning that
$\rho_{c}(\tau)\in(0,1)$.

On the other hand, there exists $\beta_{0}>0$ such that, for
$\beta\in[0,\beta_{0})$ and $\tau$ large enough (depending on $\beta$), the
configuration $\sigma^{\rho}_{\tau}$ is subcritical uniformly on
$\rho\in[0,1]$. More precisely, there exists $\epsilon\in(0,1)$ and
$c_{\ref{c:t_large_decay}}=c_{\ref{c:t_large_decay}}(\beta)>0$ such that, for
every $\rho\in[0,1]$ and $\tau\geq c_{\ref{c:t_large_decay}}$,

$\operatorname{P}_{\rho}\big{[}B_{n}\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\partial B_{2n}\big{]}\leq
e^{-c_{\ref{c:t_large_decay}}n^{\epsilon}},\text{ for every $n\geq 1$}.$ (1.2)

In particular, the set in (1.1) is empty.

Figure 3: Heatmap representing the required minimum value of the parameter so
that the state of a site is $+1$ in a box $300\times 300$ at time $\tau=6$ for
$\beta=3$. Figure 4: Heatmap representing the required minimum value of the
parameter so that the state of a site is $+1$ in a box $300\times 300$ at time
$\tau=6$ for $\beta=0.1$.

Our next theorem is the most central part of this work and it gives a detailed
description of the phase transition that happens as we vary $\rho$. Informally
speaking, this theorem gives detailed descriptions of the three phases of the
system: critical, subcritical and supercritical.

######  Theorem 1.2.

For any value of $\beta\in[0,\infty]$, if $\tau>0$ is such that
$\rho_{c}(\tau)\in(0,1)$, then

$\operatorname{P}_{\rho_{c}(\tau)}\big{[}B_{n}\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\partial
B_{2n}\big{]}\in(c_{\ref{c:critical}},1-c_{\ref{c:critical}}),$ (1.3)

for some constant $c_{\ref{c:critical}}>0$ (that depends only on $\beta$ and
$\tau$), and all $n\geq 1$. On the other hand, for all $\rho<\rho_{c}(\tau)$
and $\epsilon>0$, there exists
$c_{\ref{c:sub_critical}}=c_{\ref{c:sub_critical}}(\rho,\epsilon,\beta,\tau)>0$
such that

$\operatorname{P}_{\rho}\big{[}B_{n}\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\partial
B_{2n}\big{]}\leq\exp\big{\\{}-c_{\ref{c:sub_critical}}n/\log^{\epsilon}(n)\big{\\}},\text{
for every $n\geq 1$},$ (1.4)

and if $\rho>\rho_{c}(\tau)$, there exists
$c_{\ref{c:super_critical}}=c_{\ref{c:super_critical}}(\rho,\epsilon,\beta,\tau)>0$
such that

$\operatorname{P}_{\rho}\big{[}B_{n}\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\partial B_{2n}\big{]}\geq
1-\exp\big{\\{}-c_{\ref{c:super_critical}}n/\log^{\epsilon}(n)\big{\\}},\text{
for every $n\geq 1$}.$ (1.5)

######  Remark 1.

Here are a few observations that complement the above stated results.

  1. a)

We have stated our main Theorem 1.2 for the Glauber dynamics of the Ising
model. However, our results work in much greater generality as we explain in
Section 8, requiring a few hypothesis on the monotonicity of the underlying
dynamics.

  2. b)

Although the simulations suggest that $\rho_{c}(\tau)$ is monotone as a
function of $\tau$, this is currently unknown as the model lacks microscopic
monotonicity in $\tau$.

  3. c)

Note that our results apply to the zero temperature regime $\beta=\infty$. In
this case there is indeed a non-trivial transition and the three distinct
phases can be observed.

  4. d)

Our results can also be applied in the (self-dual) triangular lattice, where
they imply that $p_{c}(\tau)=\frac{1}{2}$, for all $\tau>0$, if one considers
a symmetric dynamic, such as Glauber dynamics or majority dynamics (see [5]).

Let us now introduce the second model considered here, that we call Dynamic
Elliptic Bootstrap. Once again, this system is constructed through the
evolution of a particle system, but in this case the phase transition will
occur as a function of time.

At time zero, all sites have opinion -1 and as time passes they can flip their
opinion to 1 and become frozen afterwards. Each site changes opinion from $-1$
to $1$ with rate given by a monotone increasing function of its neighborhood.
Furthermore, we assume that this function is invariant under
($\pi/2$)-rotations, has finite range, and a positive lower bound
$\epsilon>0$. The last assumption implies that each vertex, independent of its
neighborhood configuration, flips with a positive rate. The precise definition
of the model can be found in Section 7. Let $\sigma_{t}(x)$ denote the opinion
of $x$ at time $t$, and denote by $\operatorname{P}_{t}$ the distribution of
the configuration $\sigma_{t}$.

In this case, we examine the percolation threshold as a function of time by
introducing, analogously to (1.1),

$t_{c}=\inf\Big{\\{}t\in[0,\infty):\operatorname{P}_{t}\big{[}0\ext@arrow
<EMAIL_ADDRESS>(1.6)

A simple argument relying on stochastic domination by Bernoulli percolation
implies that

$\begin{array}[]{c}\parbox[c]{433.62pt}{\centering$t_{c}$ is non-trivial,
meaning that $t_{c}\in(0,\infty)$.\@add@centering}\end{array}$ (1.7)

Our main result here is analogous to Theorem 1.2, where we once again obtain
descriptions of the three distinct phases this process undergoes.

######  Theorem 1.3.

There exists a positive constant $c_{\ref{c:bep_critical}}>0$ such that, for
any $n\geq 1$,

$\operatorname{P}_{t_{c}}\big{[}B_{n}\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\partial
B_{2n}\big{]}\in(c_{\ref{c:bep_critical}},1-c_{\ref{c:bep_critical}}).$ (1.8)

Furthermore, for all $t<t_{c}$ and $\epsilon>0$, there exists
$c_{\ref{c:bep_sub_critical}}=c_{\ref{c:bep_sub_critical}}(t,\epsilon)>0$ such
that

$\operatorname{P}_{t}\big{[}B_{n}\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\partial
B_{2n}\big{]}\leq\exp\big{\\{}-c_{\ref{c:bep_sub_critical}}n/\log^{\epsilon}(n)\big{\\}},\text{
for every $n\geq 1$},$ (1.9)

and, if $t>t_{c}$, there exists
$c_{\ref{c:bep_super_critical}}=c_{\ref{c:bep_super_critical}}(t,\epsilon)>0$
such that

$\operatorname{P}_{\rho}\big{[}B_{n}\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\partial B_{2n}\big{]}\geq
1-\exp\big{\\{}-c_{\ref{c:bep_super_critical}}n/\log^{\epsilon}(n)\big{\\}},\text{
for every $n\geq 1$}.$ (1.10)

Previous results that more closely relate to this work have dealt with Voronoi
percolation [7, 43], contact-process percolation [44], confetti percolation
[27] and Boolean percolation [2].

The above works begin to establish a general framework for dealing with planar
dependent percolation in a systematic way. Nonetheless, there is still no
general result that can treat large classes of models using the same proof
strategy.

Roughly speaking, the proof of continuity and sharpness for percolation models
on the plane typically involves three steps: finite-size criteria, Russo-
Seymour-Welsh estimates and sharpness. While the first two steps have reached
a very high degree of generality already, notably with the very recent
development in [30], the sharpness of the phase transition still requires
model dependent arguments.

One of the objectives of this article is to further the techniques to prove
sharpness of the phase transition, solving two previously open problems.
Moreover, we emphasize several of the limitations of the currently known
techniques and dedicate a whole section to stating open problems that explore
the field and its shortcomings.

### History of sharpness.

The first result showing sharpness of the phase transition of a percolation
process on $\mathbb{Z}^{2}$ was proved by Kesten in [28], which in particular
proved that the critical point of bond percolation in $\mathbb{Z}^{2}$ is
$1/2$. In his paper, Kesten relied on the work of Harris [23] and the
techniques from Russo [37] and Seymour and Welsh [39], which give lower bounds
for the probabilities of rectangles being crossed by the percolation process
in the hard direction in terms of the crossing probabilities of squares. These
bounds are at the heart of the study of critical percolation of many planar
percolation processes, and are generally called the _Russo-Seymour-Welsh
theory_ , or RSW for short.

An important example of a result proving sharp thresholds for a planar
dependent percolation model is given by [7], which concerns Voronoi
percolation. Similar techniques relying on RSW-type results and differential
inequalities based on influence bounds were also used in the study of confetti
percolation [27] and the contact process [44].

The field of dependent planar percolation received renewed attention after a
major improvement on the techniques used to prove RSW bounds was obtained in
[43], a framework which was later applied in other works such as [2]. Very
recently, another major development in the field was proved in [30], which
obtained RSW bounds under minimal hypotheses, such as positive association
(FKG) and invariance of the underlying percolation measure under reflections
and ($\pi/2$)-rotations.

Results about Bernoulli percolation on “almost planar” graphs were also
obtained in [17], which shows that there is no percolation at the critical
parameter when the underlying graph is a _slab_ , the product
$\mathbb{Z}^{2}\times\\{1,\dots,n\\}$ for some $n\in\mathbb{N}$. We can also
cite [6], which studies the planar random cluster model, as an example of the
study of sharp thresholds on planar dependent percolation processes.

### Limitations of current techniques.

The recent techniques that have been developed for proving sharpness of
percolation phase transitions on the plane have made the field advance quite
rapidly. At the same time, these new discoveries have greatly amplified the
number and the diversity of problems that can be now attempted in the area. To
emphasize this recent shift, we dedicate Section 8 to listing various open
problems that we expect to progress in the near future.

However, there are still obstacles that the community must overcome before a
general theory of sharp thresholds of planar dependent percolation processes
is obtained. Though the techniques for proving a _finite-size criterion_ and
establishing a _RSW theory_ are now very general, the final key consisting in
_sharp threshold results_ from _differential inequalities_ is very model-
dependent. Finding correct analogies for Russo’s formula and obtaining related
differential inequalities exhibiting this threshold behavior continues to be,
for the time being, a labor that must be repeated for each distinct model.

### Proof overview.

As alluded to above, our proof, like the proof in the seminal paper of Kesten
[28], relies on three pillars:

  * •

A finite-size criterion, meaning that if the probability that the cluster
crosses a large rectangle is sufficiently small (resp., sufficiently close to
$1$), then we are in the subcritical (resp., supercritical) phase. The proof
relies on a model-agnostic multiscale renormalization scheme, as well as a
decoupling inequality.

  * •

Russo-Seymour-Welsh theory, meaning lower bounds of crossing probabilities of
rectangles in terms of the crossing probabilities of squares. We use the
general result obtained in [30].

  * •

Sharp thresholds, meaning that crossing probabilities of large rectangles are
either very close to $0$, or very close to $1$, and that the intermediate
interval of the parameter in which this probability is far away from both $0$
and $1$ collapses to a point (the critical point) as the size of these
rectangles grow, if the process undergoes a phase transition.

The first two pillars are not new: our main contribution is showing sharp
thresholds for the models considered here. Our argument relies on randomized
algorithms and the OSSS inequality [36]. This is not straightforward though,
since we have to deal with the randomness coming from the evolution together
with the randomness of the initial condition, which is the focus of our
analysis. To avoid degeneracies that might appear due to badly behaved
realizations of the dynamics, we provide an alternative graphical construction
that has a richer structure and allows us to work on the quenched setting, by
comparing it to its annealed counterpart. Once properties of this construction
are established, the core of our argument lies in a way of relating the
influences of different types of variables present in this alternative
construction of the processes we study.

### Acknowledgments -

CA was supported by the Noise-Sensitivity Everywhere ERC Consolidator Grant
772466 and the DFG Grant SA 3465/1-1. G.A. was supported by the Israel Science
Foundation Grant 957/20. RB has counted on the support of the Mathematical
Institute of Leiden University. During this period, AT has also been supported
by grants “Projeto Universal” (406250/2016-2) and “Produtividade em Pesquisa”
(304437/2018-2) from CNPq and “Jovem Cientista do Nosso Estado”,
(202.716/2018) from FAPERJ.

##  2 Main model and auxiliary results

We start with the general notation. We will consider $\mathbb{Z}^{2}$ with the
nearest-neighbor graph structure. We write $x\sim y$ in case
$x,y\in\mathbb{Z}^{2}$ are neighbors, and in general let $|x-y|_{1}$ denote
the $\ell_{1}$-distance between $x$ and $y$. The $\ell_{\infty}$-ball with
radius $n>0$ and center at $x\in\mathbb{Z}^{2}$ will be denoted by $B(x,n)$.
Given a set $A\subset\mathbb{Z}^{2}$, we denote its complement by $A^{c}$; and
define its internal boundary as

$\partial A:=\big{\\{}x\in A;\text{ there exists $y\in A^{c}$ such that $x\sim
y$}\big{\\}}.$ (2.1)

We also define the external boundary of $A$ as $\partial_{ext}A:=\partial
A^{c}$, and its cardinality (which may be $\infty$ otherwise specified) as
$|A|$. Given $A,B\subset\mathbb{Z}^{2}$, define their mutual distance
$\operatorname{dist}(A,B)$ as the infimum of the $\ell_{\infty}$-distance
between their elements.

We denote the uniform distribution over an interval $[a,b]\subset\mathbb{R}$
by $U[a,b]$, the exponential distribution with parameter $\lambda>0$ by
$\operatorname{Exp}(\lambda)$, the Poisson distribution with parameter
$\lambda>0$ by $\operatorname{Poisson}(\lambda)$, and the Bernoulli
distribution with parameter $p\in[0,1]$ by $\operatorname{Ber}(p)$. The symbol
$\mathbb{N}$ will denote the natural numbers, and $\mathbb{N}_{0}$ will denote
the set of natural numbers _including_ $0$.

Let us now define the model. We construct in standard fashion a collection of
time-indexed random variables associated to each point of $\mathbb{Z}^{2}$

$\big{(}\sigma_{t}(x);\,t\geq 0,x\in\mathbb{Z}^{2}\big{)}$ (2.2)

such that $\sigma_{t}(x)\in\\{1,-1\\}$, for every $t\geq 0$ and
$x\in\mathbb{Z}^{2}$. Given $\rho\in[0,1]$, we let
$(\sigma_{0}(x))_{x\in\mathbb{Z}^{2}}$ be an i.i.d. family of random variables
such that

$\sigma_{0}(x)\stackrel{{\scriptstyle
d}}{{=}}\rho\delta_{1}+(1-\rho)\delta_{-1},$ (2.3)

where $\delta_{y}$ denotes the Dirac measure at $y\in\mathbb{R}$. For each
$x\in\mathbb{Z}^{2}$, define now an independent Poisson point process with
rate $1$ on $\mathbb{R}_{+}$, whose marks, to which we will informally refer
as time marks or clock ticks, will be denoted by
$\mathcal{P}^{x}=\\{T_{0}^{x},T_{1}^{x},\dots\\}$. To each pair
$(x,T^{x}_{i})$, $i\in\mathbb{N}_{0}$, we associate an independent mark

$R^{x}_{i}\stackrel{{\scriptstyle d}}{{=}}U[0,1].$ (2.4)

We write $x\overset{t}{\rightarrow}y$ if there exists a sequence of nearest-
neighbor sites

$x=x_{0}\sim\dots\sim x_{k}=y$ (2.5)

and a sequence of marks $T^{x_{0}}_{i_{0}}<\dots<T^{x_{k}}_{i_{k}}<t$ in the
respective Poisson point processes. We will need the following result, which
will help measure how much the state of the process in one site can influence
the process’ state in a distant site:

######  Lemma 2.1.

For every $t>0$, there exists $c_{\ref{c:light}}(t)>0$
($\exp\\{2^{11}t\log(8t)\\}$ suffices) such that, for all $x\neq
y\in\mathbb{Z}^{2}$,

$\mathbb{P}\big{[}x\overset{t}{\rightarrow}y\big{]}\leq
c_{\ref{c:light}}(t)\exp\bigg{\\{}-\frac{1}{4}|x-y|_{1}\log(|x-y|_{1})\bigg{\\}}.$
(2.6)

We postpone the proof of this lemma to Appendix A.

Heuristically, we will define the process so that, at each time-mark
associated to a site, said site will update its state according to the Ising
process distribution conditioned on the state of its four neighbors. For
$\beta\in[0,\infty]$ and $z_{1},\dots,z_{4}\in\\{-1,1\\}$, we denote

$S_{\beta}(z_{1},\dots,z_{4}):=\frac{\exp\big{\\{}\beta\sum_{i=1}^{4}z_{i}\big{\\}}}{\exp\big{\\{}\beta\sum_{i=1}^{4}z_{i}\big{\\}}+\exp\big{\\{}-\beta\sum_{i=1}^{4}z_{i}\big{\\}},}$
(2.7)

where the case $\beta=\infty$ should be interpreted as the appropriate limit
of the above expression. We then define the _update function_
$g_{\beta}:\\{-1,1\\}^{4}\times[0,1]\to\\{-1,1\\}$ as

$g_{\beta}(z_{1},\dots,z_{4},u):=\begin{cases}1,\qquad&\text{if }u\leq
S_{\beta}(z_{1},\dots,z_{4});\\\ -1,&\text{otherwise}.\end{cases}$ (2.8)

We can now finally define the model. Given $x\in\mathbb{Z}^{2}$, we let
$\sigma_{t}(x)=\sigma_{0}(x)$ for all $t\in[0,T^{x}_{1})$; and for each $i\geq
1$ and $t\in[T^{x}_{i},T^{x}_{i+1})$, we define

$\sigma_{t}(x)=g\big{(}(\sigma_{T^{x}_{i}}(y))_{y\sim x},R^{x}_{i}\big{)}.$
(2.9)

In principle it is not clear that this process is well defined, since in order
to define $\sigma_{t}(x)$, one might need to know the state of the process at
the neighbors of $x$. However, Lemma 2.1 guarantees that, almost surely, one
only needs to look at the state of finitely many vertices at time $0$ in order
to determine $\sigma_{t}(x)$.

From now on we will fix $\tau\geq 0$ and $\beta\in~{}[0,\infty]$. Given these
parameters, we will study the model $(\sigma_{\tau}(x)\in\mathbb{Z}^{2})$ as a
site percolation process on $\mathbb{Z}^{2}$. This means we will investigate
connectivity properties of the set of points with positive magnetization
$\\{x\in\mathbb{Z}^{2};\sigma_{\tau}(x)=1\\}$. We say that $A$ is
$+$-connected to $B$ in $C$ and write

$\Big{\\{}A\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+,\,\,C}B\Big{\\}}$ (2.10)

for the event where there exists a nearest-neighbor path $x_{0},\dots,x_{n}\in
C\subset\mathbb{Z}^{2}$ starting at $A\subset\mathbb{Z}^{2}$ and ending at
$B\subset\mathbb{Z}^{2}$ such that

$\sigma_{\tau}(x_{0})=\dots=\sigma_{\tau}(x_{n})=+1.$ (2.11)

If $C=\mathbb{Z}^{2}$, we omit it from the notation. We also write

$\Big{\\{}A\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\infty\Big{\\}}:=\bigcap_{n\geq
0}\Big{\\{}A\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\partial B(0,n)\Big{\\}}.$
(2.12)

If either $A$ or $B$ is a singleton $\\{x\\}$, $x\in\mathbb{Z}^{2}$, we write
$x$ instead of $\\{x\\}$ in the notation for the above events. We say that two
points in $\mathbb{Z}^{2}$ are $*$-neighbors if the $\ell_{\infty}$-distance
between them is $1$. We analogously define the event where $A$ is
$(-,*)$-connected to $B$ in $C$, writing

$\Big{\\{}A\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{-,\,\,*,\,\,C}B\Big{\\}}$
(2.13)

when there is a $*$-connected path, that is, a path whose consecutive sites
are $*$-neighbors, starting at $A$ and ending at $B$ whose associated states
at time $\tau$ are all $-1$.

We denote by $\operatorname{P}_{\rho}$ the probability measure associated to
the process with initial density of positive states given by $\rho\geq 0$, and
by $\operatorname{E}_{\rho}$ the associated expectation. We can then define
the probability that the origin percolates

$\theta(\rho):=\operatorname{P}_{\rho}\Big{(}0\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\infty\Big{)},$ (2.14)

and the critical parameter

$\rho_{c}:=\sup\big{\\{}\rho\geq 0;\,\theta(\rho)=0\big{\\}}.$ (2.15)

We can analogously define $\rho^{*}_{c}$, the supremum of the set of
parameters $\rho$ such that with positive probability the origin is
$(-,*)$-connected to $\infty$.

It is easy to see that

$\begin{array}[]{c}\parbox[c]{433.62pt}{\centering $\theta(\rho)$ is monotone
with respect to~{}$\rho$,\@add@centering}\end{array}$ (2.16)

given the uniform coupling between starting densities and the fact that the
function $g_{\beta}$ itself is monotone with respect to its first four
parameters (see Equation (5.1) and definitions thereafter).

We consider the usual partial order in $\\{-1,+1\\}^{\mathbb{Z}^{2}}$ defined
by vertex-wise comparison, that is,
$\omega\prec\Omega\in\\{-1,+1\\}^{\mathbb{Z}^{2}}$ if and only if
$\omega(x)\leq\Omega(x)$ for every $x\in\mathbb{Z}^{2}$. We say that an event
$A$ measurable with respect to the state of the vertices at time ${\tau}$ is
_increasing_ if

$\omega\prec\Omega\in\\{-1,+1\\}^{\mathbb{Z}^{2}},\,\omega\in
A\implies\Omega\in A.$ (2.17)

It follows from [24] that, whenever two events $A,B$ are increasing in this
manner,

$\operatorname{P}_{\rho}\big{(}A\cap
B\big{)}\geq\operatorname{P}_{\rho}\big{(}A\big{)}\operatorname{P}_{\rho}\big{(}B\big{)}.$
(2.18)

Let $\Lambda$ be a proper subset of $\mathbb{Z}^{2}$. Given
$\omega\in\\{-1,+1\\}^{\partial_{ext}\Lambda}$, we define the process
$\sigma_{\tau}|_{\Lambda,\omega}$ with boundary condition $\omega$ in a
similar manner to the original process: we let
$\\{\sigma_{0}|_{\Lambda,\omega}(x);x\in\Lambda\\}$ be i.i.d. random variables
with distribution $\rho\delta_{1}+(1-\rho)\delta_{-1}$; consider Poisson
clocks with intensity $1$ for each site of $\Lambda$, each clock tagged with a
uniform random variable; and at each time in which a clock rings, we update
the state of the process at the associated site using the function $g_{\beta}$
as in (2.9), the uniform random variable associated to the time, and the state
of the process at the four neighbors, some of which might be in
$\partial_{ext}\Lambda$ and therefore have states given by $\omega$. By the
same graphical construction coupling, one shows that the probability that
$\sigma_{\tau}|_{\Lambda,\omega}$ belongs to any increasing event is monotone
in the boundary condition $\omega$.

The following decoupling result is a direct consequence of Lemma 2.1. It tells
us that the state of $\sigma_{\tau}$ in two sufficiently distant sets is
almost uncorrelated. Its straightforward proof is essentially the same as the
one of Proposition 3.6 of [5].

######  Lemma 2.2.

Fix two sets $B_{1},B_{2}\subseteq\mathbb{Z}^{2}$ (at least one of which is
finite) and random variables $f_{1},f_{2}:\Omega\to[-1,1]$, such that $f_{i}$
is measurable with respect to $\big{(}\sigma_{\tau}(x)\big{)}_{x\in B_{i}}$,
for $i=1,2$. Then there exists a constant $c_{\ref{c:decouple}}>0$ depending
only on ${\tau}$ and $\beta$ such that

$\begin{split}\operatorname{E}_{\rho}(f_{1}f_{2})\leq&\operatorname{E}_{\rho}(f_{1})\operatorname{E}_{\rho}(f_{2})\\\
&+c_{\ref{c:decouple}}^{-1}\min\\{|B_{1}|,|B_{2}|\\}\exp\Big{\\{}-c_{\ref{c:decouple}}\operatorname{dist}(B_{1},B_{2})\log\big{(}\operatorname{dist}(B_{1},B_{2})\big{)}\Big{\\}}.\end{split}$
(2.19)

As usual in $2$-dimension percolation, our results strongly rely on  _crossing
events_ and  _duality_. We define the horizontal crossing event of the box
$[0,n]\times[0,m]$ by vertices with state $+1$ at time $\tau$:

$\mathcal{C}(n,m):=\Big{\\{}\\{0\\}\times[0,m]\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+,\,\,[0,n]\times[0,m]}\\{n\\}\times[0,m]\Big{\\}},$
(2.20)

and we analogously define the $(-,*)$-crossing event

$\mathcal{C}^{*}(n,m):=\Big{\\{}\\{0\\}\times[0,m]\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{-,\,\,*,\,\,[0,n]\times[0,m]}\\{n\\}\times[0,m]\Big{\\}}.$
(2.21)

By duality we mean the fact that, relying on an elementary discrete topology
consideration and invariance of the process under rotation by $\pi/2$, the
occurrence of the event $\mathcal{C}(n,m)^{c}$ implies the occurrence of a
translated version of the event $\mathcal{C}^{*}(m,n)$, yielding

$\operatorname{P}_{\rho}\big{(}\mathcal{C}(n,m)^{c}\big{)}=\operatorname{P}_{\rho}\big{(}\mathcal{C}^{*}(m,n)\big{)}.$
(2.22)

One can prove the following result using the decoupling inequality provided by
Lemma 2.2 together with a multiscale renormalization argument:

######  Lemma 2.3 (Finite-size criterion).

There exists a constant $c_{\ref{c:fsc}}=c_{\ref{c:fsc}}({\tau},\beta)$ such
that, if

$\begin{array}[]{c}\parbox[c]{433.62pt}{\centering for some $n\geq 1$ we have
$\operatorname{P}_{\rho}(\mathcal{C}(3n,n))>1-c_{\ref{c:fsc}}$,\@add@centering}\end{array}$
(2.23)

then $\rho>\rho_{c}$ and, for _every_ $n\geq 1$,

$\mathbb{P}(\mathcal{C}(3n,n))\geq
1-c_{\ref{c:fsc}}^{-1}\exp\bigg{\\{}-c_{\ref{c:fsc}}\frac{n}{\big{(}\log(n)\big{)}^{\epsilon}}\bigg{\\}}.$
(2.24)

An analogous result holds for dual crossings, that is, if

$\begin{array}[]{c}\parbox[c]{433.62pt}{\centering for some $n\geq 1$ we have
$\operatorname{P}_{\rho}(\mathcal{C}(3n,n)^{*})>1-c_{\ref{c:fsc}}$,\@add@centering}\end{array}$
(2.25)

then $\rho<\rho^{*}_{c}$ and, for every $n\geq 1$,

$\mathbb{P}(\mathcal{C}^{*}(3n,n))\geq
1-c_{\ref{c:fsc}}^{-1}\exp\bigg{\\{}-c_{\ref{c:fsc}}\frac{n}{\big{(}\log(n)\big{)}^{\epsilon}}\bigg{\\}}.$
(2.26)

The proof of this result follows in an elementary way from Proposition $8.1$
of [5] and the duality of the process.

######  Remark 2.

There are a few observations about the lemma above that will be useful to us.

  1. a)

We first remark that the constant $c_{\ref{c:fsc}}$ in the lemma above depends
only on the speed of the correlation decay of the model. This will be useful
when proving Theorem 1.1.

  2. b)

Second, the lemma above still remains valid if one has access to weaker
decorrelation estimates: if the decay is only exponential, the same proof
still applies, but one obtains stretched-exponential bounds in (2.24) and
(2.26).

We can now define the  _fictitious regime_ of the parameter $\rho$, where
there is a uniformly positive probability in $n$ of both $+$-crossings and
$(-,*)$-crossings of rectangles $[0,3n]\times[0,n]$ in the hard direction. We
define

$\begin{split}\rho_{+}&=\inf\Big{\\{}\rho>0;\operatorname{P}_{\rho}(\mathcal{C}(3n,n))>1-c_{\ref{c:fsc}}\text{
for some $n\geq 1$}\Big{\\}}\\\
&=\sup\Big{\\{}\rho>0;\operatorname{P}_{\rho}(\mathcal{C}^{*}(n,3n))\geq
c_{\ref{c:fsc}}\text{ for all $n\geq 1$}\Big{\\}};\end{split}$ (2.27)
$\begin{split}\rho_{-}&=\sup\Big{\\{}\rho>0;\operatorname{P}_{\rho}(\mathcal{C}^{*}(3n,n))>1-c_{\ref{c:fsc}}\text{
for some $n\geq 1$}\Big{\\}}\\\
&=\inf\Big{\\{}\rho>0;\operatorname{P}_{\rho}(\mathcal{C}(n,3n))\geq
c_{\ref{c:fsc}}\text{ for all $n\geq 1$}\Big{\\}}.\end{split}$ (2.28)

It is clear by monotonicity and duality of the process that
$\rho_{-}\leq\rho_{+}$. Our whole objective is to show  _sharpness_ , that is,
that $\rho_{-}=\rho_{+}$ and the interval $[\rho_{-},\rho_{+}]$ is degenerate.
We assume from now on by contradiction that $\rho_{-}<\rho_{+}$.

One of the the main tools in the study of planar percolation has been the
Russo-Seymor-Welsh theory (RSW for short). Developed originally for Bernoulli
percolation, this theory essentially says that, if the probability of crossing
rectangles with fixed aspect-ratio but variable size in the “easy direction”
is uniformly positive in said size, then the probability of crossing in the
hard direction is also uniformly positive. In the past decade there was a
substantial development aiming at generalizing the theory for other planar
models. In [43], this theory was developed for Voronoi percolation; in [2],
the results were generalized for non-i.i.d. processes. The latest development
in this generalization effort proves the result in the greatest generality
yet: for planar models satisfying the Harris-FKG inequality and invariant
under translations and the $D_{4}$ symmetries. These results can all be
adapted to our context. In particular, the hypotheses of [30] are all
satisfied by our process, implying the existence of a constant
$c_{\ref{c:rsw}}>0$ possibly depending on $\beta,\tau$ such that

$\begin{array}[]{c}\parbox[c]{433.62pt}{\centering for all
$\rho\in[\rho_{-},\rho_{+}]$ we have\\\
$\operatorname{P}_{\rho}\big{(}\mathcal{C}(3n,n)\big{)}>c_{\ref{c:rsw}}$ and
$\operatorname{P}_{\rho}\big{(}\mathcal{C}^{*}(3n,n)\big{)}>c_{\ref{c:rsw}}$,
for all $n\geq 1$.\@add@centering}\end{array}$ (2.29)

To finish this section, we define the  _arm event_ we will need later. For
$m<n\in\mathbb{N}$, we write

$\begin{split}\operatorname{Arm}(m,n)&=\big{\\{}\partial B(0,m)\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\partial_{ext}B(0,n)\big{\\}},\\\
\operatorname{Arm}_{x}(m,n)&=\big{\\{}\partial B(x,m)\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}\partial_{ext}B(x,n)\big{\\}}.\end{split}$
(2.30)

##  3 Existence of phase transitions

We here prove Theorem 1.1. We will first treat the case of existence of phase
transition for $\beta=\infty$ or for small times. These rely heavily on Lemma
2.3 and continuity arguments. The proof of absence of phase transition for
small values of $\beta$ and large values of time is more delicate and relies
on the fact that the same is true for large-temperature Ising model.

######  Remark 3.

Exclusively in this section, we denote by $\operatorname{P}_{\rho}^{\beta}$
the distribution of the process with initial density $\rho$ and the inverse
temperature $\beta\in[0,\infty]$. For $\beta<\beta_{c}$, we let
$\operatorname{P}^{\beta}$ denote the unique infinite-volume measure.

The case $\beta=\infty$. This is the simplest case. We first fix $\tau\geq 0$
and apply Lemma 2.3, which implies the existence of a constant
$c_{\ref{c:fsc}}=c_{\ref{c:fsc}}(\tau,\beta,\tau)>0$ such that, if

$\begin{array}[]{c}\parbox[c]{433.62pt}{\centering$\operatorname{P}_{\rho}^{\infty}(\sigma_{\tau}\in\mathcal{C}(3n,n))>1-c_{\ref{c:fsc}}$,
for some $n\geq 1$,\@add@centering}\end{array}$ (3.1)

then percolation occurs. We now conclude by noticing that
$\rho\mapsto\operatorname{P}^{\infty}_{\rho}(\sigma_{\tau}\in\mathcal{C}(3n,n))$
is continuous and that this probability equals $1$ if $\rho=1$, implying
$\rho_{c}(\tau)<1$. Arguing analogously for the event $\mathcal{C}^{*}(3n,n)$,
we obtain $\rho_{c}(\tau)\in(0,1)$.

Non-trivial phase transition for small times. Assume now that
$\beta\in(0,\infty)$. Our goal is to prove that there exists a non-trivial
phase transition for small values of $\tau$. In this case, we make use of
Remark 2, since the correlation decay in Lemma 2.1 can be made uniform for
compact sets of time. Let us now restrict ourselves to $\tau\in[0,1]$. In this
case, the constant $c_{\ref{c:fsc}}$ from Lemma 2.3 is uniformly bounded.
Furthermore, the function
$\tau\mapsto\operatorname{P}^{\beta}_{\rho}(\sigma_{\tau}\in\mathcal{C}(3n,n))$
is continuous. Observe that $\tau=0$ corresponds to independent site-
percolation on the plane, which undergoes a non-trivial (and sharp) phase
transition. Therefore we conclude the existence of a non-trivial phase
transition for small values of $\tau$ as well.

Absence of phase transition. This is the case that demands a more careful
analysis. Our first goal is to prove a uniform decoupling inequality for small
values of $\beta$. This will then allow us to employ Lemma 2.3 together with
arguments that compare finite-time configurations with the limiting
distribution.

We first describe an alternative exploration process that determines the value
of $\sigma_{\tau}(0)$. Fix a realization of the graphical construction of the
process $(\sigma_{\tau})_{\tau\geq 0}$, and notice that, according to (2.7),
if $T^{i}_{x}$ is a mark at $x\in\mathbb{Z}^{2}$, then the new opinion of $x$
does not depend on the values of its neighboring vertices if $R_{i}^{x}\leq
S_{\beta}(-1,-1,-1,-1)$ or $R_{i}^{x}>S_{\beta}(1,1,1,1)$.

We now describe a random continuous graph that can be used to determine
$\sigma_{\tau}(0)$. We start by declaring the space-time point $(0,\tau)$ as
active, and go backwards in time from $(0,\tau)$ until we hit a Poisson mark,
say $T^{i}_{0}$. The mark $T^{i}_{0}$ is called a death mark of the process if
the opinion of the vertex at that time can be determined without the
information of the neighbors. In this case, we also declare the space-time
vertex $(0,T^{i}_{0})$ as a dead end. If this is not the case, the vertex
remains active and all neighboring vertices are also declared active. We now
continue the process until we reach time $0$ or until no more active vertices
remain. Observe that, given the exploration graph and the opinion of all dead
ends of the graph (which are determined by the value of the corresponding
uniform random variable), the value of $\sigma_{\tau}(0)$ is determined.

We now verify that this exploration algorithm implies that the exponential
correlation decay is uniform for all large enough $\tau$, if $\beta$ is taken
small enough. This will be achieved by comparing the exploration process with
a Galton-Watson random tree. Notice that the probability that a Poisson mark
in the exploration is not a dead end is
$S_{\beta}(1,1,1,1)-S_{\beta}(-1,-1,-,1-,1)$ and, in this case, it gives birth
to four new vertices. This implies that the distance (in $\mathbb{Z}^{2}$) one
needs to observe is bounded by the number of generations on a Galton-Watson
random tree with offspring distribution given by

$p_{\beta}(5)=1-p_{\beta}(0)=S_{\beta}(1,1,1,1)-S_{\beta}(-1,-1,-,1-,1).$
(3.2)

In particular, we obtain the following lemma.

######  Lemma 3.1.

There exists $\beta_{0}>0$ such that, for all $\beta\leq\beta_{0}$, there
exists $\tau_{0}=\tau_{0}(\beta)$ and a positive constant
$c_{\ref{c:correlation_small_beta}}=c_{\ref{c:correlation_small_beta}}(\beta)$
such that, for all $\tau\geq\tau_{0}$ and $\rho\in[0,1]$,

$\operatorname{Cov}^{\beta}_{\rho}(\sigma_{\tau}(x),\sigma_{\tau}(y))\leq
c_{\ref{c:correlation_small_beta}}e^{-c_{\ref{c:correlation_small_beta}}^{-1}|x-y|_{1}}.$
(3.3)

###### Proof.

Notice that both $S_{\beta}(1,1,1,1)$ and $S_{\beta}(-1,-1,-1,-1)$ converge to
$\frac{1}{2}$ as $\beta\to 0$. In particular, there exists $\beta_{0}$ such
that $p_{\beta}(5)<\frac{1}{5}$, for all $\beta\leq\beta_{0}$. This then
implies that the Galton-Watson tree with offspring distribution $p_{\beta}$ is
subcritical. Let $Z_{n}$ denote the number of descendants at generation $n$ of
this tree.

The distance one needs to observe in the exploration algorithm in order to
determine the opinion of a vertex $x$ is bounded from above by the maximal
nonempty generation of a Galton-Watson tree with offspring distribution
$p_{\beta}$. Besides, if two vertices have disjoint exploration graphs, their
opinions at time $\tau$ are independent. In particular, we obtain that

$\begin{split}\operatorname{Cov}^{\beta}_{\rho}(\sigma_{\tau}(x),\sigma_{\tau}(y))&\leq
2\operatorname{P}_{\rho}^{\beta}\Big{[}\inf\\{n:Z_{n}>0\\}\geq\frac{1}{2}|x-y|_{1}\Big{]}\\\
&\leq 2\operatorname{P}_{\rho}^{\beta}\big{[}Z_{\frac{1}{2}|x-y|_{1}}\geq
1\big{]}\leq 2(5p_{\beta}(5))^{\frac{1}{2}|x-y|_{1}},\end{split}$ (3.4)

concluding the proof. ∎

Let us now investigave percolation on the Ising model. In [26], the author
proves that the Ising model undergoes a sharp phase transition on the external
field for every value of $\beta$. With the results in [25], we obtain that the
case of zero external field is subcritical for $\beta$ small enough and [25]
also implies

$\operatorname{P}^{\beta}[\mathcal{C}(3n,n)]\to 0,$ (3.5)

for the Ising model with inverse temperature $\beta$ small enough. In
particular, we have

$\operatorname{P}^{\beta}[\mathcal{C}^{*}(3n,n)]\to 1.$ (3.6)

Our last ingredient is the weak convergence of the Glauber dynamics to the
Ising model (see for example [31]). We have that

$\operatorname{P}_{\rho}^{\beta}[\sigma_{\tau}\in\mathcal{C}^{*}(3n,n)]\to\operatorname{P}^{\beta}[\mathcal{C}^{*}(3n,n)],\text{
as }\tau\to\infty,$ (3.7)

uniformly in $\rho\in[0,1]$.

The proof will now be completed with the aid of Lemma 2.3. Let $\beta_{0}$ as
in Lemma 3.1, and fix $\beta\leq\beta_{0}$. Once again, thanks to Remark 2,
the constant $c_{\ref{c:fsc}}$ in Lemma 2.3 does not depend on
$\tau\geq\tau_{0}(\beta)$. Take then $n$ large enough such that

$\operatorname{P}^{\beta}[\mathcal{C}^{*}(3n,n)]\geq
1-\frac{c_{\ref{c:fsc}}}{2},$ (3.8)

and $\tau$ large so that

$\sup_{\rho\in[0,1]}\big{|}\operatorname{P}_{\rho}^{\beta}[\sigma_{\tau}\in\mathcal{C}^{*}(3n,n)]-\operatorname{P}^{\beta}[\mathcal{C}^{*}(3n,n)]\big{|}\leq\frac{c_{\ref{c:fsc}}}{2}.$
(3.9)

In particular, this implies

$\operatorname{P}_{\rho}^{\beta}[\sigma_{\tau}\in\mathcal{C}^{*}(3n,n)]\geq
1-c_{\ref{c:fsc}},$ (3.10)

and hence there is no percolation for all values of $\rho$ by Remark 2-b and
Lemma 2.3.

##  4 Sharpness using OSSS and Aizenman-Grimmett

The last decade has been marked by a frequent use of the OSSS inequality [36]
to prove sharpness results in statistical mechanics, see [15, 13, 16, 10].

The success of this technique suggests the existence of a more general result
or procedure that would apply to most models of interest. This is however not
the case, as even proofs that make use of OSSS still require model specific
arguments, as we point out below.

Before we enter into details of the flexibility (and also the limitations) of
the OSSS techniques, let us first state the main theorem.

### The OSSS inequality.

Given a natural number $n\geq 1$ and a Boolean function
$f:\\{-1,1\\}^{n}\to\\{-1,1\\}$, we now define what we mean by a _randomized
algorithm to determine $f$_. Although algorithm is usually described in terms
of decision trees, we give here a precise definition using notation from
stochastic processes. For this, fix $f$ as above and a (possibly random)
argument $\omega\in\\{-1,1\\}^{n}$ on which we want to evaluate $f$.

The first ingredient of an algorithm is a process $(X_{1},X_{2},\dots,X_{n})$,
where each $X_{j}\in\\{1,\dots,n\\}$ (this can be seen as a random permutation
of $\\{1,\dots,n\\}$, giving us the order in which we reveal $\omega$). The
second ingredient in the construction is a random time $T\in\\{1,\dots,n\\}$
at which we stop the revealment process.

Introducing the filtration
$\mathcal{F}_{j}=\sigma\big{(}X_{1},\omega(X_{1}),X_{2},\omega(X_{2}),\dots,X_{j},\omega(X_{j})\big{)}$,
we say that $\mathcal{A}=\big{(}(X_{j})_{j=1}^{n},T\big{)}$ is a _random
algorithm to determine $f$_ if:

  * •

$\mathcal{F}_{0}$ is independent of $\omega$,

  * •

$(X_{j})_{j=1}^{n}$ is predictable (meaning that $X_{j}\in\mathcal{F}_{j-1}$
for every $j=1,\dots,n$),

  * •

$T$ is a stopping time with respect to the filtration
$(\mathcal{F}_{j})_{j=0}^{n}$, and

  * •

$f$ is measurable with respect to $\mathcal{F}_{T}$.

Having this definition, we can state the OSSS inequality.

######  Theorem 4.1 ([36]).

Let $f:\\{-1,1\\}^{n}\to\\{-1,1\\}$ and let $\mathcal{A}$ be a random
algorithm to determine $f$. Then,

$\operatorname{Var}(f)\leq
4\sum_{i=1}^{n}\delta_{i}(\mathcal{A})\operatorname{Inf}_{i}(f),$ (4.1)

where $\delta_{i}$ is the probability that $\mathcal{A}$ reveals $i$ and
$\operatorname{Inf}_{i}$ is the influence of $i$, that is

$\delta_{i}(\mathcal{A})=P[X_{j}=i;\text{ for some $j\leq T$}],$ (4.2)

and

$\operatorname{Inf}_{i}(f)=P[f(\omega_{1},\dots,\omega_{i},\dots,\omega_{n})\neq
f(\omega_{1},\dots,-\omega_{i},\dots,\omega_{n})].$ (4.3)

We now move to a simple yet powerful application of OSSS to independent
percolation.

### Sharpness for Bernoulli percolation.

Let us now give a quick overview of how OSSS can be used to prove sharpness
for Bernoulli percolation. Although this is a very classical result and
technique, we have decided to include this brief overview, since our proofs
are inspired by this argument.

We start by choosing a parameter $p\in[0,1]$ and letting
$(\sigma(x))_{x\in\mathbb{Z}^{2}}$ be an i.i.d. collection of
$\operatorname{Ber}(p)$ random variables under $P_{p}$.

Recall the definition of the crossing probabilities $\mathcal{C}(n,m)$ in
(2.20). What OSSS will allow us to do is to prove that the functions
$h_{n}:[0,1]\to[0,1]$ defined through the map $p\mapsto
P_{p}[\mathcal{C}(n,n)]$ undergo a sharp transition from zero to one, as $n$
goes to infinity.

The way in which this is done is by bounding from below the derivative of
$h_{n}$ in the critical window. For simplicity, we will only establish this
bound at the critical point

$p_{c}=\inf\big{\\{}p\in[0,1];P_{p}[0\leftrightarrow\infty]>0\big{\\}}.$ (4.4)

We will also need the following bounds, which are consequences of symmetry and
the RSW’s inequality. There exist
$c_{\ref{c:bernoulli_arm}},c_{\ref{c:box_cross}}>0$ such that

$P_{p_{c}}[\operatorname{Arm}(1,n)]\leq
n^{-c_{\ref{c:bernoulli_arm}}}\qquad\text{and}\qquad
P_{p_{c}}[\mathcal{C}(n,n)]\in(c_{\ref{c:box_cross}},1-c_{\ref{c:box_cross}}),$
(4.5)

for all $n\geq 1$.

Although this is not the central point of this article, we have decided to
introduce in more detail the algorithm that is used, since it will serve as a
basis for the other algorithms that will come.

Our algorithm starts by selecting a random integer $Z$ uniformly on
$\\{1,\dots,n\\}$. We then explore all the sites that are connected by open
sites to some neighbor of the column $\\{Z\\}\times\\{1,\dots,n\\}$.

Here are few observations from the above construction:

  * •

All the sites in the column $\\{Z\\}\times\\{1,\dots,n\\}$ will be revealed,

  * •

if a site is revealed, it must be connected to the above column,

  * •

by the time we have explored all these sites, we are able to decide if
$\mathcal{C}(n,n)$ occurred.

Or in other words, this algorithm determines the crossing events.

We can now apply OSSS inequality (4.1) to obtain

$\begin{split}\operatorname{Var}_{p}(\mathcal{C}(n,n))&\leq
4\sum_{i=1}^{n^{2}}\delta_{i}(\mathcal{A})\operatorname{Inf}_{i}(\mathcal{C}(n,n))\\\
&\leq
4\sup_{i\in\\{1,\dots,n\\}^{2}}P_{p}\big{[}i\leftrightarrow\\{Z\\}\times\\{1,\dots,n\\}\big{]}\sum_{i=1}^{n^{2}}\operatorname{Inf}_{i}(\mathcal{C}(n,n)).\end{split}$
(4.6)

We estimate, for every $(i_{1},i_{2})\in\\{1,\dots,n\\}^{2}$,

$\begin{split}P_{p_{c}}\big{[}(i_{1},i_{2})&\leftrightarrow\\{Z\\}\times\\{1,\dots,n\\}\big{]}\\\
&\leq
P_{p_{c}}\big{[}|i_{1}-Z|\leq\sqrt{n}\big{]}+P_{p_{c}}\big{[}\operatorname{Arm}(1,\sqrt{n})\big{]}\overset{\eqref{e:arm_cross}}{\leq}\frac{\sqrt{n}}{n}+n^{-\frac{c_{\ref{c:bernoulli_arm}}}{2}}\leq
2n^{-c_{\ref{c:column}}}.\end{split}$ (4.7)

Finally, we recall Russo’s formula, stating that

$\begin{split}\frac{\mathop{}\\!\mathrm{d}P_{p}[\mathcal{C}(n,n)]}{\mathop{}\\!\mathrm{d}p}(p_{c})&=\sum_{i=1}^{n^{2}}\operatorname{Inf}_{i}\big{(}\mathcal{C}(n,n)\big{)}\overset{\eqref{e:upper_variance},\eqref{e:touch_column}}{\geq}\frac{n^{c_{\ref{c:column}}}}{8}\operatorname{Var}_{p_{c}}(\mathcal{C}(n,n))\\\
&=\frac{n^{c_{\ref{c:column}}}}{8}P_{p_{c}}[\mathcal{C}(n,n)]\big{(}1-P_{p_{c}}[\mathcal{C}(n,n)]\big{)}\geq\frac{c_{\ref{c:box_cross}}^{2}}{8}n^{c_{\ref{c:column}}},\end{split}$
(4.8)

which is precisely the lower bound that we wanted.

######  Remark 4.

It is important to observe here that a fortunate coincidence has helped us in
the last step of the proof. More precisely, note that the sum of influences
that appear on the right hand side of (4.6) was originated from the OSSS
inequality, but incidentally this happens to be the same sum that appears in
Russo’s formula. This fact is further commented in Remark 5 b) below.

When we apply this technique to our model, it will become clear how this can
be used to prove sharpness of the phase transition, as well as a polynomial
upper bound on the size of the critical window.

### Difficulties applying the OSSS technique to other models.

We now discuss the possible strategies and difficulties involved in
generalizing the method described above to non-independent percolation models.

The first obvious obstruction comes from the lack of independence, since the
original OSSS inequality (4.1) is stated under this assumption. It is a common
technique however to use independent Bernoulli random variables to simulate a
dependent environment.

In fact we have already introduced independent random variables (namely the
$\sigma_{0}(x)$ and $R^{x}_{i}$) graphical construction. However there are two
difficulties that need to be addressed as we comment in the following.

######  Remark 5.

In order to apply the above techniques to non-independent scenarios, we have
to take into account the following observations.

  1. a)

When devising a graphical construction, it is rarely the case that all sources
of randomness are independent Bernoulli random variables, for example in our
case the marks $T^{x}_{i}$ are distributed as a Poisson process. Our strategy
to deal with this issue is to condition on all the elements of the graphical
construction that are not independent Bernoulli variables. The challenge then
is to show that these variables that we conditioned on are not very
influential.

A very important contribution in this direction comes from the work of [1],
and we discuss this technique in detail in Lemma 4.2.

  2. b)

Suppose for simplicity that our model is constructed from independent
Bernoulli distributed random variables only. Even in that case there can be a
difficulty in applying OSSS that comes from the different roles that these
variables can play. Taking our Glauber dynamics as an example: the variables
$R^{x}_{i}$ are used for the dynamics itself, while the variables
$\sigma_{0}(x)$ give the initial condition.

To understand better why this poses a problem, let us recall that each of
these variables appear in the OSSS formula, since they can all influence our
event. However, only some of them appear in Russo’s formula (in our Glauber
example, only the $\sigma_{0}(x)$). This breaks the fortunate coincidence that
occurs in the Bernoulli case and we commented in Remark 4.

The solution to this second issue is inspired by the work of Aizenman-Grimmett
[4]. Roughly speaking, the strategy here is to compare the probability that
different types of variables are pivotal, making it possible to connect OSSS
and Russo again. This is explained in detail in Section 6.

### Homogenization by thickening.

We now briefly describe the technique developed in [1] to show that
conditioning on the location of the Poisson marks of our graphical
construction cannot have a big influence on the crossing probabilities.

The central lemma is in fact much more general and applies to several contexts
in which one wants to condition on a Poisson point process, while still
controlling the probability of certain events.

Fix an integer $k\geq 1$ and let $\mu=\sum_{i}\delta_{z_{i}}$ be a Poisson
Point Process on a Polish space $M$ with intensity measure $k\nu$.

We define a thinning $\mu_{k}$ of $\mu$, which is obtained by keeping each
point of $\mu$ independently with probability $1/k$. Clearly $\mu_{k}$ is a
Poisson Point Process with intensity $\nu$.

######  Lemma 4.2 ([1]).

In the above context, let $Y$ be a $\\{0,1\\}$-valued random variable,
measurable with respect to $\mu_{k}$, and

$Z=\operatorname{E}\big{[}Y(\mu_{k})|\mu\big{]}.$ (4.9)

In this context,

$\operatorname{Var}(Z)\leq 1/k.$ (4.10)

Moreover, if $\operatorname{E}\big{(}Y\big{)}\in(c,1-c)$, for some
$c\in[0,\frac{1}{2}]$, then

$\operatorname{P}\Big{[}\operatorname{E}\big{(}Y|\mu\big{)}\notin\Big{(}\frac{c}{2},1-\frac{c}{2}\Big{)}\Big{]}\leq\frac{4\operatorname{Var}(Z)}{c^{2}}\leq\frac{4}{kc^{2}}.$
(4.11)

###### Proof.

Let $\tilde{\mu}=\sum_{i}\delta_{(z_{i},d_{i})}$ be a Poisson Point Process on
a space $M\times[k]$, with intensity measure
$\nu\otimes\operatorname{count}_{[k]}$, where $\operatorname{count}_{[k]}$ is
the counting measure in $[k]=\\{1,\dots,k\\}$. We write
$\mu_{M}=\sum_{i}\delta_{z_{i}}$ for the projection of $\tilde{\mu}$ on the
first coordinate, which is also a PPP with intensity $k\nu$. For $j\in[k]$,
let $\tilde{\mu}_{j}=\sum_{i:d_{i}=j}\delta_{z_{i}}$. Consider also an
independent random variable $X$ uniformly distributed in $[k]$ and observe
that the pair $(\tilde{\mu}_{X},\mu_{M})$ has the same distribution as
$(\mu_{k},\mu)$. From this we compute

$\begin{split}\operatorname{Var}(Z)&=\operatorname{Var}\big{(}\operatorname{E}\big{[}Y(\mu_{X})|\mu_{M}\big{]}\big{)}\leq\operatorname{Var}\big{(}\operatorname{E}\big{[}Y(\mu_{X})|(\tilde{\mu}_{j})_{j=1}^{k}\big{]}\big{)}\\\
&\
=\operatorname{Var}\Big{(}\frac{1}{k}\sum_{j=1}^{k}Y(\tilde{\mu}_{j})\Big{)}\leq\frac{1}{k^{2}}\sum_{j=1}^{k}\operatorname{Var}\big{(}Y(\tilde{\mu}_{j})\big{)}\leq\frac{1}{k},\end{split}$
(4.12)

concluding the proof. ∎

##  5 Proof ingredients

### The thickened graphical construction.

In this section we will define a process from which we can extract the
percolation process defined in Section 2. In line with Lemma 4.2 and the
discussion from the previous section, we will first thicken the Poisson mark
process by a factor of $k\geq 1$, and keep each mark with probability $1/k$.
We will also couple all possible starting densities $\rho$ using a standard
uniform coupling.

We first associate to each $x\in\mathbb{Z}^{2}$ an independent random variable
$U_{x}\stackrel{{\scriptstyle d}}{{=}}U[0,1]$. We can then define

$\sigma_{\rho,0}(x)={\bf 1}_{\\{U^{x}<\rho\\}}-{\bf 1}_{\\{U^{x}\geq\rho\\}},$
(5.1)

so that this collection of variables becomes coupled for different
$\rho\in[0,1]$, all of them having distribution $\operatorname{Ber}(\rho)$.

Let $k\geq 1$ be some fixed integer. We define an independent Poisson point
process $\mu$ on
$\mathbb{Z}^{2}\times\mathbb{R}_{+}\times[0,1]\times\\{0,1\\}$ with underlying
intensity measure given by

$\operatorname{count}_{\mathbb{Z}^{2}}\,\otimes\,k\cdot\mathop{}\\!\mathrm{d}x\,\otimes\,U[0,1]\,\otimes\,\operatorname{Ber}(1/k),$
(5.2)

where $\operatorname{count}_{\mathbb{Z}^{2}}$ is the counting measure over
$\mathbb{Z}^{2}$, and $k\cdot\mathop{}\\!\mathrm{d}x$ represents $k$ times the
Lebesgue measure over $\mathbb{R}_{+}$. This point process will encode all the
relevant information needed in order to define the process in the manner
described above. A point $(X,T,R,D)$ in the support of the point measure will
carry the following information:

  * •

$X\in\mathbb{Z}^{2}$ will represent the site;

  * •

$T\in\mathbb{R}_{+}$ will encode the time-mark associated to $X$;

  * •

$R\in[0,1]$ will hold the uniform random variable used in order to apply the
rules of the local updates, that is, this random variable will be used in the
same manner as $R_{x}$ in (2.9);

  * •

$D\in\\{0,1\\}$ will be the Bernoulli random variable responsible for either
accepting or rejecting the update of the process given by $X$, $T$, and $R$.
This is the thinning random variable appearing in Lemma 4.2.

We fix an arbitrary manner of enumerating points in the support of the point
measure $\mu$. In this way, we can write

$\mu:=\sum_{i\geq 1}\delta_{(X_{i},T_{i},R_{i},D_{i})},$ (5.3)

so that $X_{i},T_{i},R_{i},D_{i}$ are well-defined random variables. We can
then define $\mu_{k}$, the _thinned_ version of $\mu$, by first removing all
points in the support such that $D_{i}=0$ and then projecting the remaining
points onto $\mathbb{Z}^{2}\times\mathbb{R}_{+}\times[0,1]$. Given
$x\in\mathbb{Z}^{2}$, we consider the set of time-marks of $\mu_{k}$
associated to $x$,

$\big{\\{}T_{i}\in\mathbb{R}_{+};\,(x,T_{i},R_{i})\in\operatorname{supp}(\mu_{k})\big{\\}}.$
(5.4)

These sets are distributed as the sets $\mathcal{P}^{x}$ defined in Section 2,
and the same notation will be used for them. The variables $R_{i}$ associated
to each $T_{i}\in\mathcal{P}^{x}$ are then distributed as the collection of
independent uniform random variables associated to each time-mark of $x$
defined in (2.4). We can then use these collections of Poissonian time-marks,
associated uniform random variables, and the initial state variables defined
in (5.1) in order to define the graphical construction in the same manner of
Section 2. By elementary properties of the Poisson point process, the process
defined in this manner has the same distribution as

$\big{(}\sigma_{\rho,t}(x)\big{)}_{x\in\mathbb{Z}^{2},t\in\mathbb{R}_{+}}.$

From now on we assume the new construction above was the one used in order to
define this process, and write $\operatorname{P}$ and $\operatorname{E}$ for
the probability and expectation corresponding to the new construction. We note
that all starting densities $\rho\in[0,1]$ are coupled, and the monotone
properties alluded to in Section 2 are justified by the monotonicity of
$g_{\beta}$.

### Influence and Russo’s formula.

Consider a finite set $K\subset\mathbb{Z}^{2}$ and an event $A$ measurable
with respect to the variables in the collection $(\sigma_{\rho,\tau}(x))_{x\in
K}$. This collection depends on both the initial configuration
$(U^{x})_{x\in\mathbb{Z}^{2}}$ and the graphical construction $\mu_{k}$,
constructed from $\mu=\sum_{i\geq 0}\delta_{(X_{i},T_{i},R_{i},D_{i})}$. In
order to measure how sensitive the event $A$ is to perturbations of these
random variables, we change the configuration $\sigma_{\rho,\tau}$ in
different ways:

  * •

_Change initial distribution_ : given $u\in[0,1]$ and $x\in\mathbb{Z}^{2}$,
let $\sigma^{x,u}_{\rho,\tau}$ be obtained in the same way as
$\sigma_{\rho,\tau}$, but replacing $U^{x}$ by $u$.

  * •

_Change the thinning variable_ : given $d\in\\{0,1\\}$ and $i\in\mathbb{N}$,
let $\sigma^{i,d}_{\rho,\tau}$ be obtained in the same way as
$\sigma_{\rho,\tau}$, but replacing $d_{i}$ in the expression of $\mu_{k}$ by
the value $d$.

This gives rise to different types of _pivotality_ :

  * •

_Initial distribution_ : Given $(U^{x})_{x\in\mathbb{Z}^{2}}$ and $\mu_{k}$,
we say that $x$ is $\rho$-pivotal for $A$ if the occurrence of the event $A$
changes between the configurations $\sigma^{x,0}_{\rho,\tau}$ and
$\sigma^{x,1}_{\rho,\tau}$.

  * •

_Thinning variables_ : Given $(U^{x})_{x\in\mathbb{Z}^{2}}$ and $\mu_{k}$, we
say that $i$ is $\rho$-pivotal for $A$ if the occurrence of the event $A$
changes between the configurations $\sigma^{i,0}_{\rho,\tau}$ and
$\sigma^{i,1}_{\rho,\tau}$.

######  Remark 6.

Note that both definitions above have the same notation ($\rho$-pivotal). But
one refers to some $x\in\mathbb{Z}^{2}$ and the other to some
$i\in\mathbb{N}$.

Given $\mu=\sum_{i\geq 0}\delta_{(X_{i},T_{i},R_{i},D_{i})}$, we define
$\mu^{\prime}:=\sum_{i\geq 0}\delta_{(X_{i},T_{i},R_{i})}$, that is, we forget
the knowledge about which time-marks are erased and which are kept. We will
write $\operatorname{P}[\,\cdot\,|\mu^{\prime}]$ for the conditional
probability with respect to $\mu^{\prime}$. This means that the randomness in
$\operatorname{P}[\,\cdot\,|\mu^{\prime}]$ comes from the uniform variables
$(U^{x})_{x\in\mathbb{Z}^{2}}$, which encode the initial state of the
vertices, and the variables $D_{i}$, which determine the thinning of variables
related to the graphical construction.

Suppose $A$ in a non-decreasing event. We define

  * •

_Initial condition influence_ : Given $x\in\mathbb{Z}^{2}$, let

$\operatorname{Inf}^{\textnormal{init}}_{\rho}(x,A|\mu^{\prime})=\operatorname{P}[x\text{
is $\rho$-pivotal for $A$}|\mu^{\prime}].$ (5.5)

  * •

_Thinning influence_ : Given $i\in\mathbb{N}$, let

$\operatorname{Inf}^{\textnormal{thin}}_{\rho}(i,A|\mu^{\prime})=\operatorname{P}[i\text{
is $\rho$-pivotal for $A$}|\mu^{\prime}].$ (5.6)

We can finally state Russo’s formula applied to this process, relating the
$\rho$-derivative of the probability of an increasing event, and the initial
condition influence:

######  Lemma 5.1.

Fix $\tau\geq 0$ and let $A$ be an non-decreasing event that depends on the
configuration $(\sigma_{\rho,\tau}(x))_{x\in K}$, for a finite set
$K\subseteq\mathbb{Z}^{2}$. Then

$\frac{\mathop{}\\!\mathrm{d}\operatorname{P}\big{(}\sigma_{\rho,\tau}\in
A\big{|}\mu^{\prime}\big{)}}{\mathop{}\\!\mathrm{d}\rho}=\sum_{x\in\mathbb{Z}^{2}}\operatorname{Inf}^{\textnormal{init}}_{\rho}\big{(}x,A|\mu^{\prime}\big{)},$
(5.7)

for $\operatorname{P}$-almost every $\mu^{\prime}$.

###### Proof sketch.

The main difference between the proof of the above result and the proof of the
original formula is that, in principle, the initial state of an infinite
number of vertices might influence the occurrence of $A$. But we know from
Lemma 2.2 that the probability of the state of a faraway vertex influencing
what happens inside $K$ goes to zero faster than exponentially in the distance
between said vertex and $K$. This observation and a limiting argument are
enough to conclude the result.

∎

### An OSSS inequality for the model.

Given an event $E$ depending on the states of the process at time $\tau$, we
can write

${\bf
1}_{E}=f\Big{(}\big{(}\sigma_{\rho,0}(x)\big{)}_{x\in\mathbb{Z}^{2}},(D_{i})_{i\geq
0},\mu^{\prime}\Big{)},$ (5.8)

for some function $f$ with appropriate domain. As discussed in Section 4, we
want to define an algorithm $\mathcal{A}$ that determines $f$.

In order to use the powerful machinery related to the study of Boolean
functions, we will let the algorithm know $\mu^{\prime}$ from the beginning,
having only to reveal the Bernoulli variables. Given $\mu^{\prime}$, the above
function $f(\cdot,\cdot,\mu^{\prime})$ can then be regarded as a Boolean
function.

Define revealments $\delta_{\rho}(\mathcal{A},x)$ and
$\delta_{\rho}(\mathcal{A},i)$, in the same manner as in (4.2):
$\delta_{\rho}(\mathcal{A},x)$ as the probability that the algorithm
$\mathcal{A}$ reveals the variable $\sigma_{\rho,0}(x)$ before stopping, and
$\delta_{\rho}(\mathcal{A},i)$ as the probability that the algorithm reveals
$D_{i}$ before stopping. We note that $\rho$ appears in the definition of the
entries for $f$.

Considering $\operatorname{P}_{\rho},\operatorname{E}_{\rho}$ and
$\operatorname{Var}_{\rho}$ as relating to the law of

$\Big{(}\big{(}\sigma_{\rho,0}(x)\big{)}_{x\in\mathbb{Z}^{2}},(d_{i})_{i\geq
0},\mu^{\prime}\Big{)},$ (5.9)

we obtain the following result:

######  Lemma 5.2.

Assume $f$ defined as above only depends on the states
$(\sigma_{\rho,\tau}(x))_{x\in K}$ for a finite set $K\subset\mathbb{Z}^{2}$.
Then for any algorithm $\mathcal{A}$ that determines $f$,

$\operatorname{Var}_{\rho}(E|\mu^{\prime})\leq
4\bigg{(}\sum_{x\in\mathbb{Z}^{2}}\operatorname{Inf}^{\textnormal{init}}_{\rho}(x,E|\mu^{\prime})\delta_{\rho}(\mathcal{A},x)+\sum_{i\geq
0}\operatorname{Inf}^{\textnormal{thin}}_{\rho}(i,E|\mu^{\prime})\delta_{\rho}(\mathcal{A},i)\bigg{)}.$
(5.10)

The proof of this lemma follows immediately from Theorem 4.1 once one realizes
that, $\mu^{\prime}$-a.s., there are only finitely many variables of the type
$\sigma_{\rho,0}(x)$ and $D_{i}$ that can affect the outcome of $f$. The
factor $4$ in the RHS of the above equation comes from the fact that $f$ takes
values in $\\{0,1\\}$.

######  Remark 7.

Observe that the influences of both the initial configuration and the thinning
variables appear in the upper bound of the variance. In Section 6, we will
bound the influence of the thinning variables in terms of the initial
condition in order to show sharp thresholds with respect to $\rho$.

### The algorithm.

We write $\big{[}y\overset{\tau,\,k}{\rightarrow}x\big{]}$ for the event where
there exists in $\mu^{\prime}$ a sequence of time-marks smaller than $\tau$ in
increasing order associated to points in a path from $y$ to $x$. In other
words, the event is similar to $\big{[}y\overset{\tau}{\rightarrow}x\big{]}$,
but defined for the point measure with thickened clock ticks.

For every $x\in\mathbb{Z}^{2}$ and fixed $\mu^{\prime}$, we can define
$\mathcal{B}(x)$, the random “support region of $x$”, as

$\mathcal{B}(x):=\big{\\{}y;y\overset{\tau,\,k}{\rightarrow}x\big{\\}}.$
(5.11)

Recall that our main goal is to prove a sharp threshold for the occurrence of
the event $\mathcal{C}(3n,n)$. With that in mind, we fix the function $f$ to
be determined as ${\bf 1}_{\mathcal{C}(3n,n)}$ and will now describe the
algorithm. We first note that this $f$ depends only on
$(\sigma_{\rho,T}(x))_{x\in[0,3n]\times[0,n]}$. We then define the _good event
for $x$_:

$\mathcal{G}_{x}=\big{[}\mu^{\prime};\mathcal{B}(x)\subseteq B(x,\log
n)\big{]},$ (5.12)

and the _good box event_

$\mathcal{G}(3n,n)=\bigcap_{x\in[0,3n]\times[0,n]}\mathcal{G}_{x},$ (5.13)

We recall that our algorithm will start already knowing $\mu^{\prime}$. Noting
that the good box event $\mathcal{G}(3n,n)$ depends only on $\mu^{\prime}$, we
start the algorithm _already knowing if this event happens or not_. Our first
step is asking if we are indeed in the good event.

  * (i)

If $\mathcal{G}(3n,n)$ does not occur, we reveal the state of every relevant
random variable: the initial states $U^{y}$ and the thinning variables $D_{i}$
associated to points in $\cup_{x\in[0,3n]\times[0,n]}\mathcal{B}(x)$.

We will need a sub-algorithm $\mathcal{A}_{x}$ to query the value of
$\sigma_{\rho,\tau}(x)$. This algorithm, which will only be called if we
already know that $\mathcal{G}_{x}$ occurred, performs the actions:

  * $x$.i

Reveal every $U^{y}$ for $y\in\mathcal{B}(x)$;

  * $x$.ii

Reveal every $D_{i}$ for which $X_{i}\in\mathcal{B}(x)$ and
$T_{i}\in[0,\tau]$.

Note that the above is enough to determine the value of
$\sigma_{\rho,\tau}(x)$.

We can now define the algorithm $\mathcal{A}$ in a similar way to the
algorithm defined in Section 4 to explore crossings in Bernoulli percolation:

  * (ii)

Choose an integer $Z$ uniformly among $\\{0,\dots,3n\\}$;

  * (iii)

Explore, in a previously defined order, the connected components of vertices
with state $+1$ in $\sigma_{\rho,\tau}$ intersecting the line
$\\{Z\\}\times[0,n]$. Whenever the state of a vertex $x$ needs to be queried,
we use $\mathcal{A}_{x}$.

Since every crossing of $[0,3n]\times[0,n]$ by pluses must cross the randomly
selected line, the algorithm $\mathcal{A}$ will determine the occurrence of
$\mathcal{C}(3n,n)$.

### Estimating the revealment.

Consider the event where the state of a given variable associated to a site
$x$ is revealed by the algorithm. This implies that $x$ belongs to the support
of a point $y$ (not necessarily different from $x$) which is connected to the
random line selected by the algorithm by a path of vertices with plus states.
In particular, when $x$ is far from this line, we get the occurrence of an
$\operatorname{Arm}$ event. With that in mind, we state the following lemma,
which will help us estimate the revealment of the relevant variables.

######  Lemma 5.3.

There exists an exponent $\nu>0$ such that, for all $\gamma>0$, there exists
$k_{\ref{k:one_arm}}=k_{\ref{k:one_arm}}(\gamma)\geq 2$ such that, for any
$k\geq k_{\ref{k:one_arm}}$ and for any $\rho\leq\rho_{+}$,

$\operatorname{P}\Big{[}\operatorname{P}_{\rho}\big{[}\operatorname{Arm}(n^{1/4},n^{1/2})\big{|}\mu^{\prime}\big{]}\geq
n^{-\nu}\Big{]}\leq n^{-\gamma},$ (5.14)

for every $n\geq n_{0}(k)$.

######  Remark 8.

We note that the outer probability in the RHS of (5.14) gives only
$\mu^{\prime}$, and conditioned on $\mu^{\prime}$ the probability space is an
almost surely finite hypercube. Intuitively speaking, this lemma says that the
Poisson processes $\mu^{\prime}$ for which the quenched arm event does not
decay as a small power of $n$ are very rare.

###### Proof Idea.

The proof of this result follows mutatus mutandi from the proof of Proposition
$3.4$ of [5]: the Poisson point processes are slightly changed, but the
argument is the same. The main idea is to decompose the annulus with inner
radius $n^{1/4}$ and outer radius $n^{1/2}$ into $\sim\log n$ annuli with
fixed aspect ratio. Crossing each of these annuli when $\rho\leq\rho_{+}$
costs a fixed amount, so that multiplying all the costs yield a polynomial
probability in $n$. Dependence prevents one from straight up multiplying these
probabilities, but taking these annuli sufficiently spaced and using a
decoupling argument finishes the proof. ∎

We can now finally estimate the revealment of the relevant random variables.

######  Lemma 5.4.

There exists an exponent $\nu>0$ and some $k_{\ref{k:revealment}}\geq 2$ such
that for any $k\geq k_{\ref{k:revealment}}$ and for any $\rho\leq\rho_{+}$, we
have

$\operatorname{P}\big{[}\sup_{x\in\mathbb{Z}^{2}}\delta_{\rho}(\mathcal{A},x)\geq
n^{-\nu}\big{]}\leq n^{-50},$ (5.15)
$\operatorname{P}\big{[}\sup_{i\in\mathbb{N}_{0}}\delta_{\rho}(\mathcal{A},i)\geq
n^{-\nu}\big{]}\leq n^{-50},$ (5.16)

for every $n\geq n_{0}(k)$.

###### Proof.

Both thinning and initial variables are only queried if their associated point
in $\mathbb{Z}^{2}$ belongs to the support $\mathcal{B}(x)$ of a point
connected to the line selected by the algorithm. We will prove (5.15), the
other bound following analogously.

We will consider two _bad events_ , the first of which being the event where
_some_ vertex in $[0,3n]\times[0,n]$ has large support

$\mathrm{BigSupp}_{n}:=\Bigg{\\{}\begin{array}[]{c}\text{there exists
}x\in[0,3n]\times[0,n]\text{ such that}\\\ \mathcal{B}(x)\cap
B(x,\log(n))\neq\emptyset\end{array}\Bigg{\\}}.$ (5.17)

The second of the bad events happens when, conditioned on $\mu^{\prime}$,
there exists a point of $[0,3n]\times[0,n]$ for which an associated arm event
has large probability. Let $\nu^{\prime}>0$ be such that Lemma 5.3 is valid
with $\gamma=100$, then define

$\mathrm{BadArm}_{n}:=\Bigg{\\{}\begin{array}[]{c}\text{there exists
}x\in[0,3n]\times[0,n]\text{ such that}\\\
\operatorname{P}_{\rho}\big{[}\operatorname{Arm}_{x}(n^{1/4},n^{1/2})\big{|}\mu^{\prime}\big{]}\geq
n^{-\nu^{\prime}}\end{array}\Bigg{\\}}.$ (5.18)

We assume $k\geq k_{0}(100)$ given by Lemma 5.3, which together with the union
bound implies

$\operatorname{P}\big{[}\mathrm{BadArm}_{n}\big{]}\leq 3n^{2}n^{-100}\leq
3n^{-98}.$ (5.19)

Lemma 2.1 then implies, again together with an union bound,

$\operatorname{P}\big{[}\mathrm{BigSupp}_{n}\big{]}\leq
c_{\ref{c:light}}(k\tau)3n^{2}\exp\bigg{\\{}-\frac{1}{4}\log(n)\log(\log(n))\bigg{\\}}.$
(5.20)

We now bound the revealment on the event
$\mathrm{BigSupp}_{n}^{c}\cap\mathrm{BadArm}_{n}^{c}$. Either the vertex $x$
is at a distance smaller than $2\sqrt{n}$ of the line randomly selected by the
algorithm, or some vertex in the support of $x$ connects to this randomly
selected line via a $+$-path. Both are very unlikely on
$\mathrm{BigSupp}_{n}^{c}\cap\mathrm{BadArm}_{n}^{c}$, the first case is
bounded from above by $2\sqrt{n}/3n$, the second by the probability of
$\mathrm{Arm}_{x}(n^{1/4},n^{1/2})$. We have

$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\delta(\mathcal{A},x)\bf{1}_{\mathrm{BigSupp}_{n}^{c}\cap\mathrm{BadArm}_{n}^{c}}$\mbox{}\hfil\quad\\\
&\leq\Bigg{(}\sup_{x\in[0,3n]\times
n}P_{\rho}\big{[}\operatorname{Arm}_{x}(n^{1/4},n^{1/2})\big{|}\mu^{\prime}\big{]}\Bigg{)}{\bf
1}_{\mathrm{BigSupp}_{n}^{c}\cap\mathrm{BadArm}_{n}^{c}}+\frac{1}{\sqrt{n}}\\\
&\leq n^{-\nu^{\prime}}+n^{-1/2}\\\ &\leq n^{-\nu},\end{split}$ (5.21)

after choosing $\nu$ sufficiently small. We have then proved that

$\begin{split}\operatorname{P}\big{[}\sup_{x\in\mathbb{Z}^{2}}\delta_{\rho}(\mathcal{A},x)\geq
n^{-\nu}\big{]}&\leq\operatorname{P}\big{[}\mathrm{BigSupp}_{n}\cap\mathrm{BadArm}_{n}\big{]}\\\
&\leq
c_{\ref{c:light}}(k\tau)3n^{2}\exp\bigg{\\{}-\frac{1}{4}\log(n)\log(\log(n))\bigg{\\}}+3n^{-98}\\\
&\leq n^{-50},\end{split}$ (5.22)

for sufficiently large $n$, after choosing $\nu$ appropriately. This finishes
the proof of the result. ∎

##  6 Aizenman-Grimmett

In this section we establish the relation between the two concepts of
influence introduced in Section 5, namely, the influence of an initial opinion
and the influence of a clock tick. This will rely on an Aizenman-Grimmett type
of argument, where the pivotality of a clock tick is transferred to pivotality
of initial conditions through local changes in the quenched configuration.

Since each tick has a random region of influence, it is not necessarily true
that we can relate its influence to the initial condition of its corresponding
site. In fact, we prove that if there exists a pivotal clock tick, then it is
possible to modify the configuration in a neighborhood of the tick and obtain
a pivotal initial opinion on a nearby site.

Before stating our result, we need some additional notation. We define
inductively $\log^{(1)}(j)=\log(j)$ and, for $k\in\mathbb{N}$,
$\log^{(k+1)}(j)=\log\big{(}\log^{(k)}(j)\big{)}$. Given $j\geq 1$, let

$h_{\alpha}(j)=\left\lceil\alpha\frac{\log(j)}{\log^{(2)}(j)}\right\rceil,$
(6.1)

where $\alpha\in\mathbb{R}_{+}$ is a value that will be chosen large enough
afterwards. Recall (5.11), the definition of the support $\mathcal{B}(x)$ of a
vertex $x$, and consider the event

$\tilde{\mathcal{G}}_{x}^{k}(\alpha,\zeta)=\bigg{\\{}\mu^{\prime};\begin{array}[]{c}{\mathcal{B}(x)\subset
B(x,h_{\alpha}(n))\text{ and }}\\\ {\mu^{\prime}\big{(}\partial
B(x,h_{\alpha}(n))\times[0,\tau]\times\\{0,1\\}\big{)}\leq\zeta\log
n}\end{array}\bigg{\\}},$ (6.2)

as well as the good event

$\Omega(k,n)=\bigcap_{x\in[-n,4n]\times[-n,2n]}\tilde{\mathcal{G}}^{k}_{x}(\alpha,\zeta).$
(6.3)

Even though the event above depends on the values of $\alpha$ and $\zeta$, we
suppress them in order to make notation cleaner.

The event $\Omega(k,n)$ is composed of realizations $\mu^{\prime}$ such that
there are no large supports on $[-n,4n]\times[-n,2n]$ and that the number of
clock ticks before time $\tau$ in the shells $\partial B(x,h_{\alpha}(n))$ is
bounded. Furthermore, since $\mathcal{B}(x)\subset B(x,h_{\alpha}(n))$, for
all $x\in[-n,4n]\times[-n,2n]$, we obtain that the opinion at time $\tau$ of
any $x\in[1,3n]\times[1,n]$ is determined by the initial opinions and clock
selections inside $B(x,h_{\alpha}(n))$.

Let us now bound the probability of ${\Omega(k,n)}^{c}$. We begin by bounding
the probability that a given vertex has a large support. From Lemma 2.1, we
immediately get

$\begin{split}\operatorname{P}\Big{(}\mathcal{B}(x)\cap
B(x,h_{\alpha}(n))^{c}\neq\emptyset\Big{)}&\leq\sum_{y\notin
B(x,h_{\alpha}(n))}\operatorname{P}\big{(}y\overset{\tau,k}{\to}x\big{)}\leq\sum_{\ell\geq
h_{\alpha}(n)}8c_{\ref{c:light}}(k\tau)\ell e^{-\frac{1}{4}\ell\log\ell}\\\
&\leq e^{-ch_{\alpha}(n)\log h_{\alpha}(n)}\leq
e^{-c_{\ref{c:bound_omega}}\alpha\log n},\end{split}$ (6.4)

for some $c_{\ref{c:bound_omega}}=c_{\ref{c:bound_omega}}(k,\tau)>0$.

We now proceed to bound the probability that a shell $\partial
B(x,h_{\alpha}(n))$ has many clock ticks. Notice that the random variable
$\mu^{\prime}\big{(}\partial
B(x,h_{\alpha}(n))\times[0,\tau]\times\\{0,1\\}\big{)}$ has distribution
$\operatorname{Poisson}\big{(}8k\tau h_{\alpha}(n)\big{)}$. From this, we can
bound, for any $\lambda>0$,

$\begin{split}\operatorname{P}\Big{(}\mu^{\prime}\big{(}\partial&B(x,h_{\alpha}(n))\times[0,\tau]\times\\{0,1\\}\big{)}>\zeta\log
n\Big{)}\\\ &\leq\exp\bigg{\\{}8k\tau
h_{\alpha}(n)(e^{\lambda}-1)-\lambda\zeta\log n\bigg{\\}}.\end{split}$ (6.5)

We now take $\lambda=\log\Big{(}\frac{\zeta\log n}{8k\tau
h_{\alpha}(n)}\Big{)}>0$ to obtain

$\begin{split}\operatorname{P}\Big{(}\mu^{\prime}&\big{(}\partial
B(x,h_{\alpha}(n))\times[0,\tau]\times\\{0,1\\}\big{)}>\zeta\log n\Big{)}\\\
&\leq\exp\bigg{\\{}\zeta\log n\Big{(}-\log\Big{(}\frac{\zeta\log n}{8k\tau
h_{\alpha}(n)}\Big{)}+1\Big{)}\bigg{\\}}\\\ &\leq\exp\bigg{\\{}\zeta\log
n\Big{(}-\log^{(3)}n-\log\frac{\zeta}{16k\tau\alpha}+1\Big{)}\bigg{\\}}\leq
e^{-\frac{\zeta}{2}\log n\log^{(3)}n},\end{split}$ (6.6)

for all $n$ large enough, depending on $\zeta$, $k$, $\tau$ and $\alpha$.

Combining Equations (6.4) and (6.6) with union bound yields that, for any
$k\geq 2$, there exists $\alpha$ large enough such that the following holds:
for any $\zeta>0$, there exists $n_{0}\in\mathbb{N}$ such that

$\operatorname{P}\big{(}\Omega(k,n)^{c}\big{)}\leq
30n^{2}\big{(}e^{-c_{\ref{c:bound_omega}}\alpha\log n}+e^{-\frac{\zeta}{2}\log
n\log^{(3)}n}\big{)}\leq n^{-\frac{1}{2}},$ (6.7)

for all $n\geq n_{0}$.

The next lemma is the central piece that relates thinning influences to
influences of initial opinions. In it, we prove that, provided we are in the
good event $\Omega(k,n)$, the influence of a clock tick $i$ that happens in
$X_{i}\in[-n,4n]\times[-n,2n]$ can be related to the influence of the initial
opinion of vertices in $\partial B(X_{i},h_{\alpha}(n))$ with a polynomially
small correction.

######  Lemma 6.1.

Given $k\geq 2$, there exists a constant $c_{\ref{c:AG}}=c_{\ref{c:AG}}(k)>0$
such that the following holds for any $\zeta>0$. Given
$\mu^{\prime}\in\Omega(k,n)$ and some $\rho\in[\rho_{-},\rho_{+}]$, for every
$i$ such that $T_{i}\leq\tau$ and
$X_{i}\in[-n,4n]\times[-n,2n]\cap\mathbb{Z}^{2}$, we have

$\operatorname{Inf}^{\textnormal{thin}}_{\rho}\big{(}i,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\leq
c_{\ref{c:AG}}^{\zeta\log n}\sum_{x\in\partial
B(X_{i},h_{\alpha}(n))}\operatorname{Inf}^{\textnormal{init}}_{\rho}\big{(}x,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}.$
(6.8)

We first assume the statement of the lemma above and conclude the proof of
sharpness. The idea is to prove that the derivative of
$\operatorname{P}_{\rho}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{)}$ has
polynomial growth as a function of $n$ when restricted to
$[\rho_{-},\rho_{+}]$. For this, we use the OSSS inequality together with the
influence relation given by Lemma 6.1 and Russo’s formula to lower bound such
derivative by a polynomial multiple of the variance. Proving that the variance
is uniformly bounded from below will then conclude the proof. Theorem 1.2
follows then from Lemma 2.3.

###### Proof of Theorem 1.2.

Recall first that, for all $\rho\in[\rho_{-},\rho_{+}]$, we have

$\operatorname{P}_{\rho}\big{(}\mathcal{C}(3n,n)\big{)}>c_{\ref{c:rsw}}\quad\text{and}\quad\operatorname{P}_{\rho}\big{(}\mathcal{C}^{*}(3n,n)\big{)}>c_{\ref{c:rsw}},$
(6.9)

for all $n\geq 1$.

Choose now $k$ large enough such that Lemma 5.4 applies and

$\frac{1}{kc_{\ref{c:rsw}}^{2}}\leq\frac{1}{4}.$ (6.10)

In particular, if we define

$A=\bigg{\\{}\operatorname{P}_{\rho}\big{(}\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\notin\Big{(}\frac{c_{\ref{c:rsw}}}{2},1-\frac{c_{\ref{c:rsw}}}{2}\Big{)}\bigg{\\}},$
(6.11)

then (4.11) together with the choice of $k$ implies

$\operatorname{P}\big{(}A\big{)}\leq\frac{1}{kc_{\ref{c:rsw}}^{2}}\leq\frac{1}{4}.$
(6.12)

Second, let $\nu>0$ as in Lemma 5.4 and $c_{\ref{c:AG}}$ as in Lemma 6.1.
Choose $\zeta>0$ such that

$c_{\ref{c:AG}}\zeta\leq\frac{\nu}{2}.$ (6.13)

Define the event

$B=\bigg{\\{}\sup_{x}\big{\\{}\delta_{\rho}(\mathcal{A},x)\big{\\}}\geq
n^{-\nu}\text{ or }\sup_{i}\big{\\{}\delta_{\rho}(\mathcal{A},i)\big{\\}}\geq
n^{-\nu}\bigg{\\}},$ (6.14)

and notice that Lemma 5.4 implies

$\operatorname{P}\big{(}B\big{)}\leq 2n^{-50},$ (6.15)

assuming that $n$ is large enough.

Assume now that we take $\mu^{\prime}\in\big{(}A\cup
B\big{)}^{c}\cap\Omega(k,n)$. From the fact that $\mu^{\prime}\notin A$, we
obtain the lower bound

$\operatorname{Var}_{\rho}(\mathcal{C}(3n,n)|\mu^{\prime})=\operatorname{P}_{\rho}\big{(}\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\big{[}1-\operatorname{P}_{\rho}\big{(}\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\big{]}\geq\frac{c_{\ref{c:rsw}}^{2}}{4}.$
(6.16)

Now, since $\mu^{\prime}\notin B$, we can apply the OSSS inequality to
estimate

$\begin{split}\operatorname{Var}_{\rho}(&\mathcal{C}(3n,n)|\mu^{\prime})\\\
&\leq\bigg{(}\sum_{x\in\mathbb{Z}^{2}}\operatorname{Inf}^{\textnormal{init}}_{\rho}(x,\mathcal{C}(3n,n)|\mu^{\prime})\delta_{\rho}(\mathcal{A},x)+\sum_{i\geq
0}\operatorname{Inf}^{\textnormal{thin}}_{\rho}(i,\mathcal{C}(3n,n)|\mu^{\prime})\delta_{\rho}(\mathcal{A},i)\bigg{)}\\\
&\leq
n^{-\nu}\bigg{(}\sum_{x\in\mathbb{Z}^{2}}\operatorname{Inf}^{\textnormal{init}}_{\rho}(x,\mathcal{C}(3n,n)|\mu^{\prime})+\sum_{i\geq
0}\operatorname{Inf}^{\textnormal{thin}}_{\rho}(i,\mathcal{C}(3n,n)|\mu^{\prime})\bigg{)}.\end{split}$
(6.17)

We will now use Lemma 6.1 together with the fact that
$\mu^{\prime}\in\Omega(k,n)$ to bound the last sum in the equation above.
Notice first that
$\operatorname{Inf}^{\textnormal{thin}}_{\rho}(i,\mathcal{C}(3n,n)|\mu^{\prime})=0$,
for all $i$ such that $X_{i}\notin[-n,4n]\times[-n,2n]$ or $T_{i}\geq\tau$. We
abuse notation in the following, by omitting this restriction on the values of
$i$. Lemma 6.1 together with the choice of $\zeta$ readily imply

$\begin{split}\sum_{i\geq
0}\operatorname{Inf}^{\textnormal{thin}}_{\rho}(i|\mu^{\prime})&\leq
c_{\ref{c:AG}}^{\zeta\log n}\sum_{i}\sum_{x\in\partial
B(X_{i},h_{\alpha}(n))}\operatorname{Inf}^{\textnormal{init}}_{\rho}\big{(}x,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\\\
&\leq n^{\frac{\nu}{2}}\sum_{i}\sum_{x\in\partial
B(X_{i},h_{\alpha}(n))}\operatorname{Inf}^{\textnormal{init}}_{\rho}\big{(}x,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}.\end{split}$
(6.18)

Once again, since $\mu^{\prime}\in\Omega(k,n)$, a counting argument implies
that, for each $x\in[-n,4n]\times[-n,2n]$, the number of times it appears in
the last summation above is bounded by $8\zeta h_{\alpha}(n)\log n$, which
gives the bound

$\sum_{i\geq
0}\operatorname{Inf}^{\textnormal{thin}}_{\rho}(i|\mu^{\prime})\leq 8\zeta
h_{\alpha}(n)\big{(}\log
n\big{)}n^{\frac{\nu}{2}}\sum_{x\in\mathbb{Z}^{2}}\operatorname{Inf}^{\textnormal{init}}_{\rho}(x,\mathcal{C}(3n,n)).$
(6.19)

Combining Equations (6.16), (6.17), and (6.19) with Russo’s formula (Lemma
5.1) implies that, for each $\mu^{\prime}\in\big{(}A\cup
B\big{)}^{c}\cap\Omega(k,n)$,

$\begin{split}\frac{c_{\ref{c:rsw}}^{2}}{4}&\leq n^{-\nu}\big{(}8\zeta
h_{\alpha}(n)\big{(}\log
n\big{)}n^{\frac{\nu}{2}}+1\big{)}\sum_{x\in\mathbb{Z}^{2}}\operatorname{Inf}^{\textnormal{init}}_{\rho}\big{(}x,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\\\
&=n^{-\nu}\big{(}8\zeta h_{\alpha}(n)\big{(}\log
n\big{)}n^{\frac{\nu}{2}}+1\big{)}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\rho}\operatorname{P}_{\rho}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\\\
&\leq
n^{-\frac{\nu}{4}}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\rho}\operatorname{P}_{\rho}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)},\end{split}$
(6.20)

if $n$ is chosen large enough, since $h_{\alpha}$ has sub-logarithmic growth
(see Equation (6.1)). From this,

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\rho}\operatorname{P}_{\rho}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\geq\frac{c_{\ref{c:rsw}}^{2}}{4}n^{\frac{\nu}{4}},$
(6.21)

for all $\mu^{\prime}\in\big{(}A\cup B\big{)}^{c}\cap\Omega(k,n)$.

We can now estimate

$\begin{split}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\rho}\operatorname{P}_{\rho}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{)}&=\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\rho}\operatorname{E}\Big{[}\operatorname{P}_{\rho}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\Big{]}\\\
&=\operatorname{E}\bigg{[}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\rho}\operatorname{P}_{\rho}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\bigg{]}\\\
&\geq\operatorname{E}\bigg{[}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\rho}\operatorname{P}_{\rho}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}{\bf
1}_{(A\cup B)^{c}\cap\Omega(k,n)}(\mu^{\prime})\bigg{]}\\\
&\geq\frac{c_{\ref{c:rsw}}^{2}}{4}n^{\frac{\nu}{4}}\operatorname{P}\Big{(}\big{(}A\cup
B\big{)}^{c}\cap\Omega(k,n)\Big{)}.\end{split}$ (6.22)

We now combine (6.7), (6.12), and (6.15) to conclude that

$\begin{split}\operatorname{P}\Big{(}A\cup
B\cup\Omega(k,n)^{c}\Big{)}&\leq\operatorname{P}\big{(}A\big{)}+\operatorname{P}\big{(}B\big{)}+\operatorname{P}\big{(}\Omega(k,n)^{c}\big{)}\\\
&\leq\frac{1}{4}+2n^{-50}+n^{-\frac{1}{2}}\leq\frac{1}{2}.\end{split}$ (6.23)

This implies that

$\begin{split}\frac{c_{\ref{c:rsw}}^{2}}{8}n^{\frac{\nu}{4}}|\rho_{+}-\rho_{-}|&\leq\int_{\rho_{-}}^{\rho_{+}}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\rho}\operatorname{P}_{\rho}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{)}\mathop{}\\!\mathrm{d}\rho\\\
&=\operatorname{P}_{\rho_{+}}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{)}-\operatorname{P}_{\rho_{-}}\big{(}\sigma_{T}\in\mathcal{C}(3n,n)\big{)}\leq
1\end{split}$ (6.24)

which yields $|\rho_{+}-\rho_{-}|\leq cn^{-\frac{\nu}{4}}$, for all $n$ large
enough. This implies $\rho_{+}=\rho_{-}$ and concludes the proof. ∎

We now proceed to the proof of Lemma 6.1. Here, assuming that a clock tick $i$
is pivotal for a configuration $\mu^{\prime}\in\Omega(k,n)$, we change the
initial configuration and clock-tick selections on the shell $\partial
B(X_{i},h_{\alpha}(n))$ in order to obtain a vertex with pivotal initial
condition. Due to the fact that $\mu^{\prime}\in\Omega(k,n)$, we can bound the
Radon-Nikodym derivative of this change of configurations and obtain estimates
that lead to (6.8).

###### Proof of Lemma 6.1.

Fix $\mu^{\prime}\in\Omega(k,n)$. Recall that we write
$\mu^{\prime}=\sum_{j\geq 0}\delta_{(X_{j},T_{j},R_{j})}$, and denote by

$\mathcal{S}=[-n,4n]\times[-n,2n]\,\text{ and
}\,\mathcal{T}=\Big{\\{}j:(X_{j},T_{j})\in\mathcal{S}\times[0,\tau]\Big{\\}}$
(6.25)

the collections of sites and clock ticks whose selection can influence the
configuration $\sigma_{T}$ restricted to the rectangle $[1,3n]\times[1,n]$.

Assume that the clock tick $i$ is pivotal, meaning that its selection
determines the existence of crossings at time $\tau$. Without loss of
generality, we will consider only the case when the acceptance of this clock
tick yields a crossing of the rectangle $[1,3n]\times[1,n]$. Since
$\mu^{\prime}\in\Omega(k,n)$, we have $X_{i}\in[-n,4n]\times[-n,2n]$.
Furthermore, the only vertices whose opinions can be changed through the
selection of the tick $i$ are the ones in $B(X_{i},h_{\alpha}(n))$, since the
support of any $y\notin B(X_{i},h_{\alpha}(n))$ cannot contain $X_{i}$.

We will now define maps
$\varphi_{-}^{i},\varphi_{+}^{i}:\\{-1,1\\}^{\mathcal{S}\cup\mathcal{T}}\to\\{-1,1\\}^{\mathcal{S}\cup\mathcal{T}}$
that locally modify the initial configuration and clock selections on the
boundary $\partial B(X_{i},h_{\alpha}(n))$ (we assume here, in order to ease
the notation, that -1 in a clock tick means that it is not accepted, while a
+1 signals its acceptance). The map $\varphi_{-}^{i}$ changes the initial
opinion $\sigma_{0}|_{\partial B(X_{i},h_{\alpha}(n))}$ to the constant $-1$
and forbids all clock-tick selections that happen before time $\tau$ in any
vertex of $\partial B(X_{i},h_{\alpha}(n))$. Analogously, $\varphi_{+}^{i}$
changes the initial configuration on $\partial B(X_{i},h_{\alpha}(n))$ to the
all one and disregards the same clock ticks as in $\varphi_{-}^{i}$.

Given a selection configuration
$\omega\in\\{-1,1\\}^{\mathcal{S}\cup\mathcal{T}}$ for which $i$ is pivotal,
we now argue that $\sigma_{\tau}(\varphi_{-}^{i}(\omega))$ does not contain a
horizontal crossing of the rectangle $[1,3n]\times[1,n]$. In order to do so,
it suffices to verify that
$\sigma_{\tau}(\varphi_{-}^{i}(\omega))(y)\leq\sigma_{\tau}(\omega^{i,-})(y)$,
for all $y\notin B(X_{i},h_{\alpha}(n))$, where $\omega^{i,-}$ is obtained
from $\omega$ by changing the selection of the clock tick $i$ to -1. This will
readily imply the absence of crossings, since
$\sigma_{\tau}(\varphi_{-}^{i}(\omega))(y)=-1$, for all $y\in\partial
B(X_{i},h_{\alpha}(n))$, and $i$ is pivotal for $\omega$.

We proceed in two steps to go from $\omega^{i,-}$ to
$\varphi_{-}^{i}(\omega)$. First, change the initial configuration for
$\omega^{i,-}$ in $\partial B(X_{i},h_{\alpha}(n))$ to the constant equal to
-1 and obtain a intermediate configuration $\omega^{-}$ which satisfies
$\sigma_{\tau}(\omega^{-})\prec\sigma_{\tau}(\omega^{i,-})$ due to
monotonicity with respect to the initial condition. Second, from $\omega^{-}$
we cancel all the clock ticks that happen before time $\tau$ in $\partial
B(X_{i},h_{\alpha}(n))$. This can only decrease the configuration at time
$\tau$, since we remove the chance that an opinion on $\partial
B(X_{i},h_{\alpha}(n))$ would eventually turn to 1. We thus obtain the
domination
$\sigma_{\tau}(\varphi_{-}^{i}(\omega^{i,-}))\prec\sigma_{\tau}(\omega^{i,-})$.
The proof of the claim is then completed by noticing that
$\sigma_{\tau}(\varphi_{-}^{i}(\omega^{i,-}))$ and
$\sigma_{\tau}(\varphi_{-}^{i}(\omega))$ coincide outside the ball
$B(X_{i},h_{\alpha}(n))$. A similar argument holds for
$\sigma_{\tau}(\omega^{i,+})\prec\sigma_{\tau}(\varphi_{+}^{i}(\omega))$,
where $\omega^{i,+}$ is obtained from $\omega$ by changing the selection of
the clock tick $i$ to 1.

From the discussion above, we obtain that, if the clock tick $i$ is pivotal
for a configuration $\omega$, then there are two configurations that differ
only in the initial condition, namely, $\varphi_{-}^{i}(\omega)$ and
$\varphi_{+}^{i}(\omega)$, such that the initial opinions on the set $\partial
B(X_{i},h_{\alpha}(n))$ are jointly pivotal. This means that

$\begin{array}[]{c}\parbox[c]{433.62pt}{\centering if $\sigma_{0}|_{\partial
B(X_{i},h_{\alpha}(n))}\equiv-1$, then there is no crossing, while a crossing
is present at time $\tau$ if $\sigma_{0}|_{\partial
B(X_{i},h_{\alpha}(n))}\equiv 1$.\@add@centering}\end{array}$ (6.26)

Let us now bound the Radon-Nikodym derivative of this change of measure.
Observe first that, due to the fact that $\mu^{\prime}\in\Omega(k,n)$, in
order to go from a configuration
$\omega\in\\{-1,1\\}^{\mathcal{S}\cup\mathcal{T}}$ to
$\varphi_{-}^{i}(\omega)$, one needs to change at most $8h_{\alpha}(n)$
initial positions and at most $\zeta\log n$ clock-tick selections. In
particular, this implies that, if $\varphi_{-}^{i}(\omega)=\tilde{\omega}$,

$\operatorname{P}_{\rho}\big{(}\omega|\mu^{\prime}\big{)}\leq k^{\zeta\log
n}\left(\frac{1}{1-\rho}\right)^{8h_{\alpha}(n)}\operatorname{P}_{\rho}\big{(}\tilde{\omega}\big{|}\mu^{\prime}\big{)}\leq\left(\frac{k}{1-\rho}\right)^{\zeta\log
n}\operatorname{P}_{\rho}\big{(}\tilde{\omega}\big{|}\mu^{\prime}\big{)}.$
(6.27)

Furthermore, for each
$\tilde{\omega}\in\\{-1,1\\}^{\mathcal{S}\cup\mathcal{T}}$, there are at most
$2^{8h_{\alpha}(n)}2^{\zeta\log n}\leq 4^{\zeta\log n}$ configurations
$\omega\in\\{-1,1\\}^{\mathcal{S}\cup\mathcal{T}}$ such that
$\varphi_{-}^{i}(\omega)=\tilde{\omega}$. From this, we can estimate

$\begin{split}\operatorname{Inf}^{\textnormal{thin}}_{\rho}\big{(}&i,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}=\sum_{\omega\in\\{-1,1\\}^{\mathcal{S}\cup\mathcal{T}}}\operatorname{P}\big{(}\omega\big{|}\mu^{\prime}\big{)}{\bf
1}_{i\text{ is pivotal}}(\omega)\\\
&\leq\sum_{\tilde{\omega}}\sum_{\omega\in(\varphi_{-}^{i})^{-1}(\tilde{\omega})}\operatorname{P}\big{(}\omega\big{|}\mu^{\prime}\big{)}{\bf
1}_{\partial B(X_{i},h_{\alpha}(n))\text{ is pivotal}}(\tilde{\omega})\\\
&\leq 4^{\zeta\log n}\left(\frac{k}{1-\rho}\right)^{\zeta\log
n}\sum_{\tilde{\omega}\in\\{-1,1\\}^{\mathcal{S}\cup\mathcal{T}}}\operatorname{P}_{\rho}\big{(}\tilde{\omega}\big{|}\mu^{\prime}\big{)}{\bf
1}_{\partial B(X_{i},h_{\alpha}(n))\text{ is pivotal}}(\tilde{\omega})\\\
&\leq\left(\frac{4k}{1-\rho}\right)^{\zeta\log
n}\operatorname{P}_{\rho}\big{(}\partial B(X_{i},h_{\alpha}(n))\text{ is
pivotal}\big{|}\mu^{\prime}\big{)}.\end{split}$ (6.28)

Finally, let us consider the event on the RHS of the equation above. Assume
that $\partial B(X_{i},h_{\alpha}(n))$ is pivotal for a configuration
$\omega\in\\{-1,1\\}^{\mathcal{S}\cup\mathcal{T}}$ and that
$\sigma_{\tau}(\omega)$ does not have a crossing. If this happens, we can
progressively change the initial opinions of $\omega$ in $\partial
B(X_{i},h_{\alpha}(n))$ from -1 to 1 and eventually arrive at a configuration
with one such vertex being pivotal. This implies

$\begin{split}\operatorname{P}_{\rho}\big{(}B(X_{i},h_{\alpha}(n))&\text{ is
closed pivotal}\big{|}\mu^{\prime}\big{)}\\\
&\leq\left(\frac{1}{\rho}\right)^{8h_{\alpha}(n)}\sum_{x\in\partial
B(X_{i},h_{\alpha}(n))}\operatorname{Inf}^{\textnormal{init}}_{\rho}\big{(}x,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}.\end{split}$
(6.29)

An analogous bound holds for the case when $\sigma_{\tau}(\omega)$ has a
crossing, but we change the vertices whose opinions are 1. In this case, the
only change is in the basis of the exponential price, which changes to
$\frac{1}{1-\rho}$. This concludes the proof, since
$\rho\in(\rho_{-},\rho_{+})$ is bounded away from zero and one. ∎

##  7 Elliptic bootstrap percolation

In this section, we consider elliptic bootstrap percolation and point out how
our proof can be adapted for this case. Let us first start by precisely
defining the model we consider.

Definition of the model. We will denote the configuration at time $t\geq 0$ by
$\sigma_{t}\in\\{-1,1\\}^{\mathbb{Z}^{2}}$. In order to match with the rest of
the paper, we set the unusual choice of opinions to be -1 and 1.

Consider the rate function

$\lambda:\\{-1,1\\}^{B(0,1)}\to\mathbb{R}_{+},$ (7.1)

where $B(0,1)=\\{x\in\mathbb{Z}^{2}:||x||_{1}\leq 1\\}$. This function will
control the update rate of any site in the process. We assume that $\lambda$
is a rotationally-invariant monotone non-decreasing function and that
$\lambda(-\bar{1})=\epsilon>0$, where $-\bar{1}$ denotes the constant
configuration equal to $-1$.

We set the evolution of the process in the following way. At time zero, define
$\sigma_{0}(x)=-1$, for all $x\in\mathbb{Z}^{2}$. A site $x\in\mathbb{Z}^{2}$
turns from -1 to 1 with rate $\lambda(\tau_{x}(\sigma_{t}))$, where
$\tau_{x}:\mathbb{Z}^{2}\to\mathbb{Z}^{2}$ is the translation
$\tau_{x}(y)=y-x$. A site with opinion 1 never changes back to -1.

The evolution can be described via its infinitesimal generator: for any local
function $f:\\{-1,1\\}^{\mathbb{Z}^{2}}\to\mathbb{R}$, we define

$\mathcal{L}f(\sigma)=\sum_{x\in\mathbb{Z}^{2}}\lambda(\tau_{x}(\sigma))\big{(}f(\sigma^{x})-f(\sigma)\big{)},$
(7.2)

where

$\sigma^{x}(y)=\begin{cases}1,&\text{if }y=x,\\\
\sigma(y),&\text{otherwise.}\end{cases}$ (7.3)

By performing a time change, we can assume without loss of generality that
$\lambda(\bar{1})=1$. We will denote by $\operatorname{P}_{t}$ the
distribution of $\sigma_{t}$ when $\sigma_{0}=-\bar{1}$.

Percolation. Due to the fact that $\lambda(\sigma)\in[\epsilon,1]$, for all
$\sigma\in\\{-1,1\\}^{B(0,1)}$,the configuration $\sigma_{t}$ dominates and is
dominated by independent Bernoulli percolation with parameters
$p_{t}=1-e^{\epsilon t}$ and $q_{t}=1-e^{-t}$, respectively. In particular,
this implies that the critical percolation time

$t_{c}=\inf\big{\\{}t\geq 0:\operatorname{P}\big{(}\sigma_{t}\text{
percolates}\big{)}>0\big{\\}}$ (7.4)

is well defined, finite and positive.

For each $t\geq 0$ fixed, the distribution of $\sigma_{t}$ is translation and
rotation invariant. Positive assossiation once again follows from [24], and
Lemmas 2.2 and 2.3 remain valid.

Our goal here is to prove that the process undergoes a sharp phase transition
as time increases. Analogously to (2.27) and (2.28), define the fictitious
regime

$\begin{split}t_{+}&=\inf\Big{\\{}t\in\mathbb{R}_{+};\operatorname{P}_{t}(\mathcal{C}(3n,n))>1-c_{\ref{c:fsc}}\text{
for some $n\geq 1$}\Big{\\}}\\\
&=\sup\Big{\\{}t\in\mathbb{R}_{+};\operatorname{P}_{t}(\mathcal{C}^{*}(n,3n))\geq
c_{\ref{c:fsc}}\text{ for all $n\geq 1$}\Big{\\}},\end{split}$ (7.5)
$\begin{split}t_{-}&=\sup\Big{\\{}t\in\mathbb{R}_{+};\operatorname{P}_{t}(\mathcal{C}^{*}(3n,n))>1-c_{\ref{c:fsc}}\text{
for some $n\geq 1$}\Big{\\}}\\\
&=\inf\Big{\\{}t\in\mathbb{R}_{+};\operatorname{P}_{t}(\mathcal{C}(n,3n))\geq
c_{\ref{c:fsc}}\text{ for all $n\geq 1$}\Big{\\}}.\end{split}$ (7.6)

Since our process dominates independent Bernoulli percolation with parameter
$1-e^{-\epsilon t}$, we have $t_{-}\leq t_{+}<\infty$ and
$t_{c}\in[t_{-},t_{+}]$. Furthermore, since $t_{+}$ is finite, we fix from now
on a finite time $\tau>t_{+}$ and restrict our analysis to times
$t\in[0,\tau]$. Similarly as in the previous case, there exists
$c_{\ref{c:rsw_bootstrap}}>0$ such that, for all $t\in(t_{-},t_{+})$

$\begin{array}[]{c}\parbox[c]{433.62pt}{\centering$\operatorname{P}_{t}\big{(}\mathcal{C}(3n,n)\big{)}>c_{\ref{c:rsw_bootstrap}}$
and
$\operatorname{P}_{t}\big{(}\mathcal{C}^{*}(3n,n)\big{)}>c_{\ref{c:rsw_bootstrap}}$,
for all $n\geq 1$.\@add@centering}\end{array}$ (7.7)

The strategy for proving the Theorem 1.3 above is the same as the one
presented for Theorem 1.2. However, instead of carrying out the proof
completely, we will only state the modifications on the main tools and focus
more heavily on the Aizenman-Grimmett argument that relates the two different
notions of pivolatily we have here.

Graphical construction. Let us begin by defining the graphical construction
that is used for this process. We will skip directly to the construction with
the denser collection of clock rings.

As in Section 5, we consider a Poisson point process on
$\mathbb{Z}^{2}\times\mathbb{R}_{+}\times[0,1]\times\\{0,1\\}$ with intensity
$\operatorname{count}_{\mathbb{Z}^{2}}\otimes
k\cdot\mathop{}\\!\mathrm{d}x\otimes U[0,1]\otimes\operatorname{Ber}(1/k)$. We
will continue to write $\mu=\sum_{i\geq 0}\delta_{(X_{i},T_{i},R_{i},D_{i})}$,
but we change the notation $\mu^{\prime}=\sum_{i\geq
0}\delta_{(X_{i},T_{i})}$.

Each point $(X_{i},T_{i},R_{i},D_{i})$ encodes the same as in the previous
case:

  * •

$X_{i}\in\mathbb{Z}^{2}$ will represent the site where the time tick occurs;

  * •

$T_{i}\in\mathbb{R}_{+}$ will encode the time when each mark arrives;

  * •

$R_{i}\in[0,1]$ will hold the uniform random variable that is used in order to
apply the rules of the local updates;

  * •

$D_{i}\in\\{0,1\\}$ is a variable used to thin the point process.

Suppose a clock tick $(X_{i},T_{i},R_{i},D_{i})$ happens. Then the opinion in
the site $X_{i}$ changes to 1 at time $T_{i}$ if
$R_{i}\leq\lambda\big{(}\tau_{X_{i}}(\sigma_{T_{i}-})\big{)}$ and $D_{i}=1$.

Notice that Lemma 2.1 still holds with this construction, since the only
change in the construction of the dynamics for this case is in the updating
rule, while the Poisson point process considered still remains the same.

Pivotality and Russo’s formula. We cannot obtain an explicit formula for the
derivative of the crossing probabilities with respect to time. However,
combining the expression of the generator with the fact that the rate function
$\lambda$ satisfies $\lambda\geq\epsilon$, we can lower bound such derivative.
For a monotone non-decreasing event $A$ depending on the states of vertices in
a finite subset $K\in\mathbb{Z}^{2}$, we have

$\begin{split}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\operatorname{P}\big{(}\sigma_{t}\in
A\big{)}&=\sum_{x\in
K}\operatorname{E}\Big{(}\lambda\big{(}\tau_{x}(\sigma_{t})\big{)}\big{(}{\bf
1}_{A}(\sigma_{t}^{x})-{\bf 1}_{A}(\sigma_{t})\big{)}\Big{)}\\\ &=\sum_{x\in
K}\operatorname{E}\Big{(}\lambda\big{(}\tau_{x}(\sigma_{t})\big{)}{\bf
1}_{A}(\sigma_{t}^{x}){\bf 1}_{A^{c}}(\sigma_{t})\Big{)}\\\
&\geq\epsilon\sum_{x\in K}\operatorname{P}\big{(}\sigma_{t}\notin
A,\sigma_{t}^{x}\in A\big{)}.\end{split}$ (7.8)

In the bound above, the notion of pivotality that appears is different from
pivotality for the initial condition introduced in Section 5. We here have
closed pivotailty with respect to the configuration at time $t$.

######  Definition 7.1.

We say that $x\in\mathbb{Z}^{2}$ is closed time-$t$ pivotal for the monotone
non-decreasing event $A$ if $\sigma_{t}\notin A$ and $\sigma_{t}^{x}\in A$.

In particular, closed time-$t$ pivotality can only happen if
$\sigma_{t}(x)=-1$. We define the conditional time-$t$ influence of a vertex
$x\in\mathbb{Z}^{2}$ as

$\operatorname{Inf}^{t}(x,A|\mu^{\prime})=\operatorname{P}\big{(}x\text{ is
closed time-$t$ pivotal for $A$}\big{|}\mu^{\prime}\big{)}.$ (7.9)

We will also consider the conditional thinning influence, but we need to
slightly change the definition. Fix a realization $\mu^{\prime}$ and $i\geq
0$. Denote by $\sigma_{t}^{i,0}$ and $\sigma_{t}^{i,1}$ the configurations
obtained by setting $D_{i}$ to 0 or 1, respectively.

######  Definition 7.2.

We say that the clock tick $i$ is pivotal for the non-decreasing event $A$ if
$\sigma_{t}^{i,1}\in A$ and $\sigma_{t}^{i,0}\notin A$, for some choice of
$R_{i}$.

In the above, since we are dealing with non-decreasing events $A$, it suffices
to choose $R_{i}=0$. Furthermore, we define the influence of a clock tick $i$
as

$\operatorname{Inf}^{\textnormal{thin}}(i,A|\mu^{\prime})=\operatorname{P}[i\text{
is pivotal for $A$}|\mu^{\prime}].$ (7.10)

Once again, we can consider randomized algorithms $\mathcal{A}$ that determine
a function $f={\bf 1}_{A}$. In our new setting, the algorithm has access to
$\mu^{\prime}$ and has to examine the variables $(R_{i},D_{i})_{i\geq 0}$ in
order to determine the occurrence of the event $A$. In this setting, we let
$\mu^{\prime}$ forget the values of $R_{i}$, as they will be used on our
enhancement argument. Of course, the influence of a random variable $R_{i}$
will be dominated by the influence of the corresponding $D_{i}$, since $R_{i}$
can only be pivotal if $D_{i}$ is. From this, we obtain the following version
of the OSSS inequality

######  Lemma 7.3.

For any algorithm $\mathcal{A}$ that determines $f={\bf 1}_{A}$,

$\operatorname{Var}(f|\mu^{\prime})\leq 8\sum_{i\geq
0}\delta(\mathcal{A},i)\operatorname{Inf}^{\textnormal{thin}}(i,A|\mu^{\prime}).$
(7.11)

Algorithm and revealment bounds. We now proceed to introduce the algorithm and
bound its revealment. This will follow the same lines as the previous case,
and we just list the properties pointing out where modifications are
necessary.

We start by recalling the definition of support of a site in (5.11) and the
good event (5.12). Once again, the support of a given vertex $x$ depends only
on the realization of the Poisson point process $\mu^{\prime}$ and the value
of $\sigma_{t}(x)$ is determined if all clock-tick selections in the support
of $x$ are revealed using the suitable modification of the algorithm
$\mathcal{A}_{x}$. We can then apply the same algorithm in order to determine
the existence of crossings of $\mathcal{C}(3n,n)$ at time $t<\tau$.

Here once again, Lemma 5.4 can be applied to estimate the revealment and we
obtain an analogous result.

Aizenman-Grimmett. The last ingredient we need is one analogous to Lemma 6.1,
which we state and prove here.

For $x\in\mathbb{Z}^{2}$ and $\ell\in\mathbb{N}$, denote by $J_{x}(\ell)$ the
set

$J_{x}(\ell)=\big{\\{}j\in\mathbb{N}:(X_{j},T_{j})\in\partial
B(x,\ell)\times[0,\tau]\big{\\}}.$ (7.12)

Recall the definition of $h_{\alpha}$ in (6.1). Similarly to (6.2) and (6.3)
we define

$\tilde{\mathcal{G}}_{x}^{k}=\bigg{\\{}\mu^{\prime}:\begin{array}[]{c}\mathcal{B}(x)\subset
B(x,h_{\alpha}(n))\text{ and }|J_{x}(\ell)|\leq\zeta\log n,\\\ \text{ for
}\ell=h_{\alpha}(n)-1,h_{\alpha}(n),h_{\alpha}(n)+1\end{array}\bigg{\\}},$
(7.13)

and

$\Omega(k,n)=\bigcap_{x\in[-n,4n]\times[-n,2n]}\tilde{\mathcal{G}}^{k}_{x}.$
(7.14)

Differently of (6.2), the event $\tilde{\mathcal{G}}_{x}^{k}$ controls not
only the number of clock ticks in the boundary $\partial B(x,h_{\alpha}(x))$,
but also for other values of radii. The bound (6.7) still holds in this case.

######  Lemma 7.4.

Given $k\geq 1$, there exists a constant
$c_{\ref{c:AG_bootstrap}}=c_{\ref{c:AG_bootstrap}}(k)>0$ such that the
following holds for any $\zeta>0$. Given $\mu^{\prime}\in\Omega(k,n)$ and some
$t\in[t_{-},t_{+}]$, for every $i$ such that $T_{i}\leq t$ and
$X_{i}\in[-n,4n]\times[-n,2n]\cap\mathbb{Z}^{2}$ we have

$\operatorname{Inf}^{\textnormal{thin}}\big{(}i,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\leq
c_{\ref{c:AG}}^{\zeta\log n}\sum_{x\in\partial
B(x_{i},h_{\alpha}(n))}\operatorname{Inf}^{t}\big{(}x,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}.$
(7.15)

The strategy for proving the lemma above is essentially the same as the one
for Lemma 6.1. The main technical difficulty is that we do not pass from a
pivotal tick to a pivotal set of initial conditions, but actually to a pivotal
set at time $t$.

###### Proof.

Fix $\mu^{\prime}\in\Omega(k,n)$, and observe that the occurrence of
$\mathcal{C}(3n,n)$ is determined by the evolution restricted to the
collection of clock ticks

$\mathcal{T}=\Big{\\{}j:(x_{j},t_{j})\in[-n,4n]\times[-n,2n]\times[0,t]\Big{\\}}.$
(7.16)

Assume that the clock tick $i$ is pivotal for a selection of decision
variables and clock ticks $\omega=(R_{j},D_{j})_{j\in\mathcal{T}}$, meaning
that its selection determines the existence of crossings at time $t$. Observe
now that, since $\mu^{\prime}\in\Omega(k,n)$, the only vertices whose opinions
can be changed through the selection of the tick $i$ are the ones in
$B(X_{i},h_{\alpha}(n))$.

As in the proof of Lemma 6.1, we now define a map
$\varphi:\big{(}[0,1]\times\\{0,1\\}\big{)}^{\mathcal{T}}\to\big{(}[0,1]\times\\{0,1\\}\big{)}^{\mathcal{T}}$
that modifies the position of the pivotality. This map will only modify the
variables $(R_{j},D_{j})$ for indices in the set

$J_{X_{i}}=J_{X_{i}}(h_{\alpha}(n)-1)\cup J_{X_{i}}(h_{\alpha}(n))\cup
J_{X_{i}}(h_{\alpha}(n)+1)\cup\\{i\\}.$ (7.17)

Given a realization $\omega=(R_{j},D_{j})_{j\in\mathcal{T}}$ for which the
clock tick $i$ is pivotal, we split the definition of
$\varphi(\omega)=(\tilde{R}_{j},\tilde{D}_{j})_{j\in\mathcal{T}}$ in three
steps.

  * •

First, we set the selection variable $D_{i}$ of the pivotal tick to
$\tilde{D}_{i}=1$. This creates open pivotality, meaning that there exists a
crossing at time $t$.

  * •

Second, we change the variables $(R_{j},D_{j})$ for $j\in
J_{X_{i}}(h_{\alpha}(n)-1)\cup J_{X_{i}}(h_{\alpha}(n)+1)$ in order to ensure
that the configuration outside $\partial B(X_{i},h_{\alpha}(n))$ is not
modified. For a fixed value of $j$, the modification will depend on whether a
flip at $(X_{j},T_{j})$ occurs or not. Assume first that the opinion of vertex
$X_{j}$ does not change. In this case, we set $\tilde{D}_{j}=0$ and do not
modify $R_{j}$. Whenever a flip occurs, then we set $\tilde{D}_{j}=1$ and
$\tilde{R}_{j}=\epsilon R_{j}$. This last step guarantees that the opinion of
this vertex turns to 1 in the new evolution, independently of the neighbors.

  * •

Finally, we change the variables with $j\in J_{X_{i}}(h_{\alpha}(n))$: freeze
all of them by declaring the selection variables $\tilde{D}_{j}=0$.

Let us first collect some information about this measure change. First of all,
setting $\tilde{D}_{i}=1$ implies that there exists a crossing at time $t$.
The second step ensures that the configuration outside $\partial
B(X_{i},h_{\alpha}(n))$ still remains the same at time $t$. Here we make use
of the fact that the rate function $\lambda$ has support of radius one.
Finally, pivotality of the clock tick $i$ implies that the configuration after
the last modification above does not have any crossings.

In particular, what we obtain from the discussion above is that, if $i$ is a
pivotal click for $\omega=(R_{j},D_{j})_{j\in\mathcal{T}}$, then the shell
$\partial B(X_{i},h_{\alpha}(n))$ is closed time-$t$ pivotal for
$\varphi(\omega)=(\tilde{R}_{j},\tilde{D}_{j})_{j\in\mathcal{T}}$, meaning
that, if all the selection variables of vertices with clock ticks before time
$t$ in $\partial B(X_{i},h_{\alpha}(n))$ are accepted, then the configuration
presents a crossing at time $t$.

We want to bound the Radon-Nikodym derivative of
$\operatorname{P}\circ\varphi^{-1}$ with respect to $\operatorname{P}$. Since
$\varphi$ restricted to
$\big{(}[0,1]\times\\{0,1\\}\big{)}^{\mathcal{T}\setminus J_{X_{i}}}$ is the
identity, we may restrict ourselves to
$\big{(}[0,1]\times\\{0,1\\}\big{)}^{J_{X_{i}}}$. Fix then a set
$A=\prod_{j\in J_{X_{i}}}[0,a_{j}]\times\\{\tilde{d}_{j}\\}$, and notice that

$\begin{split}\operatorname{P}\big{(}\varphi^{-1}(A)\big{|}\mu^{\prime}\big{)}&=\operatorname{P}\Big{(}\varphi(\omega)\in\prod_{j\in
J}[0_{j},a_{j}]\times\\{\tilde{d}_{j}\\}\Big{|}\mu^{\prime}\Big{)}\\\
&=\sum_{\vec{d}\in\\{0,1\\}^{J_{X_{i}}}}\operatorname{P}\Big{(}D_{j}=d_{j},\tilde{D}_{j}=\tilde{d}_{j},\tilde{R}_{j}\leq
a_{j},\text{ for all }j\in J_{X_{i}}\Big{|}\mu^{\prime}\Big{)}\\\
&\leq\sum_{\vec{d}\in\\{0,1\\}^{J_{X_{i}}}}\operatorname{P}\Big{(}\tilde{R}_{j}\leq
a_{j},\text{ for all }j\in J_{X_{i}}\Big{|}\mu^{\prime}\Big{)}.\end{split}$
(7.18)

In order to bound the probability of the events above, notice that
$\tilde{R}_{j}\geq\epsilon R_{j}$, for every $j\in J_{X_{i}}$. This yields the
bound

$\begin{split}\operatorname{P}\big{(}\varphi^{-1}(A)\big{|}\mu^{\prime}\big{)}&\leq\sum_{\vec{d}\in\\{0,1\\}^{J_{X_{i}}}}\prod_{j\in
J_{X_{i}}}\frac{a_{j}}{\epsilon}\leq
2^{|J_{X_{i}}|}\epsilon^{-|J_{X_{i}}|}\prod_{j\in J_{X_{i}}}a_{j}\\\
&\leq\Big{(}\frac{2k}{\epsilon}\Big{)}^{|J_{X_{i}}|}\operatorname{P}\big{(}A\big{|}\mu^{\prime}\big{)}\leq\Big{(}\frac{2k}{\epsilon}\Big{)}^{4\zeta\log
n}\operatorname{P}\big{(}A\big{|}\mu^{\prime}\big{)},\end{split}$ (7.19)

where the last inequality uses the fact that $|J_{X_{i}}|\leq 3\zeta\log
n+1\leq 4\zeta\log n$, for $n$ large enough.

With this bound on the Radon-Nikodym derivative, and from the properties of
our measure transformation, we obtain

$\begin{split}\operatorname{Inf}^{\textnormal{thin}}\big{(}i,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)}\leq\left(\frac{2k}{\epsilon}\right)^{4\zeta\log
n}\operatorname{P}\big{(}B(X_{i},h_{\alpha}(n))\text{ is closed time-$t$
pivotal}\big{|}\mu^{\prime}\big{)}.\end{split}$ (7.20)

Finally, it remains to pass from pivotality of the shell $\partial
B(X_{i},h_{\alpha}(n))$ to pivotality of a single site at time $t$. We further
modify the clock-tick selections $\tilde{D}_{j}$ in $\partial B(x,h_{2}(n))$
one by one until we eventually arrive in a pivotal vertex. At this point, we
stop the changes and do not make any further modifications. This vertex will
be closed time-$t$ pivotal. In particular, we obtain the bound

$\begin{split}\operatorname{P}\big{(}B(X_{i},h_{\alpha}(n))&\text{ is closed
time-$t$ pivotal}\big{|}\mu^{\prime}\big{)}\\\ &\qquad\qquad\qquad\leq
k^{\zeta\log n}\sum_{x\in\partial
B(X_{i},h_{\alpha}(n))}\operatorname{Inf}^{t}\big{(}x,\mathcal{C}(3n,n)\big{|}\mu^{\prime}\big{)},\end{split}$
(7.21)

which concludes the proof, when combined with (7.20). ∎

The proof of Theorem 1.1 can now be applied to conclude Theorem 1.3, combining
the relevant estimates presented in this section.

##  8 Open problems and remarks

This section states several open questions that encourage further research on
the field of planar dependent percolation. But before presenting these
problems, let us quickly mention some of the extensions of our techniques that
we believe can be readily obtained without deep new ideas:

  1. a)

Slight variations of the dynamics we have considered. In this case it is
important to make sure that the monotonicity and Harris-FKG inequality still
hold true.

  2. b)

Other lattices besides $\mathbb{Z}^{2}$ can be also considered, taking care to
guarantee that it is still planar in the sense that primal and dual crossings
cannot coexist. Observe also that invariance with respect to the isometries of
$\mathbb{Z}^{2}$ was an essential ingredient in our proofs.

  3. c)

Finite range interactions between spins should also not pose any problem to
the presented techniques.

Now that we have mentioned some trivial extensions of our techniques, let us
present some more challenging questions that each require a non-trivial
insight. This is a long list and some problems may be significantly more
challenging than others.

  1. a)

Can one establish an exponential decay of the connectivity for the off-
critical phases? In our Theorem 1.3 we fall short of obtaining an actual
exponential decay.

  2. b)

Can we replace the underlying graph $\mathbb{Z}^{2}$ with a non-planar
counterpart, like slabs of type $\mathbb{Z}^{2}\times\\{0,\dots,n\\}$? One of
the difficulties in this case would be establishing a RSW theory in the
absence of duality. A beautiful result in this direction can be found in [17],
where the authors establish the absence of percolation for the critical
independent model.

  3. c)

Our techniques do not work in case spins can interact with unbounded range.
Even in the case that the coupling constants decay exponentially fast with the
distance between sites, new ideas would be necessary to attack sharpness.

  4. d)

In order for our argument to work, it was crucial to assume that our dynamic
is attractive: more vertices with positive initial states result in more
vertices with positive states for later times. This was derived from the
coupling and the monotonicity of the function $S_{\beta}$ defined in (2.7).
The proof of our results would work with other functions used to define the
dynamics, if they were monotone and depended on finitely many variables. The
question that emerges is: how to study the case where no monotonicity is
present? In [35], the authors study a planar percolation model without
positive association, which can give a direction for future research in this
vein.

  5. e)

Another technique that was used in our proof is the so called “finite-energy
property”, that allows us to change the state of a finite region by paying a
bounded price. Can one adapt our proofs to models without finite energy, where
such surgery constructions are not possible? Examples of such models include
random interlacements process or the family of limiting distributions of the
high-dimensional voter model intersected with the plane.

  6. f)

One can investigate sharp thresholds in other related graphs lacking the
simmetries of $\mathbb{Z}^{2}$, such as book percolation [9] and the quenched
measure of percolation with defect lines [11].

  7. g)

Studying pivotality in critical and near-critical percolation allows one to
prove scaling relations for the universal exponents, should they exist [29,
12]. Can something similar be done to the processes studied here?

  8. h)

Our proof also heavily relied on the planarity of the model. An obvious and
difficult question is how to prove analogous results for $d\geq 3$ [15, 10,
16, 14].

  9. i)

The simple algorithm that we have introduced to determine the occurrence of
crossings has a small revealment. This is enough for us to prove noise
sensitivity of the crossing event on _both_ initial state variables and the
variables used to select which time-marks are used by the process. Noise
sensitivity on each of these variables in isolation is not straightforward
though, can one prove it?

  10. j)

A challenging question is asking for sharp thresholds of Glauber dynamics _in
time_. That is, start the process with a measure out of equilibrium that
converges to a measure exhibiting percolation. Is there a time $\tau_{c}$
neatly dividing a subcritical from and a supercritical phase?

  11. k)

Consider zero-temperature Glauber dynamics on $\mathbb{Z}^{2}$ starting with
an i.i.d. initial condition with initial density $\rho$. It is known that for
values of $\rho$ close to one, this process stabilizes at positive spins as
$\tau$ goes to infinity, see [19]. By symmetry, the opposite is true for small
values of $\rho$. A famous folklore conjecture states that there is a phase
transition exactly at the value $\rho=1/2$, see for instance Conjecture 1 in
[34].

  12. l)

If we consider the hexagonal lattice, our techniques yield $p_{c}(\tau)=1/2$,
for every $\tau>0$. In this case, do we have exceptional times for the
percolation event as we let $\tau$ grow? This question is related to the
previous question about noise sensitivity, see [38, 20].

  13. m)

In the divide and color model [42] one first samples a subcritical percolation
process, and then proceeds to color each finite cluster one of two colors.
This process exhibits asymptotic self-duality in Bernoulli percolation. Can we
prove that the divide and color critical parameter in our case also converges
to one half?

##  Appendix A Proof of Lemma 2.1

In this section, we provide the proof of Lemma 2.1. This lemma provides bounds
on the probability that a site $y$ is reached by a genealogical path that
starts in a far away point $x$.

###### Proof of Lemma 2.1.

Given a sequence of times $T^{x_{0}}_{i_{0}}<\dots<T^{x_{k}}_{i_{k}}<t$
realizing the event $\\{x\overset{t}{\rightarrow}y\\}$, by elementary
properties of the Poisson process we have that

$T^{x_{j+1}}_{i_{j+1}}-T^{x_{j}}_{i_{j}}\stackrel{{\scriptstyle
d}}{{=}}\operatorname{Exp}(1),\quad\text{ for }j=0,\dots,k-1.$ (A.1)

Using a union bound over all possible nearest-neighbor paths from $x$ to $y$,
we obtain

$\begin{split}\mathbb{P}\big{[}x\overset{t}{\rightarrow}y\big{]}&\leq\sum_{\begin{subarray}{c}x_{0}\sim\dots\sim
x_{k}\\\
x_{0}=x,\,x_{k}=y\end{subarray}}\mathbb{P}\left[\begin{array}[]{c}\text{there
exists an associated }\\\ \text{ sequence of Poisson marks }\\\
0<T^{x_{0}}_{i_{0}}<\dots<T^{x_{k}}_{i_{k}}<t.\end{array}\right]\\\ &\leq
e^{-t}\sum_{m\geq|x-y|_{1}}4^{m}\sum_{j\geq m}\frac{T^{j}}{j!}.\end{split}$
(A.2)

Denoting $|x-y|_{1}$ by $n$, we obtain that the right hand side of the last
equation is bounded from above by

$\begin{split}e^{-t}\sum_{m\geq n}e^{t}\frac{(4t)^{m}}{m!}&\leq
e^{4t}\frac{(4t)^{n}}{\left(\frac{n}{2}\right)^{\frac{n}{2}}}\\\
&\leq\exp\Big{\\{}4t+n\log(4t)-\frac{n\log(n)}{2}+\frac{n}{2}\log(2)\Big{\\}},\\\
&\leq\exp\Big{\\{}4t-n\big{(}\log\big{(}\sqrt{n}\big{)}-\log\big{(}4\sqrt{2}t\big{)}\big{)}\Big{\\}}.\end{split}$
(A.3)

Separating into the cases where $n>2^{10}t$ and $n\leq 2^{10}t$, one sees that
the right hand side of the above equation is bounded from above by

$\exp\bigg{\\{}2^{11}t\log(8t)-\frac{1}{4}n\log(n)\bigg{\\}}.$ (A.4)

This concludes the proof. ∎

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