# Cutoff for the Glauber-Exclusion process in the full high-temperature
regime: an information percolation approach

Hong-Quan Tran<EMAIL_ADDRESS>CEREMADE, CNRS, Université Paris-
Dauphine, PSL University 75016 Paris, France

###### Abstract

The Glauber-Exclusion process is a superposition of a Glauber dynamics and the
Symmetric Simple Exclusion Process (SSEP) on the lattice. The model was shown
to admit a reaction-diffusion equation as the hydrodynamic limit. In this
article, we define a notion of temperature regimes via the reaction function
in the equation and prove cutoff in the full high-temperature regime for the
attractive model in dimensions $1$ and $2$ with periodic boundary condition.
Our results show that the equation in the hydrodynamic limit reflects the
mixing behavior of the large but finite system. Besides, cutoff is proved
under the lack of reversibility and an explicit formula for the invariant
measure. We also provide the spectral gap and prove pre-cutoff in all
dimensions. Our proof involves a new interpretation of attractiveness, the
information percolation framework introduced by Lubetzky and Sly, anti-
concentration of simple random walk on the lattice, and a coupling inspired by
excursion theory. We hope that this approach can find new applications in the
future.

###### Contents

  1. 1 Introduction
  2. 2 Results
  3. 3 Monotone boolean functions and attractiveness
    1. 3.1 Monotone boolean functions
    2. 3.2 Interpretation of attractiveness
  4. 4 Dual coupling
    1. 4.1 Graphical Construction
    2. 4.2 The branching exclusion process (BEP)
    3. 4.3 The idealized branching process (IBP)
    4. 4.4 The coupling
    5. 4.5 Analysis of the IBP
    6. 4.6 Success of the coupling
  5. 5 Application of the dual coupling
  6. 6 Upper bound
    1. 6.1 Proof of the upper bound
    2. 6.2 Proof of Lemma 30: the bad set is rarely visited
    3. 6.3 First step of the proof of Proposition 31: reformulation
    4. 6.4 Second step of the proof of Proposition 31: the excursion coupling
  7. Appendix A

##  1 Introduction

The Glauber-Exclusion process, introduced in [9, 10], is a superposition of a
Glauber dynamics and an appropriately accelerated symmetric simple exclusion
process (SSEP) on a lattice. The model is also called Glauber-Kawasaki
dynamics in [11]. Here, we use the term Glauber-Exclusion as in [60]. The
model can be described as follows.

#### Notations.

Fix a number $d\in\mathbb{Z}_{+}$. For any $L\in\mathbb{Z}_{+}$, let
$\Lambda^{d}_{L}:=(\mathbb{Z}/L\mathbb{Z})^{d}$ be the torus of side-length
$L$ in dimension $d$. For any finite set $\Omega$, we call the elements of
$\\{-1,1\\}^{\Omega}$ the spin configurations on $\Omega$, and we denote by
$\leq$ the usual coordinate-wise order on $\\{-1,1\\}^{\Omega}$. Our state
space is $\mathcal{X}:=\\{-1,1\\}^{\Lambda^{d}_{L}}$, the set of all spin
configurations on $\Lambda^{d}_{L}$. For any $x\in\mathcal{X}$,
$u,u^{\prime}\in\Lambda^{d}_{L}$, we denote by $x^{u\leftrightarrow
u^{\prime}}$ the configuration obtained from $x$ by exchanging the values at
two sites $u$ and $u^{\prime}$, and $x^{u,1}$ (resp. $x^{u,-1}$) the
configuration obtained from $x$ by replacing the value at site $u$ by $1$
(resp. $-1$). We also denote by $\exp(\theta)$ the exponential distribution
with parameter $\theta$, i.e. the distribution with density
$\mathds{1}_{\left\\{s>0\right\\}}\theta e^{-\theta s}\mathrm{d}s$.

The generator of the symmetric simple exclusion process (SSEP)
$\mathcal{L}_{E}$ acts on an observable $\varphi:\mathcal{X}\to\mathbb{R}$ by

$(\mathcal{L}_{E}\varphi)(x):=\sum_{u\sim
u^{\prime}}\left(\varphi(x^{u\leftrightarrow u^{\prime}})-\varphi(x)\right),$
(1)

where the sum is taken over all pairs of neighbors $\\{u,u^{\prime}\\}$ on the
lattice. This means that $\mathcal{L}_{E}$ exchanges the contents of two
neighboring sites $u,u^{\prime}$ at rate $1$.

To define the Glauber dynamics, we first need the following definition.

#### Flip-rate function.

Throughout the paper, we fix a number $m\in\mathbb{Z}_{+}$ and a function
$c:\\{-1,1\\}^{B(0,m)}\to\mathbb{R}_{>0}$, where $B(0,m)$ is the subset of
$\mathbb{Z}^{d}$ consisting of all sites of distance at most $m$ from $0$. We
call this function $c$ the _local flip-rate function_.

For any $L$ large enough, we identify $B(0,m)$ with the subset of
$\Lambda^{d}_{L}$ consisting of all sites of distance at most $m$ from site
$0$. The function $c$ extends to a _global flip-rate function_
$\hat{c}:\Lambda^{d}_{L}\times\mathcal{X}\to\mathbb{R}_{>0}$ as follows.

  1. 1.

$\hat{c}(0,\cdot)$ is local and given by $c$:

$\displaystyle\forall
x\in\mathcal{X},\;\hat{c}(0,x):=c\left(x|_{B(0,m)}\right).$

  2. 2.

$\hat{c}$ is invariant by translation:

$\displaystyle\forall
x\in\mathcal{X},u\in\Lambda^{d}_{L},\;\hat{c}(u,x):=\hat{c}(0,x_{u+\cdot}),$

where

$\displaystyle x_{u+\cdot}(u^{\prime}):=x(u+u^{\prime}),\;\forall
x\in\mathcal{X},\,u,u^{\prime}\in\Lambda^{d}_{L}.$

The generator $\mathcal{L}_{G}$ of the Glauber dynamics associated with the
local flip-rate function $c(\cdot)$ acts on an observable
$\varphi:\mathcal{X}\to\mathbb{R}$ by

$(\mathcal{L}_{G}\varphi)(x):=\sum_{u\in\Lambda^{d}_{L}}\hat{c}(u,x)(\varphi(x^{u,-x(u)})-\varphi(x)).$
(2)

This means that $\mathcal{L}_{G}$ flips the spin on site $u$ of configuration
$x$ at rate $\hat{c}(u,x)$, for any
$(u,x)\in\Lambda^{d}_{L}\times\mathcal{X}$. We suppose that the Glauber
dynamics is attractive.

######  Hypothesis 1 (Attractiveness).

For any $x,y\in\\{-1,1\\}^{B(0,m)}$ such that $x\leq y$,

$\displaystyle c(x)$ $\displaystyle\geq c(y)\text{ if }x(0)=y(0)=1,$ (3)
$\displaystyle c(x)$ $\displaystyle\leq c(y)\text{ if }x(0)=y(0)=-1.$ (4)

Hypothesis 1 implies that, for any $x,y\in\mathcal{X}$ such that $x\leq y$,
for any $u\in\Lambda^{d}_{L}$,

$\displaystyle\hat{c}(u,x)$ $\displaystyle\geq\hat{c}(u,y)\text{ if
}x(u)=y(u)=1,$ (5) $\displaystyle\hat{c}(u,x)$
$\displaystyle\leq\hat{c}(u,y)\text{ if }x(u)=y(u)=-1.$ (6)

These conditions are necessary and sufficient to construct a Markovian
coupling of two processes whose generators are $\mathcal{L}_{G}$ that
preserves order. More precisely, suppose that $(Y^{1}_{t})_{t\geq 0}$ and
$(Y^{2}_{t})_{t\geq 0}$ are two Markov processes with generator
$\mathcal{L}_{G}$ starting at $y_{1},y_{2}$, respectively. Then there is a
Markovian coupling of $(Y^{1}_{t})_{t\geq 0}$ and $(Y^{2}_{t})_{t\geq 0}$ such
that, almost surely,

$y_{1}\leq y_{2}\Rightarrow\forall t\geq 0,\;Y^{1}_{t}\leq Y^{2}_{t}.$

See Theorem 4.11, p.143 in [38] for more details. When such an order-
preserving coupling exists, we say that the process $Y$ is attractive.

Although we will work with a general function $c(\cdot)$, we usually use the
following example considered by De Masi et al in [10] to illustrate our
method.

######  Example 1 (Example of De Masi et al).

The model considered is in dimension one, i.e. $d=1$, with the local flip-rate
function $c$ given by

$\forall x\in\\{-1,1\\}^{\\{1,0,-1\\}},\;c(x):=1-\gamma
x(0)(x(1)+x(-1))+\gamma^{2}x(1)x(-1),$

for some $\gamma\in[0,1]$. This corresponds to the global flip-rate function
$\hat{c}$ given by

$\forall x\in\mathcal{X},u\in\Lambda_{L},\;c(u,x):=1-\gamma
x(u)(x(u+1)+x(u-1))+\gamma^{2}x(u+1)x(u-1).$

The _Glauber-Exclusion process_ is a continuous time Markov process
$\textbf{X}=(X_{t})_{t\geq 0}$, taking values in $\mathcal{X}$, whose
generator is

$\mathcal{L}_{GE}:=L^{2}\mathcal{L}_{E}+\mathcal{L}_{G}.$ (7)

This is a superposition of a Glauber dynamics and a SSEP accelerated by a
diffusive factor $L^{2}$.

For $\rho\in[-1,1]$, we denote by $\mathfrak{R}_{\rho}$ the Rademacher
probability measure on $\\{-1,1\\}$ with average $\rho$. More precisely,

$\displaystyle\mathfrak{R}_{\rho}(1)$ $\displaystyle=\dfrac{1+\rho}{2},$
$\displaystyle\mathfrak{R}_{\rho}(-1)$ $\displaystyle=\dfrac{1-\rho}{2}.$

#### Reaction function.

We denote by $\nu_{\rho}$ the product measure of $\mathfrak{R}_{\rho}$ on
$\\{-1,1\\}^{B(0,m)}$. We define the polynomial $R(\cdot):[-1,1]\to\mathbb{R}$
by

$\displaystyle\forall\rho\in[-1,1],\;R(\rho):=\mathbb{E}_{\nu_{\rho}}\left[-2\xi_{0}c(\xi)\right],$

where $\xi=(\xi_{u})_{u\in B(0,m)}\sim\nu_{\rho}$. We call $R(\cdot)$ the
_reaction function_. The reason behind this name will be explained in the next
paragraph.

#### The hydrodynamic limit.

Originally, De Masi, Ferrari, and Lebowitz introduced the model in [9, 10] for
the infinite lattice $\mathbb{Z}^{d}$ and proved that the hydrodynamic limit
of the model is a reaction-diffusion equation. We recall here the analogous
result obtained by Kipnis, Olla, and Varadhan [30] for our case in a finite
box. Let the density field on $(\mathbb{R}/\mathbb{Z})^{d}$ at time $t$ be
defined by

$\mu_{t}^{L}(\mathrm{d}u)=\dfrac{1}{L^{d}}\sum_{u\in\Lambda^{d}_{L}}X_{t}(u)\delta_{u/L}(\mathrm{d}u)\;\;\text{on
}(\mathbb{R}/\mathbb{Z})^{d}.$

Then, when $L$ grows to infinity, under appropriate convergence conditions on
the initial data, the density field converges weakly to the solution
$\rho:\mathbb{R}_{+}\times(\mathbb{R}/\mathbb{Z})^{d}\to[-1,1]$ of the
following reaction-diffusion equation:

$\dfrac{\partial\rho}{\partial t}=\Delta\rho+R(\rho),$ (8)

where $\Delta$ is the Laplacian in $(\mathbb{R}/\mathbb{Z})^{d}$, and
$R(\cdot)$ is the function defined above. Equation (8) explains the name
reaction function of $R(\cdot)$. The intuition behind this result is as
follows. Applying the generator $\mathcal{L}_{GE}$ to the function
$\varphi:x\mapsto x(u)$, we see that

$\mathcal{L}\varphi(x)=\left(L^{2}\sum_{u^{\prime}\sim
u}(x(u^{\prime})-x(u))\right)-2x(u)\hat{c}(u,x).$

Therefore,

$\dfrac{d}{dt}\mathbb{E}\left[X_{t}(u)\right]=\mathbb{E}\left[\mathcal{L}\varphi(X_{t})\right]=\left(L^{2}\sum_{u^{\prime}\sim
u}\mathbb{E}\left[X_{t}(u^{\prime})-X_{t}(u)\right]\right)-2\mathbb{E}\left[X_{t}(u)\hat{c}(u,X_{t})\right].$

Suppose that $\mathbb{E}\left[X_{t}(u)\right]$ is well approximated by
$\rho(t,u/L)$ for some smooth function $\rho$, for any site $u$ and any fixed
time $t$. Then

$\displaystyle L^{2}\sum_{u^{\prime}\sim
u}\mathbb{E}\left[X_{t}(u^{\prime})-X_{t}(u)\right]\approx
L^{2}\sum_{u^{\prime}\sim
u}\left(\rho(t,u^{\prime}/L)-\rho(t,u/L)\right)\approx\Delta\rho(t,u/L).$

Moreover, the local law around a site $u$ at any fixed time $t$ is believed to
be close to that of a product of
$\mathfrak{R}_{\mathbb{E}\left[X_{t}(u)\right]}$, a phenomenon known as _local
equilibrium_ , see Subchapter $1.2$ and Chapter 3 in [29] for more details.
Then

$\displaystyle\mathbb{E}\left[X_{t}(u)\hat{c}(u,X_{t})\right]\approx\mathbb{E}_{\nu_{\mathbb{E}\left[X_{t}(u)\right]}}\left[\xi_{0}c(\xi)\right]=R\left(\mathbb{E}\left[X_{t}(u)\right]\right)\approx
R(\rho(t,u/L)).$

These intuitive arguments indicate that $\rho(\cdot,\cdot)$ satisfies equation
(8). Showing the convergence amounts to verifying the above intuition
rigorously.

Based on this result, we define the _temperature regimes_ as follows.

#### Temperature regimes.

  * •

_High-temperature regime_ : The system is at high temperature if the reaction
function $R(\cdot)$ has a unique root $\rho_{*}$ in $[-1,1]$, and
$R^{\prime}(\rho_{*})<0$.

  * •

_Critical-temperature regime_ : The system is at critical temperature if the
reaction function $R(\cdot)$ has a unique root $\rho_{*}$ in $[-1,1]$, and
$R^{\prime}(\rho_{*})=0$.

  * •

_Low-temperature regime_ : The system is at low temperature if the reaction
function $R(\cdot)$ has at least two roots in $[-1,1]$.

#### Explanation for the temperature regimes.

Let us explain why we define the temperature regimes for the model as above.
Consider the equation (8) with the initial condition $\rho(0,\cdot)$ constant
in space, say $\rho(0,\cdot)=\rho_{0}$ for some $\rho_{0}\in[-1,1]$. We then
see that $\rho(t,\cdot)$ must also be constant in space, so we can safely
denote by $\rho(t)$ the value of the function $\rho$ at time $t$. The equation
$\eqref{eq:hydro}$ then simply becomes an ODE:

$\begin{cases}\rho^{\prime}=R(\rho),\\\ \rho(0)=\rho_{0}.\end{cases}$ (9)

Though simple, the solution of $\eqref{eq:ODE}$ can exhibit very different
behaviors, depending on $R(\cdot)$. Consider the flip-rate function in Example
1. Then $R(\rho)=-2(1-2\gamma)\rho-2\gamma^{2}\rho^{3}$. One can verify that
the system is at high temperature if $0\leq\gamma<1/2$, at critical
temperature if $\gamma=1/2$, and at low temperature if $1/2<\gamma\leq 1$. We
present in Figure 1 the phase diagrams of the ODE (9) for two different values
of $\gamma$.

(a) High temperature

(b) Low temperature

Figure 1: Phase diagram for $\gamma=5/12$ and $\gamma=7/12$.

Let $\rho_{+}(\cdot)$ and $\rho_{-}(\cdot)$ denote the solutions of the ODE
(9) with initial conditions $\rho_{+}(0):=1$ and $\rho_{-}(0):=-1$. Viewing
the diagram, we see that, in the first case, for any initial condition
$\rho_{0}$, the solution of ODE (9) converges to the unique root $\rho_{*}$
exponentially fast: $\left|\rho(t)-\rho_{*}\right|\lesssim
e^{R^{\prime}(\rho_{*})t}$. In the second case, $R(\cdot)$ has 3 roots
$\rho^{(1)}<\rho^{(2)}<\rho^{(3)}$, and
$\rho_{-}(t)\xrightarrow{t\to\infty}\rho^{(1)}$, and
$\rho_{+}(t)\xrightarrow{t\to\infty}\rho^{(3)}$. Intuitively, this implies
that the system mixes rapidly at high temperature, while at low temperature,
it exhibits the phenomenon known as _metastability_ , see [6] for more
details.

In this paper, we study the model in the high-temperature regime.

######  Hypothesis 2 (High temperature).

  * •

$R(\rho)$ admits a unique root $\rho_{*}$ in $[-1,1]$,

  * •

$R^{\prime}(\rho_{*})<0$.

The primary purpose of this paper is to study the mixing times of the system
in the high-temperature regime. We begin by explaining some terminologies.

#### The mixing times and the spectral gap.

We use the notation $\mathbb{P}_{x}\left[\cdot\right]$ for the law of the
process $X$ starting from $x$. The strict positivity of $c(\cdot)$ ensures
that the generator $\mathcal{L}^{GE}$ is irreducible. This implies that there
exists a unique invariant probability distribution $\pi$, and the distribution
of the process converges to $\pi$ when the time tends to infinity:

$\forall
x,y\in\mathcal{X},\;\mathbb{P}_{x}\left[X_{t}=y\right]\xrightarrow[t\to\infty]{}\pi(y).$
(10)

This convergence is often measured by the total variation distance,
$\textrm{d}_{\textsc{tv}}\left(\cdot,\cdot\right)$, defined by

$\textrm{d}_{\textsc{tv}}\left(\mu,\nu\right)=\max_{A\subset\mathcal{X}}|\mu(A)-\nu(A)|,$
(11)

for any two probability distributions $\mu,\nu$ on $\mathcal{X}$. The speed of
convergence is quantified by the _mixing-times_ :

$\textrm{t}_{\textsc{mix}}(\epsilon):=\inf\left\\{t\geq
0:\max_{x\in\mathcal{X}}\textrm{d}_{\textsc{tv}}\left(\mathbb{P}_{x}\left[X_{t}\in\cdot\right],\pi\right)\leq\epsilon\right\\},\;\;\epsilon\in(0,1).$
(12)

The smallest real part of the non-zero eigenvalues of $-\mathcal{L}_{GE}$,
also called the _spectral gap_ of $\mathcal{L}^{GE}$, denoted by gap, governs
the asymptotic speed of convergence to equilibrium:

$\texttt{gap}=\lim_{t\to\infty}-\dfrac{1}{t}\log\max_{x\in\mathcal{X}}\textrm{d}_{\textsc{tv}}\left(\mathbb{P}_{x}\left[X_{t}\in\cdot\right],\pi\right).$
(13)

#### The cutoff phenomenon.

Note that the mixing times and the spectral gap depend on the side length $L$
of the system, though we have kept this dependency implicit to lighten the
notation. We say that the system exhibits _cutoff_ if in the limit where $L$
tends to infinity, the asymptotic behavior of
$\textrm{t}_{\textsc{mix}}(\epsilon)$ does not depend on the precision
$\epsilon$ anymore:

$\forall\epsilon\in(0,1)\;\text{fixed},\dfrac{\textrm{t}_{\textsc{mix}}(\epsilon)}{\textrm{t}_{\textsc{mix}}(1/4)}\xrightarrow[L\to\infty]{}1.$
(14)

We refer the readers to the works [2, 12, 13] for the discovery of the cutoff
phenomenon and to the book [37] for an introduction to the subject.

#### Conjectured universal behaviors.

A stochastic spin system, with an appropriate notion of temperature, under an
appropriate time scale, is conjectured to exhibit the following behaviors.

  * •

At high temperatures, the system mixes fast. The inverse-spectral-gap
$\texttt{gap}^{-1}$ is $\mathcal{O}\left(1\right)$, the mixing times are
logarithmic in the size of the lattice, and cutoff occurs.

  * •

At critical temperatures, the inverse gap is (sub)-polynomial in the size of
the lattice, and so are the mixing times.

  * •

At low temperatures, the inverse gap is exponential in the size of the
lattice, and so is the mixing time, and there is no cutoff.

See, for example, [24, 35, 65] for the conjectures.

#### Literature.

The Glauber dynamics is arguably the most studied interacting particle system
in the literature. Tons of work have been devoted to verifying the conjectured
behaviors above for it; see Subsection 1.1 in [40] for a brief review of the
development of the subject. The readers can look at [1, 17, 25, 26, 27, 39,
42, 45, 48, 47, 57, 58, 59, 67, 68] for results in the high-temperature
regime, at [15, 16, 40, 52] for results in the critical-temperature regime,
and at [8, 28, 45, 56, 61] for results in the low-temperature regime. We
especially mention here the work [42], which proves cutoff in the full high-
temperature regime in any dimension. The proof in [42] involves a new
framework called _information percolation_ , see also [21, 41, 43, 44]. For
background on Glauber dynamics, see the lecture notes [46], or Chapter 15 in
[37] for a quick introduction. In particular, Theorem $15.4$ in [37] presents
a very simple example that illustrates the difference in mixing behaviors at
low and high temperatures.

Besides Glauber dynamics, the Exclusion Process is also an emblematic
interacting particle system. It describes the relaxation of a gas of
interacting particles to equilibrium. The convergence is usually studied from
two points of view: the macroscopic evolution of the density of particles,
which is the study of hydrodynamic limits, or the microscopic evolution of the
law of the particles and its total variation distance to equilibrium, which is
the study of mixing times. The readers can see [29] for an introduction to the
study of hydrodynamic limits and [37] for an introduction to the study of
mixing times. For the results on the hydrodynamic limit of SSEP, see, for
example, [5, 22, 23, 36, 30]. For the results on the mixing times, see [14,
31, 32, 33, 34, 50, 51, 66].

We mention here that cutoff for SSEP has been proved in relatively few cases:
for the complete graph in [34], for dimension one in [31, 32, 33], see also
[23, 63] for nonconservative variants. The only result available in general
geometries for a nonconservative variant is obtained in [55]. However, it is
restricted to the case where the system is reversible w.r.t a product of
i.i.d. Bernoulli.

As mentioned before, the Glauber-Exclusion process was introduced in [10] and
also considered in [30]. To the best of our knowledge, the only works on its
mixing times, also the most related works to our paper, are [60] and [64]. In
[64], Tsunoda studies the model in dimension one and proves exponentially slow
mixing when the primitive of the reaction function (the potential function)
has two or more local minima, which implies that the system is at low
temperature. In [60], Tanaka and Tsunoda prove the upper bound
$\mathcal{O}\left(\log L\right)$ on the mixing times for any dimension under
the condition that $\max_{\rho\in[-1,1]}R^{\prime}(\rho)<0$, which
subsequently implies that the system is at high temperature. These two works
together show a phase transition of the mixing times for the model in Example
1 between low and high temperatures, consistent with the conjectured
behaviors. In particular, we expect cutoff to occur in the high-temperature
regime. However, to establish cutoff, not only do we have to prove that the
upper and lower bounds are of the same magnitude $\log L$, but we also need to
show that the pre-factors in front of $\log L$ in the two bounds are the same.
Surprisingly, not even a lower bound of matching magnitude $\log L$ is
available, even though it is often “the easy part” in proving cutoff. It was
not yet known whether the upper bound in [60] is sharp either.

#### Our contribution.

We sharpen the upper bound in [60] and prove a matching lower bound, therefore
showing cutoff for the full high-temperature regime for _any_ attractive rate
function $c$, in dimensions $1$ and $2$. Furthermore, for any dimension, in
the full high-temperature regime, we prove that the inverse gap is of order
$1$ and provide lower and upper bounds of magnitude $\log L$ on the mixing
times. Our proof involves a new interpretation of the attractiveness, the
information percolation framework introduced by Lubetzky and Sly in [42],
anti-concentration of simple random walk on the lattice, and what we call _the
excursion coupling_. We hope that this approach can find new applications in
the future.

##  2 Results

The first main result of the paper is the following.

######  Theorem 1 (Cutoff in dimensions $1$ and $2$).

Let $\epsilon\in(0,1)$ be fixed. For $d\in\\{1,2\\}$, there exists a constant
$\kappa$ such that

$\dfrac{\log|\Lambda^{d}_{L}|}{2|R^{\prime}(\rho_{*})|}-\kappa\log\log
L\leq\textrm{t}_{\textsc{mix}}(\epsilon)\leq\dfrac{\log|\Lambda^{d}_{L}|}{2|R^{\prime}(\rho_{*})|}+\kappa\log\log
L.$ (15)

######  Remark 1.

In our setup, the dynamics is not reversible, and no explicit formula for the
invariant measure $\pi$ is available, except when $c$ is constant. In fact, in
[20], Gabrielli et al showed that the system is reversible if and only if the
function $c(\cdot)$ is of the following form

$c(x)=(a_{1}+a_{2}x(0))h(x),$ (16)

where $h(x)$ is independent of $x(0)$, and in this case $\pi$ is a product of
i.i.d. Rademacher. To the best of our knowledge, it is also the only case that
an explicit formula for $\pi$ is available. However, one can show that if a
function $c(\cdot)$ satisfies the reversibility condition (16), then it is
attractive if and only if it is a constant function.

Our second main result is about the spectral gap of the system.

######  Theorem 2 (Spectral gap).

For any dimension $d$,

$\liminf_{L\to\infty}\texttt{gap}\geq\dfrac{1}{|R(\rho_{*})|}.$ (17)

Theorem 1 and Theorem 2 together confirm the conjectured behaviors of the
system in the full high-temperature regime in dimensions $1$ and $2$.
Furthermore, they connect the mixing behaviors of the Glauber-Exclusion
process with the behaviors of the solutions of the ODE (9).

Our third result investigates the mixing times in higher dimensions.

######  Theorem 3 (Pre-cutoff in higher dimensions).

Let $\epsilon\in(0,1)$ be fixed. For any $d\geq 3$, there exists a constant
$\kappa$ such that

$\dfrac{1}{d}\dfrac{\log|\Lambda^{d}_{L}|}{|R^{\prime}(\rho_{*})|}-\kappa\leq\textrm{t}_{\textsc{mix}}(\epsilon)\leq\dfrac{\log|\Lambda^{d}_{L}|}{|R^{\prime}(\rho_{*})|}+\kappa\sqrt{\log
L}.$ (18)

Theorem 3 says that the magnitude of the mixing times in all dimensions is
$\log L$. Furthermore, the pre-factor in front of $\log L$ is bounded between
two constants independent of the precision $\epsilon$. This phenomenon is
known as _pre-cutoff_ ; see Chapter 18 in [37] for more details. Establishing
cutoff for the process in dimensions $d\geq 3$ remains an open problem. We
believe that neither the lower nor the upper bounds in Theorem 3 is optimal.
Furthermore, our proof of the upper bound in dimensions $1$ and $2$ relies on
the fact that the simple random walks in dimensions $1$ or $2$ is recurrent,
which is no longer valid for higher dimensions. We do not know if this is just
a coincidence. The Interchange Process, a Markov process closely related to
the Exclusion process, is conjectured to behave somehow differently in
dimensions $d\geq 3$ compared to $d\in\\{1,2\\}$ in [62]. This fact is
verified for all dimensions $d\geq 5$ in [18]. We do not know whether this
phase transition in infinite volume is connected to a change of mixing
behavior in finite volume.

Nevertheless, the upper bound in Theorem 3 is already better than that in
[60]. In [60], the authors impose the assumption that the derivative
$R^{\prime}$ of $R$ is uniformly negative, i.e.
$\max_{\rho\in[-1,1]}R^{\prime}(\rho)<0,$ and prove the upper bound
$\dfrac{\log|\Lambda^{d}_{L}|}{\left|\max_{\rho\in[-1,1]}R^{\prime}(\rho)\right|}$
on the mixing time. Hence, our assumption is weaker, and the pre-factor in
front of $\log L$ is also sharper.

######  Example 2.

Let $\theta$ be a strictly positive number. In dimension one, we consider the
local flip-rate function $c(\cdot):\\{-1,1\\}^{\\{-1,0,1\\}}\to\mathbb{R}_{+}$
defined by

$\forall
x\in\\{-1,1\\}^{\\{-1,0,1\\}},\;c(x):=\theta+2\times\mathds{1}_{\left\\{x(0)=1,\;x(1)=-1\right\\}}.$
(19)

The corresponding global flip-rate function $\hat{c}$ is given by

$\hat{c}(u,x):=\theta+2\times\mathds{1}_{\left\\{x(u)=1,\;x(u+1)=-1\right\\}}.$
(20)

It is not hard to see that the function $c(\cdot)$ satisfies Hypothesis 1. The
corresponding reaction function is

$R(\rho)=\rho^{2}-2\theta\rho-1.$ (21)

By direct computation, one can show that $R(\rho)$ has a unique root in the
interval $[-1,1]$:

$\rho_{*}=\theta-\sqrt{\theta^{2}+1},$

and moreover,

$R^{\prime}(\rho_{*})<0.$

However, if $0<\theta<1$, then
$\forall\rho\in(\theta,1],\,R^{\prime}(\rho)>0$. This means that $R(\rho)$
satisfies our conditions but does not satisfy the condition in [60].

#### Structure of the proof.

In Section 3, we interpret the attractiveness in a more intuitive way. We also
discuss some properties of monotone boolean functions, which will be used
later in our proof. In Section 4, we construct the dual coupling, which will
subsequently be used in Section 5 to prove Theorem 2, Theorem 3, and the lower
bound in Theorem 1. Finally, in Section 6, we prove the upper bound in Theorem
1. For simplicity, we will assume that $d=1$. Nevertheless, many ingredients
are valid for other dimensions, and we will comment on how to adapt the proof
to higher dimensions when necessary.

#### Acknowledgement.

The author deeply thanks Justin Salez for suggesting the problem, for numerous
useful discussions, and for thoroughly reading the drafts.

##  3 Monotone boolean functions and attractiveness

This section aims to discuss some properties of monotone boolean functions and
to prove Proposition 7, which allows us to interpret the attractiveness more
intuitively.

###  3.1 Monotone boolean functions

######  Definition 4 (Boolean function).

A _boolean function_ is a function from some hypercube
$\\{-1,1\\}^{n},\,n\in\mathbb{Z}_{+}$, into $\\{-1,1\\}$. A boolean function
$f$ is increasing if $x\leq y\Rightarrow f(x)\leq f(y)$, for any
$x,y\in\\{-1,1\\}^{n}$.

######  Definition 5 (Pivotal set).

For an increasing boolean function $f$ on $\\{-1,1\\}^{n}$, its _pivotal set_
is given by

$\mathsf{Piv}\left(f\right):=\\{j\in[n]:\exists
x=(x_{1},\dots,x_{n})\in\\{-1,1\\}^{n}\text{ such that
}f(x^{j,1})=1;\;f(x^{j,-1})=-1\\}.$ (22)

In particular, to evaluate the value of an increasing boolean function $f$ on
a spin configuration $x$, we do not need to know every coordinate of $x$ but
only the coordinates in $\mathsf{Piv}\left(f\right)$. The following lemma is
elementary but very useful in our proof.

######  Lemma 6 (Pivotal sets of monotone boolean functions).

For any increasing boolean function $f$ on $\\{-1,1\\}^{n}$,

$\mathsf{Piv}\left(f\right)\neq\O\iff f(1,\dots,1)=1\text{ and
}f(-1,\dots,-1)=-1.$

In particular, for any increasing boolean function $f$ on $\\{-1,1\\}$,

$|\mathsf{Piv}\left(f\right)|=\dfrac{f(1)-f(-1)}{2}.$

###  3.2 Interpretation of attractiveness

We canonically identify the functions on $\mathcal{X}$ which depend only on
the coordinates in $B(0,m)$ with the functions on $\\{-1,1\\}^{B(0,m)}$. We
now give a more intuitive interpretation of the attractiveness.

######  Proposition 7 (Interpretation of attractiveness).

There exist $q\in\mathbb{Z}_{+}$ and increasing boolean functions
$f_{1},\dots,f_{q}$ on $\\{-1,1\\}^{B(0,m)}$ and positive numbers
$\lambda_{1},\dots,\lambda_{q}$, such that

$\forall
x\in\\{-1,1\\}^{B(0,m)},\;c(x)=\sum_{i=1}^{q}\lambda_{i}\mathds{1}_{\left\\{f_{i}(x)=-x(0)\right\\}}.$
(23)

In particular, for any function $\varphi$ on $\mathcal{X}$,

$\displaystyle\forall
x\in\mathcal{X},\;\mathcal{L}_{G}\varphi(x)=\sum_{u\in\Lambda_{L}}\sum_{i=1}^{q}\lambda_{i}\left(\varphi(x^{u,f_{i}(x_{u+\cdot})})-\varphi(x)\right).$
(24)

Moreover, we can choose $f_{1}\equiv 1,\;f_{2}\equiv-1$ and
$\lambda_{1},\lambda_{2}$ strictly positive.

#### Interpretation.

Proposition 7 is of independent interest. It implies that the technical
conditions (5), (6) for the attractiveness of the Glauber dynamics can be
replaced by a more intuitive interpretation: for each site $u$, $x(u)$ is
replaced by $f_{i}(x_{u+\cdot})$ at rate $\lambda_{i}$, $1\leq i\leq q$. The
monotonicity of $(f_{i})_{1\leq i\leq q}$ clearly implies that the system is
attractive. In fact, for two processes $(Y^{1}_{t})_{t\geq 0}$ and
$(Y^{2}_{t})_{t\geq 0}$ whose generator is $\mathcal{L}_{G}$, one can
construct a coupling of them by updating the same sites simultaneously using
the same functions. This coupling preserves the order thanks to the
monotonicity of $(f_{i})_{1\leq i\leq q}$. Besides, the updates given by
deterministic functions $f_{1}\equiv 1$ or $f_{2}\equiv-1$ can be carried out
without looking at the spins of any sites. We say that these updates are
_oblivious_. We will see that oblivious updates play a crucial role in our
proof.

######  Illustration by Example $1$.

For the flip-rate function considered in Example 1, one can verify that the
following functions and rates satisfy Proposition 7. For any
$x=(x(-1),x(0),x(1))\in\\{-1,1\\}^{\\{-1,0,1\\}}$,

$\displaystyle f_{1}(x)$ $\displaystyle=1,\;$ $\displaystyle\lambda_{1}$
$\displaystyle=(1-\gamma)^{2},$ $\displaystyle f_{2}(x)$ $\displaystyle=-1,\;$
$\displaystyle\lambda_{2}$ $\displaystyle=(1-\gamma)^{2},$ $\displaystyle
f_{3}(x)$ $\displaystyle=\text{sgn}(x(-1)+x(0)+x(1)),\;$
$\displaystyle\lambda_{3}$ $\displaystyle=4\gamma^{2},$ $\displaystyle
f_{4}(x)$ $\displaystyle=x(-1),\;$ $\displaystyle\lambda_{4}$
$\displaystyle=2(\gamma-\gamma^{2}),$ $\displaystyle f_{5}(x)$
$\displaystyle=x(1),\;$ $\displaystyle\lambda_{5}$
$\displaystyle=2(\gamma-\gamma^{2}).$

Here sgn denotes the sign function: $\text{sgn}(s)=\begin{cases}1&\text{if
$s>0$},\\\ 0&\text{if $s=0$},\\\ -1&\text{if $s<0$}.\end{cases}$

We first need the following lemma to prove Proposition 7.

######  Lemma 8 (Decomposition of monotone functions).

Let $\Omega$ be a finite set with a partial order $\prec$. Let
$f:\Omega\to\mathbb{R}_{+}$ be an increasing function, meaning that $x\prec
y\Rightarrow f(x)\leq f(y)$. Then there exist $q\in\mathbb{Z}_{+}$ and
increasing (not boolean) functions $f_{1},\dots,f_{q}:\Omega\to\\{0,1\\}$ and
positive numbers $\lambda_{1},\dots,\lambda_{q}$ such that

$\displaystyle f(\cdot)=\sum_{i=1}^{q}\lambda_{i}f_{i}(\cdot).$ (25)

Moreover, we can choose
$f_{1}(\cdot)=\mathds{1}_{\left\\{f(\cdot)>0\right\\}}$ and
$\lambda_{1}=\mathds{1}_{\left\\{f\not\equiv
0\right\\}}\min\limits_{\\{f(\cdot)>0\\}}f$.

Lemma 8 can be proved by induction on the size of the support of the function
$f$. We omit the proof here. Now we prove Proposition 7.

###### Proof of Proposition 7.

First we prove that (23) implies (24). Indeed, if (23) is true, then by
definition of $\hat{c}(\cdot)$, for any $u$ in $\Lambda_{L}$ and
$x\in\mathcal{X}$,

$\hat{c}(u,x)=\sum_{i=1}^{q}\lambda_{i}\mathds{1}_{\left\\{f_{i}(x_{u+\cdot})=-x(u)\right\\}}.$
(26)

Therefore, for any function $\varphi$ on $\mathcal{X}$,

$\displaystyle\mathcal{L}_{G}\varphi(x)$
$\displaystyle=\sum_{u\in\Lambda_{L}}\hat{c}(u,x)(\varphi(x^{u,-x(u)})-\varphi(x))$
$\displaystyle=\sum_{u\in\Lambda_{L}}\sum_{i=1}^{q}\lambda_{i}\mathds{1}_{\left\\{f_{i}(x_{u+\cdot})=-x(u)\right\\}}(\varphi(x^{u,-x(u)})-\varphi(x))$
$\displaystyle=\sum_{u\in\Lambda_{L}}\sum_{i=1}^{q}\lambda_{i}\mathds{1}_{\left\\{f_{i}(x_{u+\cdot})=-x(u)\right\\}}(\varphi(x^{u,f_{i}(x_{u+\cdot})})-\varphi(x))$
$\displaystyle=\sum_{u\in\Lambda_{L}}\sum_{i=1}^{q}\lambda_{i}(\varphi(x^{u,f_{i}(x_{u+\cdot})})-\varphi(x)),$

where the second equality is due to (26), the third equality is because
$x^{u,f_{i}(x_{u+\cdot})}=x^{u,-x(u)}$ whenever $f_{i}(x_{u+\cdot})=-x(u)$,
and the last equality is because
$\varphi(x^{u,f_{i}(x_{u+\cdot})})-\varphi(x)=0$ whenever
$f_{i}(x_{u+\cdot})=x(u)$. It remains to show (23).

Let $n:=\left|B(0,m)\right|-1.$ We identify the set ${B(0,m)}$ with
$\\{0,1,\dots,n\\}$ such that the site $0$ is identified with number $0$. Then
(23) is reformulated as follows: there exist increasing boolean functions
$\tilde{f}_{1},\dots,\tilde{f}_{q}$ on $\\{-1,1\\}^{n+1}$ and positive numbers
$\lambda_{1},\dots,\lambda_{q}$ such that, for any
$\xi_{0},\dots,\xi_{n}\in\\{-1,1\\}$,

$c(\xi_{0},\dots,\xi_{n})=\sum_{i=1}^{q}\lambda_{i}\mathds{1}_{\left\\{\tilde{f}_{i}(\xi_{0},\dots,\xi_{n})=-\xi_{0}\right\\}}.$
(27)

Note that Hypothesis 1 implies that $c$ is a positive function on
$\\{-1,1\\}^{n+1}$ such that $c(-1,\xi_{1},\dots,\xi_{n})$ is increasing and
$c(1,\xi_{1},\dots,\xi_{n})$ is decreasing as functions of
$(\xi_{1},\dots,\xi_{n})$.

We can now apply Lemma 8 to the function $c(-1,\cdot)$ to conclude that there
exist increasing functions $f_{1},\dots,f_{q_{1}}$ from $\\{-1,1\\}^{n}$ to
$\\{0,1\\}$ and positive numbers $\lambda_{1},\dots,\lambda_{q_{1}}$, for some
$q_{1}\in\mathbb{Z}_{+}$, such that, for any
$\xi_{1},\dots,\xi_{n}\in\\{-1,1\\}$,

$c(-1,\xi_{1},\dots,\xi_{n})=\sum_{i=1}^{q_{1}}\lambda_{i}f_{i}(\xi_{1},\dots,\xi_{n}).$
(28)

Moreover, we can choose

$\displaystyle f_{1}(\xi_{1},\dots,\xi_{n})$
$\displaystyle=\mathds{1}_{\left\\{c(-1,\xi_{1},\dots,\xi_{n})>0\right\\}}\equiv
1,$ $\displaystyle\lambda_{1}$
$\displaystyle=\min_{\xi_{1},\dots,\xi_{n}\in\\{-1,1\\}}c(-1,\xi_{1},\dots,\xi_{n})>0.$

Here, we have used the fact that $c(\cdot)$ only takes strictly positive
values. We then define the boolean functions $(\tilde{f}_{i})_{1\leq i\leq
q_{1}}$ on $\\{-1,1\\}^{n+1}$ by

$\tilde{f}_{i}(\xi_{0},\dots\xi_{n})=\begin{cases}1&\text{ if }\xi_{0}=1,\\\
2f_{i}(\xi_{1},\dots,\xi_{n})-1&\text{ if }\xi_{0}=-1.\end{cases}$

It is easy to see that $\tilde{f}_{i}$ is increasing for any $i$. Note also
that $\tilde{f}_{1}\equiv 1$.

Let $\varphi_{1}:\\{-1,1\\}^{n+1}\to\mathbb{R}_{+}$ be defined by, for any
$\xi_{0},\dots,\xi_{n}\in\\{-1,1\\}$,

$\varphi_{1}(\xi_{0},\dots,\xi_{n})=\sum_{i=1}^{q_{1}}\lambda_{i}\mathds{1}_{\left\\{\tilde{f}_{i}(\xi_{0},\dots,\xi_{n})=-\xi_{0}\right\\}}.$

Noting that
$\mathds{1}_{\left\\{\tilde{f}_{i}(1,\xi_{1}\dots,\xi_{n})=-1\right\\}}=0$ and
$\mathds{1}_{\left\\{\tilde{f}_{i}(-1,\xi_{1},\dots,\xi_{n})=1\right\\}}=\mathds{1}_{\left\\{f_{i}(\xi_{1},\dots,\xi_{n})=1\right\\}}=f_{i}(\xi_{1},\dots,\xi_{n})$,
we conclude that

$\displaystyle\varphi_{1}(1,\xi_{1},\dots,\xi_{n})$ $\displaystyle=0,$
$\displaystyle\varphi_{1}(-1,\xi_{1},\dots,\xi_{n})$
$\displaystyle=c(-1,\xi_{1},\dots,\xi_{n}).$

Similarly $c(1,\xi_{1},\dots,\xi_{n})$ is a positive decreasing function of
$(\xi_{1},\dots,\xi_{n})$. We can still apply Lemma 8, by replacing the
relation $\leq$ by $\geq$, to conclude that there exist decreasing functions
(increasing functions w.r.t relation $\geq$) $g_{1},\dots,g_{q_{2}}$ and
positive numbers $\lambda^{\prime}_{1},\dots,\lambda^{\prime}_{q_{2}}$ such
that

$c(1,\xi_{1},\dots,\xi_{n})=\sum_{i=1}^{q_{2}}\lambda^{\prime}_{i}g_{i}(\xi_{1},\dots,\xi_{n}).$
(29)

By a similar argument, we claim that the boolean functions
$(\tilde{f}_{q_{1}+i})_{1\leq i\leq q_{2}}$ defined by

$\tilde{f}_{q_{1}+i}(\xi_{0},\dots\xi_{n})=\begin{cases}1-2g_{i}(\xi_{1},\dots,\xi_{n})&\text{
if }\xi_{0}=1,\\\ -1&\text{ if }\xi_{0}=-1,\end{cases}$

are increasing. Similarly, we can choose $\lambda^{\prime}_{1}>0$ and
$g_{1}\equiv 1$. Hence $\tilde{f}_{q_{1}+1}\equiv-1$. Similarly, we define
$\varphi_{2}:\\{-1,1\\}^{n+1}\to\mathbb{R}_{+}$ by, for all
$\xi_{0},\dots,\xi_{n}\in\\{-1,1\\}$,

$\varphi_{2}(\xi_{0},\dots,\xi_{n})=\sum_{i=1}^{q_{2}}\lambda^{\prime}_{i}\mathds{1}_{\left\\{\tilde{f}_{q_{1}+i}(\xi_{0},\dots,\xi_{n})=-\xi_{0}\right\\}}.$

$\varphi_{2}$ satisfies

$\displaystyle\varphi_{2}(1,\xi_{1},\dots,\xi_{n})$
$\displaystyle=c(1,\xi_{1},\dots,\xi_{n}),$
$\displaystyle\varphi_{2}(-1,\xi_{1},\dots,\xi_{n})$ $\displaystyle=0.$

Hence

$c=\varphi_{1}+\varphi_{2}.$

We can set $\lambda_{q+i}=\lambda^{\prime}_{i},\;1\leq i\leq q_{2}$ to write

$c(\xi_{0},\dots,\xi_{n})=\sum_{i=1}^{q_{1}+q_{2}}\lambda_{i}\mathds{1}_{\left\\{\tilde{f}_{i}(\xi_{0},\dots,\xi_{n})=-\xi_{0}\right\\}}.$
(30)

We can renumber these functions and their coefficients such that
$\tilde{f}_{1}\equiv 1,\,\tilde{f}_{2}\equiv-1$. So we have established (27).
This finishes our proof. ∎

######  Remark 2.

We can adapt this proposition to _any_ flip rate function satisfying (5), (6),
on any graph. In this case, each site $u$ has its own update functions
$f_{u,1},\dots f_{u,q_{u}}$, which are increasing and applied at rate
$\lambda_{u,1},\dots,\lambda_{u,q_{u}}$ respectively. In our case, those
functions are local and invariant by translation because the function
$\hat{c}$ is.

From now on, we fix a choice of $(f_{i})_{1\leq i\leq q}$ and
$(\lambda_{i})_{1\leq i\leq q}$ that satisfies Proposition 7. For our purpose,
we will rewrite the reaction function $R$ and its derivative $R^{\prime}$ in
terms of $(f_{i})_{1\leq i\leq q}$ and $(\lambda_{i})_{1\leq i\leq q}$. To do
this, let $\lambda:=\sum_{i=1}^{q}\lambda_{i}$. For any $1\leq i\leq q$, $j\in
B(0,m)$, $\xi_{-m},\dots,\xi_{m}\in\\{-1,1\\}$, let us define

$\nabla_{j}f_{i}(\xi_{-m},\dots,\xi_{m}):=f_{i}(\xi_{-m},\dots,\xi_{j-1},1,\xi_{j+1},\dots,\xi_{m})-f_{i}(\xi_{-m},\dots,\xi_{j-1},-1,\xi_{j+1},\dots,\xi_{m}).$

######  Lemma 9 ($R$ and $R^{\prime}$ in terms of $(f_{i})_{1\leq i\leq q}$
and $(\lambda_{i})_{1\leq i\leq q}$).

For any $\rho\in[-1,1]$,

$\displaystyle R(\rho)$
$\displaystyle=\left(\sum_{i=1}^{q}\lambda_{i}\mathbb{E}_{\nu_{\rho}}\left[f_{i}\right]\right)-\lambda\rho,$
(31) $\displaystyle R^{\prime}(\rho)$
$\displaystyle=\left(\sum_{i=1}^{q}\sum_{j\in
B(0,m)}\lambda_{i}\mathbb{E}_{\nu_{\rho}}\left[\dfrac{1}{2}\nabla_{j}f_{i}\right]\right)-\lambda.$
(32)

###### Proof.

Recall that $R(\rho)=\mathbb{E}_{\nu_{\rho}}\left[-2\xi_{0}c(\xi)\right]$.
Hence, by (23).

$\displaystyle R(\rho)$
$\displaystyle=\mathbb{E}_{\nu_{\rho}}\left[-2\xi_{0}\sum_{i=1}^{q}\lambda_{i}\mathds{1}_{\left\\{f_{i}(\xi)=-\xi_{0}\right\\}}\right]$
$\displaystyle=\sum_{i=1}^{q}\lambda_{i}\mathbb{E}_{\nu_{\rho}}\left[-2\xi_{0}\mathds{1}_{\left\\{f_{i}(\xi)=-\xi_{0}\right\\}}\right]$
$\displaystyle=\sum_{i=1}^{q}\lambda_{i}\mathbb{E}_{\nu_{\rho}}\left[(f_{i}(\xi)-\xi_{0})\mathds{1}_{\left\\{f_{i}(\xi)=-\xi_{0}\right\\}}\right]$
$\displaystyle=\sum_{i=1}^{q}\lambda_{i}\mathbb{E}_{\nu_{\rho}}\left[f_{i}(\xi)-\xi_{0}\right]$
$\displaystyle=\left(\sum_{i=1}^{q}\lambda_{i}\mathbb{E}_{\nu_{\rho}}\left[f_{i}(\xi)\right]\right)-\lambda\rho.$

So we have established (31). We also know that, for any $\rho\in[-1,1]$,

$\mathfrak{R}_{\rho}=\dfrac{1+\rho}{2}\delta_{1}+\dfrac{1-\rho}{2}\delta_{-1},$

and hence

$\dfrac{d}{d\rho}\mathfrak{R}_{\rho}=\dfrac{1}{2}\delta_{1}-\dfrac{1}{2}\delta_{-1}.$

Therefore, for $\nu_{\rho}=\mathfrak{R}_{\rho}^{\otimes B(0,m)}$,

$\dfrac{d}{d\rho}\nu_{\rho}=\sum_{j\in
B(0,m)}\mathfrak{R}_{\rho}^{\otimes\\{-m,\dots,j-1\\}}\otimes\left(\dfrac{1}{2}\delta_{1}-\dfrac{1}{2}\delta_{-1}\right)\otimes\mathfrak{R}_{\rho}^{\otimes\\{j+1,\dots,m\\}}.$

This equality and (31) lead to (32). ∎

##  4 Dual coupling

###  4.1 Graphical Construction

We call $\Lambda_{L}\times\mathbb{R}_{+}$ the _space-time slab_ , where each
slice $\Lambda_{L}\times\\{t\\}$ is equipped with the graph structure of
$\Lambda_{L}$. We identify $\Lambda_{L}$ with the subset $\\{0,\dots,L-1\\}$
of $\mathbb{Z}_{+}$.

We introduce a notation essential to our proof.

#### A collection of marks.

A collection of marks $\mathcal{C}$ is a pair
$(\mathcal{C}^{\text{exclusion}},\mathcal{C}^{\text{glauber}})$, where
$\mathcal{C}^{\text{exclusion}}$ and $\mathcal{C}^{\text{glauber}}$ are
subsets of $\Lambda_{L}\times\mathbb{R}_{+}$ and
$[q]\times\Lambda_{L}\times\mathbb{R}_{+}$, respectively, that satisfy the
following conditions.

  * •

There is at most one mark at a time:

$\displaystyle\forall
t>0,\,\left|\mathcal{C}^{\text{exclusion}}\cap\Lambda_{L}\times\\{t\\}\right|+\left|\mathcal{C}^{\text{glauber}}\cap[q]\times\Lambda_{L}\times\\{t\\}\right|\leq
1.$

  * •

$\mathcal{C}$ is locally finite in time: for any $0<t_{1}<t_{2}$,

$\displaystyle\left|\mathcal{C}^{\text{exclusion}}\cap\Lambda_{L}\times[t_{1},t_{2}]\right|+\left|\mathcal{C}^{\text{glauber}}\cap[q]\times\Lambda_{L}\times[t_{1},t_{2}]\right|<\infty.$

In other words, $\mathcal{C}$ can be identified with a counting measure on
$(\Lambda_{L}\times\mathbb{R}_{+})\cup([q]\times\Lambda_{L}\times\mathbb{R}_{+})$
whose projection onto $\mathbb{R}_{+}$ has no multiple point and is locally
finite. We call an element $(u,t)\in\mathcal{C}^{\text{exclusion}}$ the
exclusion mark on the edge $(u,u+1)$ at time $t$ and an element
$(i,u,t)\in\mathcal{C}^{\text{glauber}}$ the Glauber mark of type $i$ on site
$u$ at time $t$.

#### Effect of a collection of marks.

Given a collection of marks $\mathcal{C}$ and a configuration
$x_{0}\in\mathcal{X}$. The process $(X^{x_{0}}_{t})_{t\geq 0}$ is defined
recursively according to the following rules.

  * •

$X^{x_{0}}_{0}=x_{0}$ ($x_{0}$ is the initial configuration).

  * •

The process $(X^{x_{0}}_{t})_{t\geq 0}$ is piecewise constant and can only
jump when a mark appears.

  * •

When we see an exclusion mark on the edge $(u,u+1)$, we make the transition
$x\mapsto x^{u\leftrightarrow u+1}$ (exchange the spins at sites $u$ and
$u+1$).

  * •

When we see a Glauber mark of type $i$ at site $u$, we make the transition
$x\mapsto x^{u,f_{i}(x_{u+\cdot})}$ (update site $u$ using the function
$f_{i}$).

Roughly speaking, the marks tell us how the process $(X^{x_{0}}_{t})_{t\geq
0}$ is updated. We have kept the dependency on $\mathcal{C}$ implicit to
lighten the notation. The Glauber-Exclusion process associated with the
generator $\mathcal{L}_{GE}$ can be constructed as follows.

#### Graphical Construction 1.

Let the background process

$\Xi=\left((\Xi^{\text{exclusion}}_{u})_{0\leq u\leq
L-1},(\Xi^{\text{Glauber}}_{i,u})_{1\leq i\leq q,\,0\leq u\leq L-1}\right)$
(33)

be defined as follows. $(\Xi^{\text{exclusion}}_{u})_{0\leq u\leq
L-1},(\Xi^{\text{Glauber}}_{i,u})_{1\leq i\leq q,\,0\leq u\leq L-1}$ are
independent homogeneous Poisson processes, and

  * •

$\Xi^{\text{exclusion}}_{u}$ is of intensity $L^{2}$, $0\leq u\leq L-1$,

  * •

$\Xi^{\text{Glauber}}_{i,u}$ is of intensity $\lambda_{i}$, $1\leq i\leq
q,\,0\leq u\leq L-1$.

Almost surely, a realization of $\Xi$ naturally defines a collection of marks
$\mathcal{C}$ as follows. For any
$(i,u,t)\in[q]\times\Lambda_{L}\times\mathbb{R}_{+}$,

  * •

$(u,t)\in\mathcal{C}^{\text{exclusion}}$ iff the process
$\Xi^{\text{exclusion}}_{u}$ jumps at time $t$,

  * •

$(i,u,t)\in\mathcal{C}^{\text{glauber}}$ iff the process
$\Xi^{\text{Glauber}}_{i,u}$ jumps at time $t$.

Then the process $(X^{x_{0}}_{t})_{t\geq 0}$ constructed by the rules above is
a Markov process with generator $\mathcal{L}^{GE}$ starting from configuration
$x_{0}$. An illustration is given in Figure 2, where each exclusion mark is
naturally drawn as a left-right arrow on the corresponding edge, and each
Glauber mark is drawn as $\times$ on the corresponding site, with the
corresponding update function next to it.

######  Remark 3.

In the future, we may construct the (random) collection of marks $\mathcal{C}$
differently, but as long as it has the same distribution as the one given by
$\Xi$, the process $(X^{x_{0}}_{t})_{t\geq 0}$ constructed by the above rules
is still a Markov process associated with generator $\mathcal{L}^{GE}$
starting at $x_{0}$.

#### The alphabet.

Let $\mathcal{A}:=\bigcup\limits_{i=0}^{\infty}(\mathbb{Z}_{+}\times
B(0,m)^{i})$ be the set of all words beginning with an element in
$\mathbb{Z}_{+}$ followed by elements in $B(0,m)$. We define a lexicographical
order on $\mathcal{A}$ using the usual orders on $\mathbb{Z}_{+}$ and
$B(0,m)$. For any word $\mathsf{w}\in\mathcal{A}$, we call the elements of
$\\{\mathsf{w}\\}\times B(0,m)$ the children of $\mathsf{w}$. The alphabet
$\mathcal{A}$ will be used to label the branching processes that naturally
arise when we study the _update history_ of our process.

#### Update history.

Consider a collection of marks $\mathcal{C}$ and the resulting process
$(X^{x}_{t})_{t\geq 0}$ for some $x\in\mathcal{X}$. Let $t\geq 0$, and let
$E\subset\Lambda_{L}$ be a set whose spins we want to know at time $t$. A
natural way to reveal the spins of $E$ at time $t$ is to go backward in time,
tracing the history of the information involved. At each site of $E$ at time
$t$, we put a particle labeled by the corresponding element in $\mathcal{A}$
(we have $E\subset\Lambda_{L}\subset\mathbb{Z}_{+}\subset\mathcal{A}$). The
history of $E$, going backward in time, is as follows.

  1. 1.

Every time we see an exclusion mark, the particles at the endpoints of the
corresponding edge, if there are any, jump to the other endpoints.

  2. 2.

Every time we see a Glauber mark, say at site $u$, if there is a particle
$\mathsf{w}$ there, we remove $\mathsf{w}$, and at each site of $B(u,m)$, we
add a new particle. We canonically label the newborn particles by the children
of $\mathsf{w}$. If a particle is born on a site already occupied by some
other particles, all of them will move together as one particle.

We follow all the particles backward until time $0$. The _update history_ of
the set $E$ at time $t$ is the sub-collection of marks $\mathcal{C}_{his}(E)$
consisting of all the marks (exclusion and Glauber) we meet on the
trajectories of all the particles described above. For our purpose, it is more
convenient to consider $\mathcal{C}^{bw}_{his}(E)$, the version of
$\mathcal{C}_{his}(E)$ when we reverse the time interval $[0,t]$:
$\mathcal{C}^{bw}_{his}(E)$ is the set of marks obtained from
$\mathcal{C}_{his}(E)$ by replacing the time $s$ in each mark by $t-s$.

We define a spin configuration $\mathfrak{s}$ on the set of all particles
appearing as follows.

  * •

For any particle $\mathsf{w}$ reaching time $0$, say at site $u$,

$\displaystyle\mathfrak{s}_{\mathsf{w}}:=x(u).$

  * •

For any particle $\mathsf{w}$ not reaching time $0$, let $i$ be the type of
the Glauber mark removing $\mathsf{w}$. Then

$\displaystyle\mathfrak{s}_{\mathsf{w}}:=f_{i}((\mathfrak{s}_{\mathsf{w}^{\prime}})_{\mathsf{w}^{\prime}\in\\{\mathsf{w}\\}\times
B(0,m)}).$

These rules allow us to define recursively the spins of all particles that
appear. In particular, by construction,

$\displaystyle\forall u\in E,\,\mathfrak{s}_{u}=X^{x}_{t}(u),$

where $\mathfrak{s}_{u}$ is the spin of the particle labeled $u$.

Figure 2: History of Glauber-Exclusion dynamics.

###  4.2 The branching exclusion process (BEP)

We fix a subset $E$ of $\Lambda_{L}$ in this subsection. We claim that the
history itself, viewed backward in time, is a Markov process, called the
_branching exclusion process_ (BEP), defined as follows.

#### Definition of the BEP.

We consider the following way of putting marks and particles in the space-time
slab. We go forward in time.

  1. 1.

At time $0$, at each site of $E$, we put a particle labeled by the
corresponding element of $\mathcal{A}$. Each particle is associated with $q$
Poisson clocks, called the Glauber clocks of type $1,\dots,q$, of intensity
$\lambda_{1},\dots,\lambda_{q}$, respectively. All the Glauber clocks are
independent.

  2. 2.

Each edge of the lattice is associated with a Poisson clock of intensity
$L^{2}$, called the exclusion clock. The exclusion clocks are independent and
independent of the Glauber clocks. Each time an exclusion clock rings, we put
an exclusion mark on the corresponding edge.

  3. 3.

The particles follow the exclusion marks until one Glauber clock rings, say
the clock of type $i$ of particle $\mathsf{w}$ at time $s$. Suppose that
$\mathsf{w}$ is at site $u$ at time $s-$. Then we put a Glauber mark of type
$i$ on site $u$ at time $s$. After that, we remove particle $\mathsf{w}$ and
add new particles to all sites in $B(u,m)$, canonically labeled by the
children of $\mathsf{w}$. We think of this as the particle $\mathsf{w}$ splits
into its neighbor sites. Each particle born on a non-occupied site is given
$q$ independent Glauber clocks, independent of all exclusion clocks and all
Glauber clocks of other particles. Each particle born on an occupied site
“moves together” with the particles already there. The particles moving
together are called a group of particles, and they are associated with the
same $q$ Glauber clocks. All the particles then continue to evolve with
similar rules.

The BEP starting from $E$ consists of two parts
$(\Psi_{BEP}(E),\mathcal{C}_{BEP}(E))$, where

  * •

$\Psi_{BEP}(E)$ is the collection of trajectories of all the particles, i.e.
the map associating each point $(u,t)$ of the space-time slab with the set of
particles occupying the site $u$ at time $t$.

  * •

$\mathcal{C}_{BEP}(E)$ is the set consisting of all the marks on the
trajectories of all the particles constructed above.

######  Remark 4.

Knowing $E$, the pair $(\Psi_{BEP}(E),\mathcal{C}_{BEP}(E))$ is determined by
$\mathcal{C}_{BEP}(E)$.

We say that a particle $\mathsf{w}$ is _alive_ at time $t$ if it was born
before time $t$ and has not been removed by time $t$. Let $W_{t}$ be the set
consisting of the particles with the smallest labels in each group of
particles alive at time $t$, for any $t\in\mathbb{R}_{+}$. A spin
configuration on $W_{t}$ is identified with a spin configuration on the set of
all particles alive at time $t$, where the particles in a group are given the
same spin.

#### Spins of particles in BEP.

Each spin configuration $\mathfrak{s}$ on $W_{t}$ is extended to
$\bigcup_{0\leq s\leq t}W_{s}$ by the following rules. For any
$\mathsf{w}\notin W_{t}$, $\mathsf{w}$ must be removed at some point in
$[0,t]$ by a Glauber mark, say of type $i$. Then

$\displaystyle\mathfrak{s}_{\mathsf{w}}:=f_{i}\left((\mathfrak{s}_{\mathsf{w}^{\prime}})_{\mathsf{w}^{\prime}\in\\{\mathsf{w}\\}\times
B(0,m)}\right).$

This allows us to extend the spin configuration to all the particles alive at
some point in the time interval $[0,t]$.

By the homogeneity of the Poisson processes, we can see that
$\mathcal{C}_{BEP}(E)$ and $\mathcal{C}_{his}^{bw}(E)$ have the same
distribution. In fact, we can show that their first marks have the same
distribution, then conditionally on their first marks, their second marks have
the same distribution, etc. This leads to the following result.

######  Proposition 10 (History is BEP).

Let $t\in\mathbb{R}_{+}$, $x\in\mathcal{X}$. Consider the BEP starting from
$E$ and the history of $E$ at time $t$ given by a collection of mark generated
by $\Xi$. Then the point processes $\mathcal{C}_{BEP}(E)$ and
$\mathcal{C}_{his}^{bw}(E)$ have the same distribution. In particular, for
some $x\in\mathcal{X}$, if we define the spin configuration $\mathfrak{s}$ on
$W_{t}$ by

$\displaystyle\forall\mathsf{w}\in
W_{t},\;\mathfrak{s}_{\mathsf{w}}:=x(u),\text{ if $\mathsf{w}$ is at site $u$
at time $t$},$

then

$X^{x}_{t}(E)\overset{d}{=}(\mathfrak{s}_{u})_{u\in E}.$ (34)

######  Example 3.

Figure 2 also illustrates the BEP starting from $E=\\{1,2\\}$. The time axis
upward is the one of the Glauber-Exclusion process, and the one downward is
the one of the BEP.

The idea of looking at the history backward in time is not new. It is known as
_duality_ in the context of interacting particle systems. For our case, the
BEP was already introduced by De Masi et al in the original paper [10] (for
the proof of Proposition 10, see Theorem 3.1 in [10]). The primary motivation
is that it makes the study of correlation functions easier. To study the
$k-$points correlation function, we only need to follow the history of $k$
particles backward instead of realizing the whole process forward. This idea
also relates to the famous “coupling from the past” algorithm in [53] and is
also an essential ingredient of the information percolation framework in [42].
Proposition 10 says that the history itself can be viewed as a Markov process
(the BEP). This will be useful for the construction of our coupling.

We introduce some further necessary notations.

#### Update functions.

By construction, for any particle $\mathsf{w}$ alive at some point in the time
interval $[0,t]$, there exists an increasing boolean function
$F_{\mathsf{w},t}$ on $\\{-1,1\\}^{W_{t}}$ such that for any spin
configuration $\mathfrak{s}$ on $W_{t}$, its extension to $\bigcup_{0\leq
s\leq t}W_{s}$ satisfies

$\mathfrak{s}_{\mathsf{w}}=F_{\mathsf{w},t}((\mathfrak{s}_{\mathsf{w}^{\prime}})_{\mathsf{w}^{\prime}\in
W_{t}}).$ (35)

The function $F_{\mathsf{w},t}$ describes the dependence of the spin of
$\mathsf{w}$ on the spins of the particles alive at time $t$.

#### Pivotal sets of the update functions.

For any $t\geq 0$, for any particle $\mathsf{w}$ that is alive at some time
$s<t$, we denote by

$\mathsf{Piv}\left(\mathsf{w},t\right):=\mathsf{Piv}\left(F_{\mathsf{w},t}\right).$
(36)

We write $\mathsf{Piv}\left(E,t\right)$ for $\bigcup_{\mathsf{w}\in
E}\mathsf{Piv}\left(\mathsf{w},t\right)$.

######  Observation 11.

For any $s<t$,

$\mathsf{Piv}\left(\mathsf{w},t\right)\subset\bigcup_{\mathsf{w}^{\prime}\in\mathsf{Piv}\left(\mathsf{w},s\right)}\mathsf{Piv}\left(\mathsf{w}^{\prime},t\right).$
(37)

#### Update times of the pivotal set.

Let the sequence $(T_{j})_{j\geq 0}$ be defined recursively, as follows.

$\displaystyle T_{0}$ $\displaystyle=0,$ $\displaystyle T_{j+1}$
$\displaystyle=\inf\left\\{t\geq T_{j}:\text{a Glauber clock of a particle in
$\mathsf{Piv}\left(E,T_{j}\right)$ rings at time $t$}\right\\}.$

######  Observation 12.

$(\mathsf{Piv}\left(E,t\right))_{t\geq 0}$ is piece-wise constant and only
jumps at the times $(T_{j})_{j>0}$. The same conclusion is true for
$(F_{\mathsf{w},t})_{t\geq 0}$, for any $\mathsf{w}\in E$.

#### Information percolation.

We now briefly explain the idea that we borrow from [42], which will be made
more precise later. First, to determine the spin of $\mathsf{w}\in E$, we do
not need to follow the whole history, but only the part that is “essential”,
say $(\mathsf{Piv}\left(\mathsf{w},s\right))_{0\leq s\leq t}$. In particular,
if $\mathsf{Piv}\left(\mathsf{w},s\right)=\O$ for some $s<t$, then the update
function $F_{\mathsf{w},s}$ degenerates, i.e. becomes a constant function.
Then, we can determine $\mathfrak{s}_{\mathsf{w}}$ without tracking further
the history in the past. Second, we regard
$(\mathsf{Piv}\left(\mathsf{w},t\right))_{t\geq 0}$ as a percolation cluster
in the space-time slab. Surprisingly, we prove in Lemma 17 below that the
information percolation cluster defined this way is “subcritical” in the full
high-temperature regime. This subcriticality means that the update functions
quickly degenerate, so the spins at time $t$ quickly become independent of
those at time $0$ as $t$ grows to infinity, and the system mixes quickly.

###  4.3 The idealized branching process (IBP)

We will compare the BEP with its idealized version, the _idealized branching
process_ (IBP). The IBP starting from a particle $\mathsf{w}\in\mathcal{A}$ is
defined as follows.

  * •

Initially, $\mathsf{w}$ has $q$ independent Poisson clocks, of intensities
$\lambda_{1},\dots,\lambda_{q}$, called the Glauber clocks of type $1,\dots,q$
respectively.

  * •

Each time a clock of a particle $\mathsf{w}^{\prime}$ rings, we remove
$\mathsf{w}^{\prime}$ and add the children of $\mathsf{w}^{\prime}$ into the
set of particles. Each of them is given $q$ independent Glauber clocks, and
the clocks of different particles are independent.

The IBP starting from $\mathsf{w}$ is a pair $(\Psi_{IBP},\mathcal{C}_{IBP})$,
where

  * •

$\Psi_{IBP}$ is the process associating each time $t\in\mathbb{R}_{+}$ with
the set of particles alive at time $t$, i.e. the set of particles born before
time $t$ and not having been removed by time $t$.

  * •

$\mathcal{C}_{IBP}$ is the counting measure on $[q]\times\mathbb{R}_{+}$ that
tracks the type of the Glauber clocks that ring.

######  Remark 5.

In the IBP, the particles no longer live on the lattice. All the particles are
different, and all their clocks are independent. In particular, it is a
dimension-free object, i.e. it does not depend on the size $L$ of the system.

For any finite subset $E$ of $\mathcal{A}$ such that for all
$\mathsf{w}_{1},\mathsf{w}_{2}\in E$, $\mathsf{w}_{1}$ is not a descendant of
$\mathsf{w}_{2}$, the IBP starting from $E$ consists of $|E|$ independent IBPs
starting from particles labeled by the elements of $E$.

#### Spins and update functions of the IBP.

Let $\widetilde{W}_{t}$ be the set of particles of the IBP alive at time $t$.
Similarly to what we have done for the BEP, for any spin configuration
$\tilde{\mathfrak{s}}$ on $\widetilde{W}_{t}$, we extend
$\tilde{\mathfrak{s}}$ to $\bigcup_{0\leq s\leq t}\widetilde{W}_{t}$ by
defining for each $\mathsf{w}\notin\widetilde{W}_{t}$,

$\tilde{\mathfrak{s}}_{\mathsf{w}}:=f_{i}((\tilde{\mathfrak{s}}_{\mathsf{w}^{\prime}})_{\mathsf{w}^{\prime}\in\\{\mathsf{w}\\}\times
B(0,m)}).$

We can extend $\tilde{\mathfrak{s}}$ to all particles alive at some time $s<t$
with this procedure. In particular, we can define the spin of $\mathsf{w}$. We
define the update functions $\tilde{F}_{\mathsf{w},t}$ , the pivotal sets
$(\widetilde{\mathsf{Piv}}(\tilde{F}_{\mathsf{w},t}))_{t\geq 0}$, and the
update times $(\tilde{T}_{j})_{j\in\mathbb{Z}_{+}}$ analogously as for the
BEP.

###  4.4 The coupling

Let $E$ be a subset of $\Lambda_{L}$. We will construct a coupling between the
BEP and the IBP starting from $E$. Let there be one exclusion clock at each
edge of the lattice and an infinite number of Glauber clocks of types
$1,\dots,q$. All of them are independent. We will use these clocks to
construct the two processes. We say that the coupling is _succesful_ until
time $t$ if $\mathsf{Piv}\left(E,t\right)=\widetilde{\mathsf{Piv}}(E,t)$, and
for each $\mathsf{w}\in\mathsf{Piv}\left(E,t\right)$, the Glauber clocks
associated with it at time $t$ are the same in the BEP and the IBP. We now
construct the coupling, which should be thought of as embedding the IBP in the
space-time slab $\Lambda_{L}\times\mathbb{R}_{+}$.

  1. 1.

Whenever an exclusion clock rings, we put an exclusion mark at the
corresponding edge. When a particle of the BEP meets an exclusion mark, it
jumps to the other endpoint of the edge.

  2. 2.

At time $0$, each particle $\mathsf{w}\in E$ is associated with the same $q$
Glauber clocks in the BEP and the IBP.

  3. 3.

For each particle, the set of clocks associated with it stays unchanged in the
interval $(T_{j},T_{j+1}]$, where $(T_{j})_{j\geq 0}$ are the update times of
$\mathsf{Piv}\left(E,\cdot\right)$.

  4. 4.

Suppose that the coupling is still successful until time $T_{j}$. Then
$T_{j+1}$ corresponds to the ring of a Glauber clock of some particle
$\mathsf{w}\in\mathsf{Piv}\left(E,T_{j}\right)$. Note that
$T_{j+1}=\tilde{T}_{j+1}$ because the same clock of $\mathsf{w}$ rings in the
IBP. Suppose that $\mathsf{w}$ is at site $u$ at time $T_{j+1}-$. Then one of
two cases can happen.

    1. (a)

Case 1: At time $T_{j+1}-$, there is no particle in $B(u,m)$ except
$\mathsf{w}$. This means that the children of $\mathsf{w}$ are born on sites
not occupied by other particles in the pivotal set. Then we update
$\mathsf{Piv}\left(E,T_{j+1}\right)$ and
$\widetilde{\mathsf{Piv}}(E,\tilde{T}_{j+1})$: we let the particles in
$\mathsf{Piv}\left(E,T_{j+1}-\right)\setminus\\{\mathsf{w}\\}$ keep their
clocks (in IBP and BEP), and we remove from
$\mathsf{Piv}\left(E,T_{j+1}-\right)$ and
$\widetilde{\mathsf{Piv}}(E,\tilde{T}_{j+1}-)$ the particles whose spins are
no longer necessary to determine the spins atop, and then we associate the
children of $\mathsf{w}$ in BEP and IBP with the same Glauber clocks. Note
that
$\mathsf{Piv}\left(E,T_{j+1}\right)\subset(\mathsf{Piv}\left(E,T_{j+1}-\right)\setminus\\{\mathsf{w}\\})\cup(\\{\mathsf{w}\\}\times
B(0,m))$. In this case, the coupling is still successful until time $T_{j+1}$.

    2. (b)

Case 2: At time $T_{j+1}-$, there is some particles in $B(u,m)$ other than
$\mathsf{w}$. Then from time $T_{j+1}$, all the particles in the two pivotal
sets are given different Poisson clocks. The coupling becomes unsuccessful
since then.

We say that the coupling is successful until infinity if it is successful
until $t$ for any finite $t$. The interest in the above coupling is the
following.

######  Observation 13 (Spins atop coincide).

If the coupling is successful until time $t$, and if
$\mathfrak{s}_{\mathsf{w}}=\tilde{\mathfrak{s}}_{\mathsf{w}},\;\forall\mathsf{w}\in\mathsf{Piv}\left(E,t\right)$,
then
$\mathfrak{s}_{\mathsf{w}}=\tilde{\mathfrak{s}}_{\mathsf{w}},\;\forall\mathsf{w}\in
E$.

We will show that the coupling is successful until infinity with high
probability.

######  Proposition 14 (The coupling is successful w.h.p).

Let $E$ be an arbitrary subset of $\Lambda_{L}$. Consider the coupling of the
BEP and the IBP starting from $E$ described above. Then

$\mathbb{P}\left[\text{The coupling is not successful until
infinity}\right]\leq\beta\dfrac{|E|^{2}}{L},$ (38)

for some constant $\beta$.

The motivation for the coupling is as follows. We are inspired by the local
equilibrium phenomenon, so we try to couple the BEP and the IBP so that when
we generate the spins of the particles at time $t$ in the two processes
according to the same distribution (assuming that those sets are the same),
then the spins atop are the same. If this is the case, as the spins atop of
the IBP are independent, the spins atop of the BEP are also independent. The
naive coupling is to match every particle in the two processes. However, the
total number of particles increases exponentially fast, so we can not expect
the naive coupling to work until the mixing times. Nevertheless, it is _not
necessary_ to match the evolution of every particle, but only the ones that
truly influence the spins atop, say $\mathsf{Piv}\left(E,\cdot\right)$.

The idea to look only at $(\mathsf{Piv}\left(E,t\right))_{t\geq 0}$ ultimately
comes from [42] (in this paper, the authors call it the update support
function, see Subsection $2.2$ in [42]). In Subsection 4.5, we analyze the
IBP, and in particular, we prove that $\widetilde{\mathsf{Piv}}(\cdot)$
behaves like a subcritical Galton Watson process. In Subsection 4.6, we use
this to prove Proposition 14.

###  4.5 Analysis of the IBP

In this subsection, we consider the IBP starting from a particle
$\mathsf{w}\in\mathcal{A}$. For convenience, we identify the set of children
of $\mathsf{w}$ with $B(0,m)$. Let the functions
$\phi,\psi:\;\mathbb{R}_{+}\to\mathbb{R}$ be defined by

$\displaystyle\phi(t)$
$\displaystyle:=\mathbb{P}\left[\widetilde{\mathsf{Piv}}(\mathsf{w},t)\neq\O\right],$
(39) $\displaystyle\psi(t)$
$\displaystyle:=\mathbb{E}\left[|\widetilde{\mathsf{Piv}}(\mathsf{w},t)|\right],$
(40)

which means that $\phi(t)$ is the probability that the process
$\widetilde{\mathsf{Piv}}(\mathsf{w},\cdot)$ still survives at time $t$ and
$\psi(t)$ is the average size of $\widetilde{\mathsf{Piv}}(\mathsf{w},t)$. The
main purpose of this subsection is to analyze several aspects of the IBP and
in particular, to prove that $\psi(t)\sim e^{R(\rho_{*})t}$.

The average size of $\widetilde{\mathsf{Piv}}(\mathsf{w},t)$ conditionally on
survival is

$\mathbb{E}\left[|\widetilde{\mathsf{Piv}}(\mathsf{w},t)|\big{|}\widetilde{\mathsf{Piv}}(\mathsf{w},t)\neq\O\right]=\dfrac{\psi(t)}{\phi(t)}.$
(41)

Note that if $\widetilde{\mathsf{Piv}}(\mathsf{w},t)=\O$, then the spin of
$\mathsf{w}$ does not depend on the spins at time $t$ anymore, so we can
safely write

$\vartheta(t)=\mathbb{E}\left[\tilde{\mathfrak{s}}_{\mathsf{w}}\big{|}\widetilde{\mathsf{Piv}}(\mathsf{w},t)=\O\right]$
(42)

for the average spin of $\mathsf{w}$ conditionally on the extinction of
$\widetilde{\mathsf{Piv}}(\mathsf{w},t)$. Clearly, the functions
$\psi,\phi,\vartheta$ do not depend on our choice of
$\mathsf{w}\in\mathcal{A}$.

First, we have a result linking the average of
$\tilde{\mathfrak{s}}_{\mathsf{w}}$ with the ODE (9).

######  Lemma 15 (Spin atop of an IBP).

Let $\rho_{0}\in[-1,1]$ and $t\geq 0$. Consider the IBP starting from
$\mathsf{w}$. Suppose that, conditionally on the IBP, the spins at time $t$
are generated independently by a product of Rademacher:
$\tilde{\mathfrak{s}}(\widetilde{W}_{t})\sim\mathfrak{R}_{\rho_{0}}^{\otimes\widetilde{W}_{t}}$.
Then

$\mathbb{E}\left[\tilde{\mathfrak{s}}_{\mathsf{w}}\right]=\rho(t),$ (43)

where $\rho$ is the solution of equation (9).

(a) $\zeta>t$

(b) $\zeta<t$

Figure 3: Conditionally on the first ring

###### Proof.

Let $g(t):=\mathbb{E}\left[\tilde{\mathfrak{s}}_{\mathsf{w}}\right]$. Let
$\zeta$ be the first time that a Glauber clock of $\mathsf{w}$ rings. Let
$\mathcal{E}(i)$ be the event that the Glauber clock ringing at time $\zeta$
is of type $i$. Then $\zeta$ and $\mathcal{E}(i)$ are independent, and
$\zeta\sim\exp(\lambda),\;\mathbb{P}\left[\mathcal{E}(i)\right]=\lambda_{i}/\lambda$.
Moreover, the branches starting from the children of $\mathsf{w}$ are
independent and independent of $\zeta$ and $\mathcal{E}(i)$, each behaving as
an IBP. We see that

$\displaystyle
g(t)=\mathbb{E}\left[\mathds{1}_{\left\\{\zeta>t\right\\}}\tilde{\mathfrak{s}}_{\mathsf{w}}\right]+\mathbb{E}\left[\mathds{1}_{\left\\{\zeta\leq
t\right\\}}\tilde{\mathfrak{s}}_{\mathsf{w}}\right].$ (44)

We estimate the two terms on the right-hand side of (6.1) separately.

  * •

First term: conditionally on $\zeta$ (see Figure 3), on the event
$\\{\zeta>t\\}$,
$\tilde{\mathfrak{s}}_{\mathsf{w}}\sim\mathfrak{R}_{\rho_{0}}$, and hence

$\displaystyle\mathbb{E}\left[\mathds{1}_{\left\\{\zeta>t\right\\}}\tilde{\mathfrak{s}}_{\mathsf{w}}\right]=\mathbb{E}\left[\mathds{1}_{\left\\{\zeta>t\right\\}}\mathbb{E}\left[\tilde{\mathfrak{s}}_{\mathsf{w}}|\zeta\right]\right]=\mathbb{E}\left[\mathds{1}_{\left\\{\zeta>t\right\\}}\times\rho_{0}\right]=\rho_{0}\mathbb{P}\left[\zeta>t\right]=\rho_{0}e^{-\lambda
t}.$

  * •

Second term: we see that

$\displaystyle\mathbb{E}\left[\mathds{1}_{\left\\{\zeta\leq
t\right\\}}\tilde{\mathfrak{s}}_{\mathsf{w}}\right]$
$\displaystyle=\sum_{i=1}^{q}\mathbb{E}\left[\mathds{1}_{\mathcal{E}(i)}\mathds{1}_{\left\\{\zeta\leq
t\right\\}}f_{i}((\tilde{\mathfrak{s}}_{\mathsf{w}^{\prime}})_{\mathsf{w}^{\prime}\in\\{\mathsf{w}\\}\times
B(0,m)})\right]$
$\displaystyle=\sum_{i=1}^{q}\mathbb{P}\left[\mathcal{E}(i)\right]\mathbb{E}\left[\mathds{1}_{\left\\{\zeta\leq
t\right\\}}f_{i}\left((\tilde{\mathfrak{s}}_{\mathsf{w}^{\prime}})_{\mathsf{w}^{\prime}\in\\{\mathsf{w}\\}\times
B(0,m)}\right)\right],$

since $\mathcal{E}(i)$ is independent of $\zeta$ and of the IBPs starting from
the children of $\mathsf{w}$. Conditionally on $\zeta$, on the event
$\\{\zeta<t\\}$, by construction,
$(\tilde{\mathfrak{s}}_{\mathsf{w}^{\prime}})_{\mathsf{w}^{\prime}\in\\{\mathsf{w}\\}\times
B(0,m)}$ are i.i.d. $\mathfrak{R}_{g(t-\zeta)}$. Hence

$\displaystyle\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}f_{i}\left((\tilde{\mathfrak{s}}_{\mathsf{w}^{\prime}})_{\mathsf{w}^{\prime}\in\\{\mathsf{w}\\}\times
B(0,m)}\right)\right]=\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\mathbb{E}_{\nu_{g(t-\zeta)}}\left[f_{i}\right]\right].$

Hence

$\displaystyle\mathbb{E}\left[\mathds{1}_{\left\\{\zeta\leq
t\right\\}}\tilde{\mathfrak{s}}_{\mathsf{w}}\right]$
$\displaystyle=\sum_{i=1}^{q}\mathbb{P}\left[\mathcal{E}(i)\right]\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\mathbb{E}_{\nu_{g(t-\zeta)}}\left[f_{i}\right]\right]$
$\displaystyle=\sum_{i=1}^{q}\dfrac{\lambda_{i}}{\lambda}\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\mathbb{E}_{\nu_{g(t-\zeta)}}\left[f_{i}\right]\right]$
$\displaystyle=\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\sum_{i=1}^{q}\dfrac{\lambda_{i}}{\lambda}\mathbb{E}_{\nu_{g(t-\zeta)}}\left[f_{i}\right]\right]$
$\displaystyle=\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\times\dfrac{1}{\lambda}\times\left[R(g(t-\zeta))+\lambda
g(t-\zeta)\right]\right],$

where the last equality comes from Lemma 9.

This implies that the first term of (6.1) equals $e^{-\lambda t}\rho_{0}$ and
the second term of (6.1) equals

$\displaystyle\int_{0}^{t}\lambda e^{-\lambda
s}\dfrac{1}{\lambda}\left(R(g(t-s))+\lambda g(t-s)\right)\mathrm{d}s$
$\displaystyle=\int_{0}^{t}e^{-\lambda s}\left(R(g(t-s))+\lambda
g(t-s)\right)\mathrm{d}s$
$\displaystyle=\int_{0}^{t}e^{-\lambda(t-s)}\left(R(g(s))+\lambda
g(s)\right)\mathrm{d}s,$

where the last inequality follows from the change of variable
$s^{\prime}=t-s$. This implies that

$\displaystyle g(t)$ $\displaystyle=e^{-\lambda
t}\rho_{0}+\int_{0}^{t}e^{-\lambda(t-s)}\left(R(g(s))+\lambda
g(s)\right)\mathrm{d}s$ $\displaystyle=e^{-\lambda
t}\left(\rho_{0}+\int_{0}^{t}e^{\lambda s}\left(R(g(s))+\lambda
g(s)\right)\mathrm{d}s\right).$

Now we can differentiate both sides with respect to $t$ to conclude that

$\displaystyle g^{\prime}(t)$ $\displaystyle=-\lambda e^{-\lambda
t}\left(\rho_{0}+\int_{0}^{t}e^{\lambda s)}\left(R(g(s))+\lambda
g(s)\right)\mathrm{d}s\right)+e^{-\lambda t}e^{\lambda t}\left(R(g(t))+\lambda
g(t)\right)$ $\displaystyle=-\lambda g(t)+R(g(t))+\lambda g(t)$
$\displaystyle=R(g(t)).$

This means that $g$ solves equation (9) with initial condition
$g(0)=\rho_{0}$. This finishes our proof. ∎

Recall that $\rho_{+}$ and $\rho_{-}$ are solutions of equation (8) with
initial conditions $1$ and $-1$. We have the following lemma.

######  Lemma 16.

$\displaystyle\phi(t)$ $\displaystyle=\dfrac{\rho_{+}(t)-\rho_{-}(t)}{2},$
(45) $\displaystyle\vartheta(t)$
$\displaystyle=\dfrac{\rho_{+}(t)+\rho_{-}(t)}{2(1-\phi(t))}.$ (46)

###### Proof.

Let us generate the IBP up to time $t$. We denote by
$\mathbb{E}_{+}\left[\cdot\right]$ the probability taken with respect to the
IBP when we assign the spins at time $t$ to be all-plus, i.e.
$\tilde{\mathfrak{s}}_{\mathsf{w}^{\prime}}:=1,\,\forall\mathsf{w}^{\prime}\in\widetilde{W}_{t}$
and $\mathbb{E}_{-}\left[\cdot\right]$ the probability taken when we assign
the spins at time $t$ to be all-minus, i.e.
$\tilde{\mathfrak{s}}_{\mathsf{w}^{\prime}}:=-1,\,\forall\mathsf{w}^{\prime}\in\widetilde{W}_{t}$.
Note that

$\displaystyle\mathbb{E}_{+}\left[\tilde{\mathfrak{s}}_{\mathsf{w}}\right]$
$\displaystyle=\mathbb{E}_{+}\left[\tilde{\mathfrak{s}}_{\mathsf{w}}\mathds{1}_{\left\\{\widetilde{\mathsf{Piv}}(\mathsf{w},t)\neq\O\right\\}}\right]+\mathbb{E}_{+}\left[\tilde{\mathfrak{s}}_{\mathsf{w}}\mathds{1}_{\left\\{\widetilde{\mathsf{Piv}}(\mathsf{w},t)=\O\right\\}}\right].$
(47)

Recall that
$\widetilde{\mathsf{Piv}}(\mathsf{w},t)=\mathsf{Piv}\left(\tilde{F}_{\mathsf{w},t}\right)$,
where
$\tilde{\mathfrak{s}}_{\mathsf{w}}=\tilde{F}_{\mathsf{w},t}\left((\tilde{\mathfrak{s}}_{\mathsf{w}^{\prime}})_{\mathsf{w}^{\prime}\in\widetilde{W}_{t}}\right)$
for some increasing boolean function $\tilde{F}_{\mathsf{w},t}$. Hence

$\displaystyle\mathbb{E}_{+}\left[\tilde{\mathfrak{s}}_{\mathsf{w}}\mathds{1}_{\left\\{\widetilde{\mathsf{Piv}}(\mathsf{w},t)\neq\O\right\\}}\right]=\mathbb{E}_{+}\left[\mathds{1}_{\left\\{\widetilde{\mathsf{Piv}}(\mathsf{w},t)\neq\O\right\\}}\right]=\mathbb{P}\left[\widetilde{\mathsf{Piv}}(\mathsf{w},t)\neq\O\right],$

where the first equality is due to Lemma 6. Moreover, when
$\widetilde{\mathsf{Piv}}(\mathsf{w},t)=\O$,
$\tilde{\mathfrak{s}}_{\mathsf{w}}$ no longer depends on the spins of
particles in $\widetilde{W}_{t}$, so the second term in (47) is equal to
$(1-\phi(t))\vartheta(t)$ by definition of the functions $\phi,\vartheta$.
This implies that

$\rho_{+}(t)=\phi(t)+(1-\phi(t))\vartheta(t).$

Similarly

$\rho_{-}(t)=-\phi(t)+(1-\phi(t))\vartheta(t).$

These two equalities imply what we want. ∎

In the following proposition, we prove that $\psi$ satisfies a particular
differential equation, implying that the pivotal set is subcritical, one of
the most important results in this paper.

######  Proposition 17 (Differential equation for $\psi$).

There exist a univariate polynomial $Q_{1}$ and a bivariate polynomial $Q_{2}$
such that

$\psi^{\prime}=\psi\left(R^{\prime}(\rho_{*})+(\vartheta-\rho_{*})Q_{1}(\vartheta)+\phi
Q_{2}(\phi,\vartheta)\right).$ (48)

###### Proof.

The idea is to proceed as in the proof of Lemma 15. See Figure 3 for
intuition. Let us generate the IBP starting from one particle $\mathsf{w}$ up
to time $t$. Let $\zeta$ be the first time that a Glauber clock of
$\mathsf{w}$ rings. Then $\zeta\sim\exp(\lambda)$. We see that

$\displaystyle\psi(t)$
$\displaystyle=\mathbb{E}\left[|\widetilde{\mathsf{Piv}}(\mathsf{w},t)|\right]$
$\displaystyle=\mathbb{E}\left[|\widetilde{\mathsf{Piv}}(\mathsf{w},t)|\mathds{1}_{\left\\{\zeta>t\right\\}}\right]+\mathbb{E}\left[|\widetilde{\mathsf{Piv}}(\mathsf{w},t)|\mathds{1}_{\left\\{\zeta<t\right\\}}\right].$
(49)

We estimate the two terms in (49) separately.

  * •

First term: when $\zeta>t$,
$\widetilde{\mathsf{Piv}}(\mathsf{w},t)=\\{\mathsf{w}\\}$ by construction, and
hence

$\displaystyle\mathbb{E}\left[|\widetilde{\mathsf{Piv}}(\mathsf{w},t)|\mathds{1}_{\left\\{\zeta>t\right\\}}\right]=\mathbb{E}\left[|\\{\mathsf{w}\\}|\mathds{1}_{\left\\{\zeta>t\right\\}}\right]=\mathbb{P}\left[\zeta>t\right].$
(50)

  * •

Second term:
For any subset $A\subset B(0,m)$, any $i\in[q]$, any spin configuration
$\eta,\xi$ on $B(0,m)$, let us define the spin configuration $\xi^{A,\eta}$ on
$B(0,m)$, the boolean function $f_{i}^{A,\eta}$ on $\\{-1,1\\}^{B(0,m)}$, and
the events $\mathcal{E}(i),\mathcal{E}(A,\eta)$ by

$\displaystyle\xi^{A,\eta}(j)$
$\displaystyle:=\eta(j)\times\mathds{1}_{\left\\{j\in
A\right\\}}+\xi(j)\mathds{1}_{\left\\{j\in A^{C}\right\\}},$ $\displaystyle
f_{i}^{A,\eta}(\xi)$ $\displaystyle:=f_{i}(\xi^{A,\eta}),$
$\displaystyle\mathcal{E}(i)$ $\displaystyle:=\\{\text{The Glauber clock of
$\mathsf{w}$ ringing at time $\zeta$ is of type $i$}\\},$
$\displaystyle\mathcal{E}(A,\eta)$
$\displaystyle:=\\{\widetilde{\mathsf{Piv}}(w^{\prime},t)\neq\O,\forall\mathsf{w}^{\prime}\in
A,\text{ and
}\widetilde{\mathsf{Piv}}(w^{\prime},t)=\O,\forall\mathsf{w}^{\prime}\in
A^{C}\\}\cap\\{(\tilde{\mathfrak{s}}_{\mathsf{w}^{\prime}})_{\mathsf{w}^{\prime}\in
A^{C}}=\eta|_{A^{C}}\\}.$

Note that, on the event $\mathcal{E}(i)\cap\mathcal{E}(A,\eta)$,

$\displaystyle\widetilde{\mathsf{Piv}}(\mathsf{w},t)=\bigcup_{\mathsf{w}^{\prime}\in\mathsf{Piv}\left(f_{i}^{A^{C},\eta}\right)}\widetilde{\mathsf{Piv}}(\mathsf{w}^{\prime},t).$

Hence

$\displaystyle\mathbb{E}\left[\left|\widetilde{\mathsf{Piv}}(\mathsf{w},t)\right|\mathds{1}_{\left\\{\zeta<t\right\\}}\mathds{1}_{\mathcal{E}(i)}\mathds{1}_{\mathcal{E}(A,\eta)}\right]=\sum_{\mathsf{w}^{\prime}\in\mathsf{Piv}\left(f_{i}^{A^{C},\eta}\right)}\mathbb{E}\left[\left|\widetilde{\mathsf{Piv}}(\mathsf{w}^{\prime},t)\right|\mathds{1}_{\left\\{\zeta<t\right\\}}\mathds{1}_{\mathcal{E}(i)}\mathds{1}_{\mathcal{E}(A,\eta)}\right].$
(51)

By construction, $\zeta,\mathcal{E}(i)$ and the branches of the IBP starting
at the children of $\mathsf{w}$ are independent. Hence, for any non empty
$A\subset B(0,m)$, $\mathsf{w}^{\prime}\in A,\eta\in\\{-1,1\\}^{B(0,m)}$,

$\displaystyle\mathbb{E}\left[\left|\widetilde{\mathsf{Piv}}(\mathsf{w}^{\prime},t)\right|\mathds{1}_{\left\\{\zeta<t\right\\}}\mathds{1}_{\mathcal{E}(i)}\mathds{1}_{\mathcal{E}(A,\eta)}\right]$
$\displaystyle=\mathbb{P}\left[\mathcal{E}(i)\right]\mathbb{E}\left[\left|\widetilde{\mathsf{Piv}}(\mathsf{w}^{\prime},t)\right|\mathds{1}_{\left\\{\zeta<t\right\\}}\mathds{1}_{\mathcal{E}(A,\eta)}\right].$
$\displaystyle=\dfrac{\lambda_{i}}{\lambda}\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\left|\widetilde{\mathsf{Piv}}(\mathsf{w}^{\prime},t)\right|\prod_{\mathsf{w}^{\prime\prime}\in
A\setminus\\{\mathsf{w}^{\prime}\\}}\mathds{1}_{\left\\{\widetilde{\mathsf{Piv}}(\mathsf{w}^{\prime\prime},t)\neq\O\right\\}}\prod_{\mathsf{w}^{\prime\prime}\in
A^{C}}\mathds{1}_{\left\\{\widetilde{\mathsf{Piv}}(\mathsf{w}^{\prime\prime},t)=\O,\tilde{\mathfrak{s}}_{\mathsf{w}^{\prime\prime}}=\eta_{\mathsf{w}^{\prime\prime}}\right\\}}\right].$

Note that the branches starting at the children of $\mathsf{w}$ at time
$\zeta$ are independent IBPs independent of $\zeta$, shifted by a time
$\zeta$. Then we can integrate out the randomness of these branches to
conclude that

$\displaystyle\mathbb{E}\left[\left|\widetilde{\mathsf{Piv}}(\mathsf{w}^{\prime},t-\zeta)\right|\mathds{1}_{\left\\{\zeta<t\right\\}}\mathds{1}_{\mathcal{E}(i)}\mathds{1}_{\mathcal{E}(A,\eta)}\right]=\dfrac{\lambda_{i}}{\lambda}\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\psi\phi^{|A|-1}\prod_{\mathsf{w}^{\prime\prime}\in
A^{C}}(1-\phi)\mathfrak{R}_{\vartheta}(\eta_{\mathsf{w}^{\prime\prime}})\right],$

where we have used the definition of $\psi,\phi,$ and $\vartheta$. In the
formula above, the functions $\psi,\phi,\vartheta$ inside the expectation are
evaluated at $t-\zeta$. Hence

$\displaystyle\mathbb{E}\left[\left|\widetilde{\mathsf{Piv}}(\mathsf{w}^{\prime},t-\zeta)\right|\mathds{1}_{\left\\{\zeta<t\right\\}}\mathds{1}_{\mathcal{E}(i)}\mathds{1}_{\mathcal{E}(A,\eta)}\right]=\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\mathcal{M}(i,A,\eta)\right],$

with

$\displaystyle\mathcal{M}(i,A,\eta):=\dfrac{\lambda_{i}}{\lambda}\psi\phi^{|A|-1}(1-\phi)^{2m+1-|A|}\prod_{\mathsf{w}^{\prime\prime}\in
A^{C}}\mathfrak{R}_{\vartheta}(\eta_{\mathsf{w}^{\prime\prime}}),$

where the functions $\psi,\phi,\vartheta$ are evaluated at $t-\zeta$. For
convention, $\mathcal{M}(i,A,\eta):=0$ if $|A|=0$. Hence

$\displaystyle\mathbb{E}\left[\left|\widetilde{\mathsf{Piv}}(\mathsf{w},t)\right|\mathds{1}_{\left\\{\zeta<t\right\\}}\right]$
$\displaystyle=\sum_{i,A,\eta}\mathbb{E}\left[\mathds{1}_{\mathcal{E}(i)}\mathds{1}_{\mathcal{E}(A,\eta)}\mathds{1}_{\left\\{\zeta<t\right\\}}|\widetilde{\mathsf{Piv}}(\mathsf{w},t)|\right]$
$\displaystyle=\sum_{i,A,\eta}\sum_{\mathsf{w}^{\prime}\in\mathsf{Piv}\left(f_{i}^{A^{C},\eta}\right)}\mathbb{E}\left[\mathds{1}_{\mathcal{E}(i)}\mathds{1}_{\mathcal{E}(A,\eta)}\mathds{1}_{\left\\{\zeta<t\right\\}}|\widetilde{\mathsf{Piv}}(\mathsf{w}^{\prime},t)|\right]$
$\displaystyle=\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\sum_{i,A,\eta}\left|\mathsf{Piv}\left(f_{i}^{A^{C},\eta}\right)\right|\mathcal{M}(i,A,\eta)\right],$
(52)

where the sums are taken on all $i\in[q],A\subset B(0,m)$, and
$\eta\in\\{-1,1\\}^{B(0,m)}$. Note that,

$\displaystyle\sum_{i,A,\eta}\mathds{1}_{\left\\{|A|=0\right\\}}\left|\mathsf{Piv}\left(f_{i}^{A,\eta}\right)\right|\mathcal{M}(i,A,\eta)=0,$
(53)

and

$\displaystyle\sum_{i,A,\eta}\mathds{1}_{\left\\{|A|\geq
2\right\\}}\left|\mathsf{Piv}\left(f_{i}^{A^{C},\eta}\right)\right|\mathcal{M}(i,A,\eta)=\psi\phi
g,$ (54)

where $g$ is a polynomial of $\phi$ and $\vartheta$, and the functions
$\psi,\phi,\vartheta$ are evaluated at $t-\zeta$. We now estimate the sum on
the subsets $A$ such that $|A|=1$. Note that if $A=\\{j\\}$, then

$\displaystyle\left|\mathsf{Piv}\left(f_{i}^{\\{j\\}^{C},\eta}\right)\right|=\dfrac{1}{2}\nabla_{j}f_{i}(\eta),$

due to Lemma 6. Hence

$\displaystyle\sum_{i,A,\eta}\mathds{1}_{\left\\{|A|=1\right\\}}\left|\mathsf{Piv}\left(f_{i}^{A^{C},\eta}\right)\right|\mathcal{M}(i,A,\eta)$
$\displaystyle=\sum_{i,j,\eta}\dfrac{1}{2}\nabla_{j}f_{i}(\eta)\mathcal{M}(i,\\{j\\},\eta)$
$\displaystyle=\sum_{i,j,\eta}\dfrac{1}{2}\nabla_{j}f_{i}(\eta)\times\dfrac{\lambda_{i}}{\lambda}\psi(1-\phi)^{2m}\prod_{\mathsf{w}^{\prime\prime}\in
B(0,m)\setminus\\{j\\}}\mathfrak{R}_{\vartheta}(\eta_{\mathsf{w}^{\prime\prime}})$
$\displaystyle=\psi(1-\phi)^{2m}\sum_{i=1}^{q}\dfrac{\lambda_{i}}{\lambda}\sum_{j\in
B(0,m)}\mathbb{E}_{\nu_{\vartheta}}\left[\dfrac{1}{2}\nabla_{j}f_{i}\right]$
$\displaystyle=\psi(1-\phi)^{2m}\dfrac{1}{\lambda}(R^{\prime}(\vartheta)+\lambda),$
(55)

where we have used Lemma 9 in the last equality. The equations (52), (53),
(54), (55) together imply that

$\displaystyle\mathbb{E}\left[\widetilde{\mathsf{Piv}}(\mathsf{w},t)\mathds{1}_{\left\\{\zeta<t\right\\}}\right]$
$\displaystyle=\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\times\left[\psi(1-\phi)^{2m}\left(\dfrac{R^{\prime}(\vartheta)}{\lambda}+1\right)+\psi\phi
g\right]\right],$ (56)

where the functions $\psi,\phi,\vartheta$ inside the expectation are evaluated
at $t-\zeta$. The equations (50), (56), (49) together imply that

$\displaystyle\psi(t)$
$\displaystyle=\mathbb{P}\left[\zeta>t\right]+\mathbb{E}\left[\mathds{1}_{\left\\{\zeta<t\right\\}}\times\left[\psi(1-\phi)^{2m}\left(\dfrac{R^{\prime}(\vartheta)}{\lambda}+1\right)+\psi\phi
g\right]\right]$ $\displaystyle=e^{-\lambda t}+\int_{0}^{t}\lambda e^{-\lambda
s}\left[\psi(1-\phi)^{2m}\left(\dfrac{R^{\prime}(\vartheta)}{\lambda}+1\right)+\psi\phi
g\right]\mathrm{d}s$

where the functions $\psi,\phi,\vartheta$ inside the integral sign are
evaluated at $t-s$. We make a change of variable $s^{\prime}=t-s$ to conclude
that

$\displaystyle\psi(t)$ $\displaystyle=e^{-\lambda t}+\int_{0}^{t}\lambda
e^{-\lambda(t-s)}\left[\psi(1-\phi)^{2m}\left(\dfrac{R^{\prime}(\vartheta)}{\lambda}+1\right)+\psi\phi
g\right]\mathrm{d}s$ $\displaystyle=e^{-\lambda t}\left(1+\int_{0}^{t}\lambda
e^{\lambda
s}\left[\psi(1-\phi)^{2m}\left(\dfrac{R^{\prime}(\vartheta)}{\lambda}+1\right)+\psi\phi
g\right]\mathrm{d}s\right),$

where the functions $\psi,\phi,\vartheta$ are now evaluated at $s$. Now, we
can differentiate both sides with respect to $t$ to conclude that

$\displaystyle\psi^{\prime}=-\lambda\psi+e^{-\lambda t}\lambda e^{\lambda
t}\left[\psi(1-\phi)^{2m}\left(\dfrac{R^{\prime}(\vartheta)}{\lambda}+1\right)+\psi\phi
g\right],$

where all functions $\psi^{\prime},\phi,\psi,\vartheta$ are evaluated at $t$.
Simplifying that formula, we obtain

$\displaystyle\psi^{\prime}=\psi\left(-\lambda+(1-2\phi)^{2m}\lambda+(1-2\phi)^{2m}R^{\prime}(\vartheta)+\phi
g\right).$

We can take $Q_{1}$ and $Q_{2}$ such that

$\displaystyle(\vartheta-\rho_{*})Q_{1}(\vartheta)$
$\displaystyle=R^{\prime}(\vartheta)-R^{\prime}(\rho_{*}),$ $\displaystyle\phi
Q_{2}(\phi,\vartheta)$
$\displaystyle=\left((1-2\phi)^{2m}-1\right)(R^{\prime}(\vartheta)+\lambda)+\phi
g,$

to finish the proof. ∎

######  Remark 6.

In the proof above, the monotonicity of the functions $(f_{i})_{1\leq i\leq
q}$ is crucial, and so is Hypothesis 1.

We derive the asymptotic of $\psi$ using Hypothesis 2.

######  Lemma 18 (Asymptotic of some important functions).

There exists a constant $\kappa$ such that, for any $t\geq 0$, all the numbers
$\log\psi(t),\log\phi(t),\log(\rho_{+}(t)-\rho_{*}),\log(\rho_{*}-\rho_{-}(t))$
lie in the interval
$[R^{\prime}(\rho_{*})t-\kappa,R^{\prime}(\rho_{*})t+\kappa]$.

###### Proof of Lemma 18.

We prove the result for $\psi(t)$. The results for other functions are proved
similarly. By Hypothesis 2, we easily see that

$\displaystyle R(\rho)$ $\displaystyle>0,\,\forall\rho\in[-1,\rho_{*}),$
$\displaystyle R(\rho)$ $\displaystyle<0,\,\forall\rho\in(\rho_{*},1].$

Hence $\rho_{+}(t)\searrow\rho_{*}$. Moreover,

$\displaystyle\lim_{t\to\infty}\dfrac{\rho_{+}^{\prime}(t)}{\rho_{+}(t)-\rho}=\lim_{t\to\infty}\dfrac{R(\rho_{+}(t))-R(\rho_{*})}{\rho_{+}(t)-\rho_{*}}=R^{\prime}(\rho_{*}).$

Then for $t$ large enough,

$\displaystyle\dfrac{\rho_{+}^{\prime}(t)}{\rho_{+}(t)-\rho_{*}}\leq\dfrac{R^{\prime}(\rho_{*})}{2}.$

Then, by Gronwall’s lemma, there exists a positive constant $\kappa_{1}$ such
that

$\displaystyle\forall t\geq
0,\;\rho_{+}(t)-\rho_{*}\leq\kappa_{1}e^{\frac{1}{2}R^{\prime}(\rho_{*})t}.$

One can prove that the inequality above is still true if we replace
$\rho_{+}(t)-\rho_{*}$ by the positive functions
$\rho_{*}-\rho_{-}(t),\,\phi(t),\,|\vartheta(t)-\rho_{*}|$. Therefore,

  1. 1.

$\phi(t),\,|\vartheta(t)-\rho_{*}|$ are absolutely integrable with respect to
$t$.

  2. 2.

$Q_{1}(\vartheta)$ and $Q_{2}(\phi,\vartheta)$ are bounded uniformly in $t$.

Hence $(\vartheta-\rho_{*})Q_{1}(\vartheta)+\phi Q_{2}(\phi,\vartheta)$ is
absolutely integrable as a function of $t$. Recall that Proposition 17 implies
that

$\displaystyle\dfrac{\psi^{\prime}}{\psi}-R^{\prime}(\rho_{*})=(\vartheta-\rho_{*})Q_{1}(\vartheta)+\phi
Q_{2}(\phi,\vartheta).$

Hence, taking integration from $0$ to $t$, and using the inequality
$\left|\int f-\int g\right|\leq\int\left|f-g\right|$, and noting that
$\psi(0)=1$, we obtain

$\displaystyle\left|\log\psi(t)-R^{\prime}(\rho_{*})t\right|\leq\int_{0}^{t}\left|(\vartheta-\rho_{*})Q_{1}(\vartheta)+\phi
Q_{2}(\phi,\vartheta)\right|\leq\kappa,$

where

$\displaystyle\kappa=\int_{0}^{\infty}\left|(\vartheta-\rho_{*})Q_{1}(\vartheta)+\phi
Q_{2}(\phi,\vartheta)\right|.$

This finishes our proof. ∎

######  Remark 7.

Hypothesis 2 is crucial for the proof above.

We present some elementary but useful results on $\widetilde{W}_{t}$.

######  Lemma 19.

Consider the IBP starting from a set $E$. Then there exists a constant
$\kappa$ such that, for any number $t>0$,

$\mathbb{E}\left[|\widetilde{W}_{t}|^{2}\right]\leq e^{\kappa t}|E|^{2}.$

###### Proof of Lemma 19.

Note that $(|\widetilde{W}_{t}|)_{t\geq 0}$ is a Markov process itself. Let
$\mathcal{L}$ be the generator associated with $(|\widetilde{W}_{t}|)_{t\geq
0}$. Then, for any function $\varphi:\mathbb{Z}_{+}\to\mathbb{R}$, for any
$n\in\mathbb{Z}_{+}$,

$\mathcal{L}\varphi(n)=\lambda
n\left[\varphi(n+2m+1-1)-\varphi(n)\right]=\lambda
n\left[\varphi(n+2m)-\varphi(n)\right].$

We also know that, see e.g. [19],

$\displaystyle\dfrac{d}{dt}\mathbb{E}\left[\varphi(|\widetilde{W}_{t}|)\right]$
$\displaystyle=\mathbb{E}\left[\mathcal{L}\varphi(|\widetilde{W}_{t}|)\right].$

Applying the formula above for $\varphi:n\mapsto n^{2}$, we obtain

$\displaystyle\dfrac{d}{dt}\mathbb{E}\left[|\widetilde{W}_{t}|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\lambda|\widetilde{W}_{t}|(4m|\widetilde{W}_{t}|+4m^{2})\right]$
$\displaystyle\leq\lambda(4m+4m^{2})\mathbb{E}\left[|\widetilde{W}_{t}|^{2}\right].$

Then, we can apply Gronwall’s lemma to conclude that

$\displaystyle\mathbb{E}\left[|\widetilde{W}_{t}|^{2}\right]\leq
e^{(4m+4m^{2})\lambda t}\mathbb{E}\left[|\widetilde{W}_{0}|^{2}\right],$

which finishes the proof. ∎

######  Lemma 20.

Consider the BEP and IBP starting from $E\subset\Lambda_{L}$. Then,
$|\widetilde{W}_{t}|$ stochastically dominates $|W_{t}|$ for any $t\geq 0$.

###### Proof of Lemma 20.

The reason is that the particles in the two processes are given $q$ Poisson
clocks with the same rates. For the BEP, each time a clock of a particle
rings, the particle is removed, and _at most_ $2m+1$ new particles are added
to $W_{t}$ because some new particles are born on the sites already occupied,
while for the process $|\widetilde{W}_{t}|$, each time a clock of a particle
rings, _exactly_ $2m+1$ new particles are added. ∎

###  4.6 Success of the coupling

Now we can prove Proposition 14. We will need a result about anti-
concentration of the Interchange Process IP(2).

#### Interchange process IP(2).

The interchange process IP(2) on $\Lambda^{d}_{L}$, whose edges have
conductance $L^{2}$, is a couple of random walk
$(\mathcal{U}_{1},\mathcal{U}_{2})$ that has the following description. The
state space is
$(\Lambda^{d}_{L}\times\Lambda^{d}_{L})\setminus\\{(u,u)|u\in\Lambda^{d}_{L}\\}$.
For any initial condition, the evolution is as follows. Each edge of
$\Lambda^{d}_{L}$ is associated with an independent Poisson clock of intensity
$L^{2}$. Whenever a clock rings, if $\mathcal{U}_{1}$ (or $\mathcal{U}_{2}$)
is at one endpoint of the corresponding edge, it jumps to the other endpoint.

From now on, we denote by $\mathsf{dist}\left(\cdot,\cdot\right)$ the
shortest-path distance on the lattice. We have the following result.

######  Lemma 21 (Anticoncentration).

For any $d\in\mathbb{Z}_{>0}$. Let $(\mathcal{U}_{1},\mathcal{U}_{2})$ be an
IP(2) on $\Lambda^{d}_{L}$, whose edges have conductance $L^{2}$. Let $\theta$
be a strictly positive number and $\zeta\sim\exp(\theta)$, independent of
$(\mathcal{U}_{1},\mathcal{U}_{2})$. Then, for any $k\in\mathbb{Z}_{+}$,

$\max_{u_{1},u_{2}\in\Lambda^{d}_{L};\,u_{1}\neq
u_{2}}\mathbb{P}_{u_{1},u_{2}}\left[\mathsf{dist}\left(\mathcal{U}_{1}(\zeta),\mathcal{U}_{2}(\zeta)\right)\leq
k\right]=\begin{cases}\mathcal{O}_{\theta,k}(1/L)&\text{if $d=1$},\\\
\mathcal{O}_{\theta,k}(\log L/L^{2})&\text{if $d=2$},\\\
\mathcal{O}_{\theta,k}(1/L^{2})&\text{if $d\geq 3$}.\end{cases}$

To lighten the notation, we write $\mathsf{Piv}\left(t\right)$ for
$\mathsf{Piv}\left(E,t\right)$ and $\widetilde{\mathsf{Piv}}(t)$ for
$\widetilde{\mathsf{Piv}}(E,t)$. The proof of Proposition 14 is divided into
two following lemmas.

######  Lemma 22.

With the same notations as in Proposition 14,

$\mathbb{P}\left[\text{The coupling is not successful until
infinity}\right]=\mathcal{O}\left(\dfrac{\sum_{j=0}^{\infty}\mathbb{E}\left[\widetilde{\mathsf{Piv}}(\tilde{T}_{j})\right]}{L}\right).$

######  Lemma 23.

$\sum_{j=0}^{\infty}\mathbb{E}\left[\left|\widetilde{\mathsf{Piv}}(\tilde{T}_{j})\right|\right]=\mathcal{O}\left(|E|^{2}\right).$

Clearly, those two lemmas imply Proposition 14. Those two lemmas separate the
effect of the exclusion process and the IBP.

###### Proof of Lemma 22.

We say that two particles $\mathsf{w}_{1}$ and $\mathsf{w}_{2}$ have an
interaction if there is a time $t$ such that:

  1. 1.

Both are in the pivotal set at time $t-$, i.e.
$\mathsf{w}_{1},\mathsf{w}_{2}\in\mathsf{Piv}\left(t-\right)$, and they do not
occupy the same site.

  2. 2.

At time $t$, a clock of one of the two particles rings, and then one of the
children is born on the site occupied by the other particle.

In short, two particles in $\mathsf{Piv}\left(\cdot\right)$ have an
interaction if one splits into the site occupied by the other. For example, in
Figure 2, $\mathsf{w}$ and $\mathsf{w}^{\prime}$ have an interaction at time
$s_{3}$. By definition, the coupling is successful up to time $t$ if there is
no interaction up to time $t$. Hence

$\displaystyle\mathbb{P}\left[\text{The coupling is not successful until
infinity}\right]\leq\mathbb{P}\left[\text{There is an interaction}\right].$

We only need to prove

$\mathbb{P}\left[\text{There is an
interaction}\right]\leq\mathcal{O}\left(\dfrac{\sum_{j=0}^{\infty}\mathbb{E}\left[\widetilde{\mathsf{Piv}}(\tilde{T}_{j})\right]}{L}\right).$
(57)

Note that

$\displaystyle\mathbb{P}\left[\text{There is an
interaction}\right]=\sum_{\mathsf{w}_{1},\mathsf{w}_{2}}\mathbb{P}\left[\text{The
first interaction is between $\mathsf{w}_{1},\mathsf{w}_{2}$}\right].$

Recall the definition of the update times $(T_{j})_{j\in\mathbb{Z}_{+}}$ of
the pivotal set. Let $\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})$ be the
event that $T_{j}$ is the first time such that both $\mathsf{w}_{1}$ and
$\mathsf{w}_{2}$ are in the pivotal set. Let $\mathcal{E}(j)$ be the event
that there is no interaction up to time $T_{j}$, and let
$\mathcal{E}(\mathsf{w}_{1},\mathsf{w}_{2})$ be the event that there is an
interaction between $\mathsf{w}_{1},\mathsf{w}_{2}$. Then

$\displaystyle\mathbb{P}\left[\text{The first interaction is of
$\mathsf{w}_{1},\mathsf{w}_{2}$}\right]\leq\sum_{j=0}^{\infty}\mathbb{P}\left[\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})\cap\mathcal{E}(j)\cap\mathcal{E}(\mathsf{w}_{1},\mathsf{w}_{2})\right].$

Let $\mathcal{F}_{j}$ be the $\sigma$-algebra generated by the Poisson
processes up to time $T_{j}$. Then

$\displaystyle\mathbb{P}\left[\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})\cap\mathcal{E}(j)\cap\mathcal{E}(\mathsf{w}_{1},\mathsf{w}_{2})\right]=\mathbb{E}\left[\mathds{1}_{\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})}\mathds{1}_{\mathcal{E}(j)}\mathbb{P}\left[\mathcal{E}(\mathsf{w}_{1},\mathsf{w}_{2})\big{|}\mathcal{F}_{j}\right]\right]$

Let $\tau$ be the time such that a Glauber clock of $\mathsf{w}_{1}$ or
$\mathsf{w}_{2}$ rings counted from $T_{j}$:

$\displaystyle\tau:=\inf\left\\{t>0:\text{a Glauber clock of $\mathsf{w}_{1}$
or $\mathsf{w}_{2}$ rings at time $T_{j}+t$}\right\\}.$

Conditionally on $\mathcal{F}_{j}$, on the event
$\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})\cap\mathcal{E}(j)$, from time
$T_{j}$ onwards, $\mathsf{w}_{1}$ and $\mathsf{w}_{2}$ move as an IP(2) on the
lattice, until one of their Glauber clocks rings at time
$\tau\sim\exp(2\lambda)$. To have an interaction, the distance between
$\mathsf{w}_{1}$ and $\mathsf{w}_{2}$ at time $\tau$ must be smaller than $m$.
Therefore, by Lemma 21, conditionally on $\mathcal{F}_{j}$, on the event
$\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})\cap\mathcal{E}(j)$,

$\displaystyle\mathbb{P}\left[\mathcal{E}(\mathsf{w}_{1},\mathsf{w}_{2})\big{|}\mathcal{F}_{j}\right]\leq\beta/L,$

for some constant $\beta$. We deduce that,

$\displaystyle\mathbb{P}\left[\text{There is an
interaction}\right]\leq\sum_{\mathsf{w}_{1},\mathsf{w}_{2}}\sum_{j=0}^{\infty}\dfrac{\beta}{L}\mathbb{P}\left[\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})\cap\mathcal{E}(j)\right].$

Note that on the event $\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})$, either
$\mathsf{w}_{1}$ or $\mathsf{w}_{2}$ (or both of them) is born at time
$T_{j}$. We denote by $\mathcal{E}(j,\mathsf{w}_{i})$ the event that
$\mathsf{w}_{i}$ is born at time $T_{j}$, $i\in\\{1,2\\}$. Hence

$\displaystyle\sum_{\mathsf{w}_{1},\mathsf{w}_{2}}\mathds{1}_{\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})}\mathds{1}_{\mathcal{E}(j)}\leq\sum_{i\in\\{1,2\\}}\sum_{\mathsf{w}_{1},\mathsf{w}_{2}}\mathds{1}_{\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})}\mathds{1}_{\mathcal{E}(j)}\mathds{1}_{\mathcal{E}(j,\mathsf{w}_{i})}.$

Moreover, note also that on the event
$\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})$,
$\mathsf{w}_{2}\in\mathsf{Piv}\left(T_{j}\right)$. Hence

$\displaystyle\sum_{\mathsf{w}_{1},\mathsf{w}_{2}}\mathds{1}_{\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})}\mathds{1}_{\mathcal{E}(j)}\mathds{1}_{\mathcal{E}(j,\mathsf{w}_{1})}$
$\displaystyle\leq\sum_{\mathsf{w}_{1},\mathsf{w}_{2}}\mathds{1}_{\left\\{\mathsf{w}_{2}\in\mathsf{Piv}\left(T_{j}\right)\right\\}}\mathds{1}_{\mathcal{E}(j,\mathsf{w}_{1})}\mathds{1}_{\mathcal{E}(j)}$
$\displaystyle\leq\sum_{\mathsf{w}_{1}}\left|\mathsf{Piv}\left(T_{j}\right)\right|\mathds{1}_{\mathcal{E}(j,\mathsf{w}_{1})}\mathds{1}_{\mathcal{E}(j)}$
$\displaystyle\leq(2m+1)\left|\mathsf{Piv}\left(T_{j}\right)\right|\mathds{1}_{\mathcal{E}(j)},$

where all these inequalities are true almost surely. The last inequality is
because at most $(2m+1)$ particles are born at time $T_{j}$. Note that, on the
event $\mathcal{E}(j)$,
$\mathsf{Piv}\left(T_{j}\right)=\widetilde{\mathsf{Piv}}(\tilde{T}_{j})$. This
implies that

$\displaystyle\sum_{\mathsf{w}_{1},\mathsf{w}_{2}}\mathds{1}_{\mathcal{E}(j,\mathsf{w}_{1},\mathsf{w}_{2})}\mathds{1}_{\mathcal{E}(j)}\leq
2(2m+1)\left|\widetilde{\mathsf{Piv}}(\tilde{T}_{j})\right|.$

Taking expectation, we conclude that

$\displaystyle\mathbb{P}\left[\text{There is an interaction}\right]\leq
2(2m+1)\dfrac{\beta}{L}\sum_{j=0}^{\infty}\mathbb{E}\left[\widetilde{\mathsf{Piv}}(\tilde{T}_{j})\right],$

which finishes the proof. ∎

We finish this section by proving Lemma 23. The left-hand side is the sum of
the expected sizes of the process
$\left|\widetilde{\mathsf{Piv}}(\cdot)\right|$ at its update times. The
intuition is that the inequality is true if we replace
$\widetilde{\mathsf{Piv}}(\cdot)$ by a subcritical Galton-Watson process. The
only problem is that $\left|\widetilde{\mathsf{Piv}}(\cdot)\right|$ is not
Markovian. Nevertheless, it is “subcritical”, and the idea is to enlarge the
process $\left|\widetilde{\mathsf{Piv}}(\cdot)\right|$ to make it Markovian
and then adapt the proof of the subcritical Galton-Watson case.

###### Proof of Lemma 23.

Let $t_{0}$ be a number such that $\psi(t_{0})<1$, which exists thanks to
Lemma 18. We rewrite the left-hand side as

$\displaystyle\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\mathbb{E}\left[\left|\widetilde{\mathsf{Piv}}(\tilde{T}_{j})\right|\mathds{1}_{\left\\{it_{0}\leq\tilde{T}_{j}<(i+1)t_{0}\right\\}}\right].$

Let the sequence $(B_{i})_{i\in\mathbb{Z}_{+}}$ be defined recursively as
follows.

$\displaystyle B_{0}$ $\displaystyle:=\widetilde{\mathsf{Piv}}(0),$
$\displaystyle B_{i}$ $\displaystyle:=\bigcup_{\mathsf{w}\in
B_{i-1}}\widetilde{\mathsf{Piv}}(\mathsf{w},it_{0}),\,\forall i\geq 1.$

It is not hard to see that $\widetilde{\mathsf{Piv}}(it_{0})\subset
B_{i},\forall u\in\mathbb{Z}_{+}$, because given the IBP up to time $it_{0}$
and the spins of the particles in $B_{i}$, by definition of $B_{i}$, we can
recursively determine the spins of the particles in $B_{i-1}$, and so on, up
to time $0$. It is also not hard to see that the sequence
$(|B_{i}|)_{i\in\mathbb{Z}_{+}}$ is a subcritical Galton-Watson process, by
its definition and the definition of $t_{0}$.

Let $\kappa$ be the constant in Lemma 19. We claim that

$\displaystyle\mathbb{E}\left[\sum_{j=0}^{\infty}\left|\widetilde{\mathsf{Piv}}(\tilde{T}_{j})\right|\mathds{1}_{\left\\{it_{0}\leq\tilde{T}_{j}<(i+1)t_{0}\right\\}}\right]\leq\dfrac{e^{\kappa
t_{0}}}{2m}\mathbb{E}\left[|B_{i}|^{2}\right].$

To see this, consider the IBP starting from $B_{i}$ (at time $it_{0}$) up to
time $(i+1)t_{0}$. We denote this process by $\texttt{Gw}_{i}$. Let
$(\mathcal{T}_{j})_{j\in\mathbb{Z}_{+}}$ be the update times of
$\texttt{Gw}_{i}$. Then $\texttt{Gw}_{i}(it_{0})=B_{i}$, and

$\displaystyle\mathbb{E}\left[\sum_{j=0}^{\infty}\left|\widetilde{\mathsf{Piv}}(\tilde{T}_{j})\right|\mathds{1}_{\left\\{it_{0}\leq\tilde{T}_{j}<(i+1)t_{0}\right\\}}\right]$
$\displaystyle\leq\mathbb{E}\left[\sum_{j=0}^{\infty}\left|\texttt{Gw}_{i}(\mathcal{T}_{j})\mathds{1}_{\left\\{it_{0}\leq\mathcal{T}_{j}<(i+1)t_{0}\right\\}}\right|\right].$

This is because $\texttt{Gw}_{i}$ can be thought of as an enlargement of
$\widetilde{\mathsf{Piv}}(\cdot)$ in the time interval $[it_{0},(i+1)t_{0}]$,
where we first enlarge the set $\widetilde{\mathsf{Piv}}(it_{0})$ to $B_{i}$,
and then we do not remove any particle of the branching process starting from
$B_{i}$.

Note that the size of $\texttt{Gw}_{i}(\cdot)$ increases almost surely, hence

$\displaystyle\left|\texttt{Gw}_{i}(\mathcal{T}_{j})\mathds{1}_{\left\\{it_{0}\leq\mathcal{T}_{j}<(i+1)t_{0}\right\\}}\right|\leq\left|\texttt{Gw}_{i}((i+1)t_{0})\right|.$

Moreover, at each update time $\mathcal{T}_{j}$, the size of $\texttt{Gw}_{i}$
increases by 2m, and hence

$\displaystyle\sum_{j=0}^{\infty}\mathds{1}_{\left\\{it_{0}\leq\mathcal{T}_{j}<(i+1)t_{0}\right\\}}\leq\dfrac{\left|\texttt{Gw}_{i}((i+1)t_{0})\right|}{2m}.$

This implies

$\displaystyle\sum_{j=0}^{\infty}\left|\texttt{Gw}_{i}(\mathcal{T}_{j})\mathds{1}_{\left\\{it_{0}\leq\mathcal{T}_{j}<(i+1)t_{0}\right\\}}\right|\leq\dfrac{\left|\texttt{Gw}_{i}((i+1)t_{0})\right|^{2}}{2m}.$

By Lemma 19,

$\displaystyle\mathbb{E}\left[\left|\texttt{Gw}_{i}((i+1)t_{0})\right|^{2}\right]\leq
e^{\kappa t_{0}}\mathbb{E}\left[|B_{i}|^{2}\right].$

This proves the claim. Now, we can take the sum over $i$ to conclude that

$\displaystyle\mathbb{E}\left[\sum_{j=0}^{\infty}\left|\widetilde{\mathsf{Piv}}(\tilde{T}_{j})\right|\right]\leq\dfrac{e^{\kappa
t_{0}}}{2m}\sum_{i=0}^{\infty}\mathbb{E}\left[|B_{i}|^{2}\right].$

Since $(|B_{i}|)_{i\in\mathbb{Z}_{+}}$ is a subcritical Galton-Watson process,
we have

$\displaystyle\sum_{i=0}^{\infty}\mathbb{E}\left[|B_{i}|^{2}\right]=\mathcal{O}\left(\mathbb{E}\left[|B_{0}|^{2}\right]\right)=\mathcal{O}\left(|E|^{2}\right).$

This finishes the proof. ∎

##  5 Application of the dual coupling

Now we can prove Theorem 2, Theorem 3, and the lower bound in Theorem 1.

######  Proposition 24 (Replacement lemma).

Suppose that $X_{0}\sim\mathfrak{R}_{\rho_{0}}^{\otimes\Lambda_{L}}$, for some
$\rho_{0}\in(-1,1)$. Then, for any subset $E\subset\Lambda_{L}$,

$\sup_{t\geq
0}\textrm{d}_{\textsc{tv}}\left(\text{Law}\left(X_{t}(E)\right),\mathfrak{R}_{\rho(t)}^{\otimes|E|}\right)\leq\dfrac{\beta|E|^{2}}{L},$
(58)

where $\rho(\cdot)$ is the solution of equation (9), and $\beta$ is the
constant in Proposition 14.

######  Remark 8.

The proposition above is of independent interest. It shows that the joint
distribution of $o\left(\sqrt{L}\right)$ arbitrary sites is close to that of a
product of i.i.d. Rademacher. This provides sharp estimates on the correlation
functions, allowing us to derive the correct lower bound on the mixing times
using distinguishing statistics.

Proposition 24 is just a direct corollary of Proposition 14.

###### Proof.

Consider the coupling of BEP and IBP starting from $E$ in Subsection 4.4. We
generate the spins at time $t$ by
$\mathfrak{s}(W_{t})\sim\mathfrak{R}_{\rho_{0}}^{\otimes W_{t}}$ and
$\tilde{\mathfrak{s}}(\widetilde{W}_{t})\sim\mathfrak{R}_{\rho_{0}}^{\otimes\widetilde{W}_{t}}$.
Conditionally on the success of the coupling until time $t$, we couple the
spins on $\mathsf{Piv}\left(E,t\right)$ and $\widetilde{\mathsf{Piv}}(E,t)$
such that
$\mathfrak{s}_{\mathsf{w}}=\tilde{\mathfrak{s}}_{\mathsf{w}},\;\forall\mathsf{w}\in\mathsf{Piv}\left(E,t\right)$.
Then Observation 13 implies that

$\mathfrak{s}_{\mathsf{w}}=\tilde{\mathfrak{s}}_{\mathsf{w}},\;\forall\mathsf{w}\in
E.$

Recall that, by Proposition 10,

$\displaystyle X_{t}(E)$
$\displaystyle\overset{d}{=}(\mathfrak{s}_{\mathsf{w}})_{\mathsf{w}\in E},$

and by Lemma 15,

$\displaystyle(\tilde{\mathfrak{s}}_{\mathsf{w}})_{\mathsf{w}\in E}$
$\displaystyle\overset{d}{=}\mathfrak{R}_{\rho(t)}^{\otimes E}.$

Hence

$\displaystyle\textrm{d}_{\textsc{tv}}\left(\text{Law}\left(X_{t}(E)\right),\mathfrak{R}_{\rho(t)}^{\otimes|E|}\right)$
$\displaystyle\leq\mathbb{P}\left[(\mathfrak{s}_{\mathsf{w}})_{\mathsf{w}\in
E}\neq(\tilde{\mathfrak{s}}_{\mathsf{w}})_{\mathsf{w}\in E}\right]$
$\displaystyle\leq\mathbb{P}\left[\text{The coupling is not successful until
time $t$}\right]$ $\displaystyle\leq\dfrac{\beta|E|^{2}}{L}.$

This finishes our proof. ∎

######  Corollary 25 (1-point and 2-point correlation functions).

Under the hypothesis of Proposition 24, for any $u,v\in\Lambda_{L}$, for any
$t\geq 0$,

$\displaystyle\left|\mathbb{E}\left[X_{t}(u)\right]-\rho(t)\right|$
$\displaystyle\leq 2\beta/L,$ (59)
$\displaystyle\left|\textrm{Cov}\left(X_{t}(u),X_{t}(v)\right)\right|$
$\displaystyle\leq 12\beta/L.$ (60)

###### Proof.

Let $u,v\in\Lambda_{L}$, and let $\xi_{1},\xi_{2}$ be i.i.d. Rademacher
$\mathfrak{R}_{\rho(t)}$. Let $\beta$ be the constant in Proposition 14.

  1. 1.

Applying Proposition 24 with $E:=\\{u\\}$, we deduce that there exists a
coupling of $X_{t}(u)$ and $\xi_{1}$ such that

$\displaystyle\mathbb{P}\left[X_{t}(u)\neq\xi_{1}\right]\leq\dfrac{\beta}{L}.$

Hence,

$\displaystyle\left|\mathbb{E}\left[X_{t}(u)\right]-\rho(t)\right|$
$\displaystyle=\left|\mathbb{E}\left[X_{t}(u)-\xi_{1}\right]\right|$
$\displaystyle=\left|\mathbb{E}\left[(X_{t}(u)-\xi_{1})\mathds{1}_{\left\\{X_{t}(u)\neq\xi_{1}\right\\}}\right]\right|$
$\displaystyle\leq\mathbb{E}\left[\left|X_{t}(u)-\xi_{1}\right|\mathds{1}_{\left\\{X_{t}(u)\neq\xi_{1}\right\\}}\right]$
$\displaystyle\leq 2\mathbb{P}\left[X_{t}(u)\neq\xi_{1}\right]$
$\displaystyle\leq 2\beta/L.$

  2. 2.

Similarly, applying Proposition 24 with $E:=\\{u,v\\}$, we deduce that there
exists a coupling of $(X_{t}(u),X_{t}(v))$ and $(\xi_{1},\xi_{2})$ such that
if we let $\mathcal{E}:=\\{(X_{t}(u),X_{t}(v))\neq(\xi_{1},\xi_{2})\\}$, then
$\mathbb{P}\left[\mathcal{E}\right]\leq\dfrac{4\beta}{L}$. Hence

$\displaystyle\left|\textrm{Cov}\left(X_{t}(u),X_{t}(v)\right)\right|$
$\displaystyle=\left|\mathbb{E}\left[X_{t}(u)X_{t}(v)\right]-\mathbb{E}\left[X_{t}(u)\right]\mathbb{E}\left[X_{t}(v)\right]\right|$
$\displaystyle\leq\left|\mathbb{E}\left[X_{t}(u)X_{t}(v)\right]-\mathbb{E}\left[\xi_{1}\xi_{2}\right]\right|+\left|\mathbb{E}\left[\xi_{1}\xi_{2}\right]-\mathbb{E}\left[X_{t}(u)\right]\mathbb{E}\left[X_{t}(v)\right]\right|.$

The first term in the last expression is

$\displaystyle\left|\mathbb{E}\left[(X_{t}(u)X_{t}(v)-\xi_{1}\xi_{2})\mathds{1}_{\mathcal{E}}\right]\right|\leq
2\mathbb{P}\left[\mathcal{E}\right]\leq 8\beta/L.$

The second term does not exceed

$\displaystyle\left|\mathbb{E}\left[\xi_{1}\right]\right|\left|\mathbb{E}\left[\xi_{2}\right]-\mathbb{E}\left[X_{t}(v)\right]\right|+\left|\mathbb{E}\left[X_{t}(v)\right]\right|\left|\mathbb{E}\left[\xi_{1}\right]-\mathbb{E}\left[X_{t}(u)\right]\right|\leq
2\beta/L+2\beta/L=4\beta/L.$

This finishes our proof. ∎

We are ready to prove the lower bound on the mixing times. We denote by
$(X^{+}_{t})_{t\geq 0}$ the Glauber-Exclusion process starting from all-plus
and recall that $\rho_{+}(\cdot)$ is the solution of (9) with the initial
condition $\rho_{+}(0)=1$.

###### Proof of the lower bound.

The proof uses the classical distinguishing statistic method (see Section
$7.3$ in [37]). Our distinguishing statistic is the total magnetization
$\sum\limits_{u\in\Lambda_{L}}x(u)$. By abuse of notation, let
$X_{\infty}\sim\pi$ (so that for any initial configuration,
$X_{t}\xrightarrow[t\to\infty]{d}X_{\infty}$). Let $\beta$ be defined as in
Proposition 14. From (59), we deduce that

$\displaystyle\left|\mathbb{E}\left[\sum_{u\in\Lambda_{L}}X^{+}_{t}(u)\right]-L\rho_{+}(t)\right|\leq
2\beta,$

and by letting $t\to\infty$,

$\left|\mathbb{E}\left[\sum_{u\in\Lambda_{L}}X_{\infty}(u)\right]-L\rho_{*}\right|\leq
2\beta.$

From (60), we see that

$\displaystyle\textrm{Var}\left[\sum_{u\in\Lambda_{L}}X^{+}_{t}(u)\right]$
$\displaystyle=\sum_{u\in\Lambda_{L}}\textrm{Var}\left[X^{+}_{t}(u)\right]+\sum_{u\neq
v}\textrm{Cov}\left(X^{+}_{t}(u),X^{+}_{t}(v)\right)$ $\displaystyle\leq
L+L(L-1)\times 12\beta/L$ $\displaystyle<(12\beta+1)L.$

By letting $t$ tend to infinity, we get

$\textrm{Var}\left[\sum_{u\in\Lambda_{L}}X_{\infty}(u)\right]\leq(12\beta+1)L.$

So by Proposition 7.9 in [37],

$\displaystyle\textrm{d}_{\textsc{tv}}\left(\mathbb{P}\left[X^{+}_{t}\in\cdot\right],\pi\right)\geq
1-8\dfrac{\max\left\\{\textrm{Var}\left[\sum\limits_{u\in\Lambda_{L}}X^{+}_{t}(u)\right],\textrm{Var}\left[\sum\limits_{u\in\Lambda_{L}}X_{\infty}(u)\right]\right\\}}{\left(\mathbb{E}\left[\sum\limits_{u\in\Lambda_{L}}X^{+}_{t}(u)\right]-\mathbb{E}\left[\sum\limits_{u\in\Lambda_{L}}X_{\infty}(u)\right]\right)^{2}}.$

Let $t:=\dfrac{\log L}{-2R(\rho_{*})}-\kappa$ for some constant $\kappa$ that
we will choose later. Let $\kappa_{1}$ be a constant satisfying Lemma 18. Then

$\displaystyle\rho_{+}(t)-\rho_{*}$ $\displaystyle\geq
e^{R(\rho_{*})t-\kappa_{1}}$
$\displaystyle=\dfrac{e^{-R(\rho_{*})\kappa-\kappa_{1}}}{\sqrt{L}}.$

Therefore, when $L$ is large enough,

$\displaystyle\mathbb{E}\left[\sum_{u\in\Lambda_{L}}X^{+}_{t}(u)\right]-\mathbb{E}\left[\sum_{u\in\Lambda_{L}}X_{\infty}(u)\right]$
$\displaystyle\geq L(\rho_{+}(t)-\rho_{*})-4\beta$
$\displaystyle=\sqrt{L}e^{-R(\rho_{*})\kappa-\kappa_{1}}-4\beta.$
$\displaystyle>\dfrac{1}{2}\sqrt{L}e^{-R(\rho_{*})\kappa-\kappa_{1}}.$

Therefore,

$8\dfrac{\max\left\\{\textrm{Var}\left[\sum_{u\in\Lambda_{L}}X^{+}_{t}(u)\right],\textrm{Var}\left[\sum_{u\in\Lambda_{L}}X_{\infty}(u)\right]\right\\}}{\left(\mathbb{E}\left[\sum_{u\in\Lambda_{L}}X^{+}_{t}(u)\right]-\mathbb{E}\left[\sum_{u\in\Lambda_{L}}X_{\infty}(u)\right]\right)^{2}}<\dfrac{32(12\beta+1)}{e^{-2R(\rho_{*})\kappa-2\kappa_{1}}}.$

Note that as $R^{\prime}(\rho_{*})<0$, for $\kappa$ large enough, the last
expression is smaller than $\epsilon$. Hence

$\textrm{d}_{\textsc{tv}}\left(\mathbb{P}\left[X^{+}_{t}\in\cdot\right],\pi\right)\geq
1-\epsilon,$

and therefore,

$\displaystyle\textrm{t}_{\textsc{mix}}(\epsilon)>t=\dfrac{\log
L}{-2R(\rho_{*})}-\kappa,$

which finishes our proof. ∎

######  Remark 9.

The result in higher dimensions is proved similarly. One can first show a
version of Proposition 14 for higher dimension, where the bound on the right-
hand side is replaced by $\mathcal{O}\left(\dfrac{|E|^{2}\log
L}{L^{2}}\right)$ for dimension $d=2$, and by
$\mathcal{O}\left(\dfrac{|E|^{2}}{L^{2}}\right)$ for $d\geq 3$, and then
propagates these changes. The only modification needed in the proof is to use
the anticoncentration in Lemma 21 for the appropriate dimension.

Now we prove Theorem 2 and the upper bound in Theorem 3. First, we need the
following lemma.

######  Lemma 26.

Consider a BEP starting with one particle $\mathsf{w}$. Then, there exist
constants $\kappa$ and $\beta$ such that for any $t\in\mathbb{R}_{+}$,

$\mathbb{E}\left[\left|\mathsf{Piv}\left(\mathsf{w},t\right)\right|\right]\leq\psi(t)+e^{\kappa
t}\dfrac{\beta}{\sqrt{L}}.$

###### Proof of Theorem 2 and upper bound in Theorem 3.

Let $x\in\mathcal{X}$ be an arbitrary configuration. We want to compare
$\mathbb{P}_{x}\left[X_{t}\in\cdot\right]$ and
$\mathbb{P}_{\pi}\left[X_{t}\in\cdot\right]$ for some $t>0$. We use the same
BEP starting from $E=\Lambda_{L}$ to generate $X^{x}_{t}$ and $X^{\pi}_{t}$,
the processes starting from $x$ and $\pi$. Let $\mathsf{w}\in\Lambda_{L}$ be
arbitrary. We see that

$\displaystyle\textrm{d}_{\textsc{tv}}\left(\mathbb{P}_{x}\left[X_{t}\in\cdot\right],\mathbb{P}_{\pi}\left[X_{t}\in\cdot\right]\right)$
$\displaystyle\leq\mathbb{P}\left[X^{x}_{t}\neq X^{\pi}_{t}\right]$
$\displaystyle\leq\sum_{u\in\Lambda_{L}}\mathbb{P}\left[X^{x}_{t}(u)\neq
X^{\pi}_{t}(u)\right]$
$\displaystyle\leq\sum_{u\in\Lambda_{L}}\mathbb{P}\left[\mathsf{Piv}\left(u,t\right)\neq\O\right]$
$\displaystyle=L\mathbb{P}\left[\mathsf{Piv}\left(\mathsf{w},t\right)\neq\O\right]$
$\displaystyle\leq
L\mathbb{E}\left[\left|\mathsf{Piv}\left(\mathsf{w},t\right)\right|\right].$

By Observation 11, it is not hard to see that,

$\mathbb{E}\left[\left|\mathsf{Piv}\left(\mathsf{w},mt\right)\right|\right]\leq\mathbb{E}\left[\left|\mathsf{Piv}\left(\mathsf{w},t\right)\right|\right]^{m}.$

Hence, with the constant $\kappa$ and $\beta$ as in Lemma 26,

$\textrm{d}_{\textsc{tv}}\left(\mathbb{P}_{x}\left[X_{mt}\in\cdot\right],\mathbb{P}_{\pi}\left[X_{mt}\in\cdot\right]\right)\leq
L\mathbb{E}\left[\left|\mathsf{Piv}\left(\mathsf{w},mt\right)\right|\right]\leq
L\left(\psi(t)+e^{\kappa t}\dfrac{\beta}{\sqrt{L}}\right)^{m}.$ (61)

Provided that $\left(\psi(t)+e^{\kappa t}\dfrac{\beta}{\sqrt{L}}\right)<1$, we
can take $m=\left\lceil\dfrac{\log
L-\log\epsilon}{-\log\left(\psi(t)+e^{\kappa
t}\dfrac{\beta}{\sqrt{L}}\right)}\right\rceil$ to make the right-hand side
smaller than $\epsilon$. This implies that

$\textrm{t}_{\textsc{mix}}(x;\epsilon)\leq mt\leq\left(\dfrac{\log
L-\log\epsilon}{-\log\left(\psi(t)+e^{\kappa
t}\dfrac{\beta}{\sqrt{L}}\right)}+1\right)t,$

for any $t$ such that $\left(\psi(t)+e^{\kappa
t}\dfrac{\beta}{\sqrt{L}}\right)<1$. It remains to choose a suitable $t$ that
leads to our results. We will choose $t$ such that $e^{\kappa
t}\dfrac{\beta}{\sqrt{L}}\ll\psi(t)L^{-1/3}$. This will imply

$\displaystyle\psi(t)+e^{\kappa
t}\dfrac{\beta}{\sqrt{L}}\leq\psi(t)(1+L^{-1/3}).$

Hence, with $\kappa_{1}$ the constant in Lemma 18, we have

$\displaystyle\log\left(\psi(t)+e^{\kappa
t}\dfrac{\beta}{\sqrt{L}}\right)\leq\log\psi(t)+\log(1+L^{-1/3})\leq
R(\rho_{*})t+\kappa_{1}+L^{-1/3}.$

Therefore, if $R(\rho_{*})t+\kappa_{1}+L^{-1/3}<0$,

$\displaystyle\textrm{t}_{\textsc{mix}}(\epsilon)$
$\displaystyle\leq\dfrac{\log
L-\log\epsilon}{-(R(\rho_{*})t+\kappa_{1}+L^{-1/3})}t+t$
$\displaystyle=\dfrac{\log
L-\log\epsilon}{-(R(\rho_{*})+\kappa_{1}/t+1/(tL^{1/3})}+t$
$\displaystyle=\dfrac{\log L}{-R(\rho_{*})}+\mathcal{O}\left(\dfrac{\log
L}{t}\right)+t.$

We can take $t=\sqrt{\log L}$, which satisfies all the conditions we impose on
$t$, to finish the proof of the upper bound in Theorem 3 for $d=1$. The proof
for higher dimensions is similar.

It remains to prove Theorem 2. Recall that

gap
$\displaystyle=\lim_{t\to\infty}-\dfrac{1}{t}\log\max_{x\in\Omega}\textrm{d}_{\textsc{tv}}\left(\mathbb{P}_{x}\left[X_{t}\in\cdot\right],\pi\right).$
$\displaystyle=\lim_{m\to\infty}-\dfrac{1}{mt}\log\max_{x\in\Omega}\textrm{d}_{\textsc{tv}}\left(\mathbb{P}_{x}\left[X_{mt}\in\cdot\right],\pi\right).$

On the other hand, equation (61) implies that, for any $t\geq 0$,
$m\in\mathbb{Z}_{+}$,

$\log\max_{x\in\Omega}\textrm{d}_{\textsc{tv}}\left(\mathbb{P}_{x}\left[X_{mt}\in\cdot\right],\pi\right)\leq\log
L+m\log\left(\psi(t)+e^{\kappa t}\dfrac{\beta}{\sqrt{L}}\right).$

Therefore,

$\texttt{gap}\geq\lim_{m\to\infty}-\dfrac{1}{mt}\left(\log
L+m\log\left(\psi(t)+e^{\kappa
t}\dfrac{\beta}{\sqrt{L}}\right)\right)=-\dfrac{1}{t}\log\left(\psi(t)+e^{\kappa
t}\dfrac{\beta}{\sqrt{L}}\right).$

This implies

$\displaystyle\liminf_{L\to\infty}\texttt{gap}\geq-\dfrac{1}{t}\log\psi(t)\geq-\dfrac{1}{t}(R(\rho_{*})t+\kappa_{1})=-R(\rho_{*})-\kappa_{1}/t.$

Note that this inequality is true for any positive $t$. Letting $t$ tend to
infinity, we obtain the conclusion of Theorem 2. ∎

Now we prove Lemma 26.

###### Proof of Lemma 26.

We use the coupling of BEP and IBP starting from one particle $\mathsf{w}$. We
write $\mathsf{Piv}\left(t\right)$ for $\mathsf{Piv}\left(\mathsf{w},t\right)$
and $\widetilde{\mathsf{Piv}}(t)$ for
$\widetilde{\mathsf{Piv}}(\mathsf{w},t)$. Let $\mathcal{E}$ denote the event
that the coupling is successful until time $t$. Then, on $\mathcal{E}$, we
have

$\left|\mathsf{Piv}\left(t\right)\right|=\left|\widetilde{\mathsf{Piv}}(t)\right|.$

Let $\beta_{1}$ be as in Proposition 14. Then
$\mathbb{P}\left[\mathcal{E}^{C}\right]\leq\beta_{1}/L$, where
$\mathcal{E}^{C}$ denotes the complement of $E$. Hence

$\displaystyle\mathbb{E}\left[|\mathsf{Piv}\left(t\right)|\right]$
$\displaystyle=\mathbb{E}\left[|\mathsf{Piv}\left(t\right)|(\mathds{1}_{\mathcal{E}}+\mathds{1}_{\mathcal{E}^{C}})\right]$
$\displaystyle=\mathbb{E}\left[|\mathsf{Piv}\left(t\right)|\mathds{1}_{\mathcal{E}}\right]+\mathbb{E}\left[|\mathsf{Piv}\left(t\right)|\mathds{1}_{\mathcal{E}^{C}}\right].$

Note that

$\mathbb{E}\left[|\mathsf{Piv}\left(t\right)|\mathds{1}_{\mathcal{E}}\right]=\mathbb{E}\left[|\widetilde{\mathsf{Piv}}(t)|\mathds{1}_{\mathcal{E}}\right]\leq\mathbb{E}\left[|\widetilde{\mathsf{Piv}}(t)|\right]=\psi(t).$

Let $\kappa_{1}$ be the constant in Lemma 19. Note also that

$\mathbb{E}\left[|\mathsf{Piv}\left(t\right)|\mathds{1}_{\mathcal{E}^{C}}\right]^{2}\leq\mathbb{E}\left[|W_{t}|\mathds{1}_{\mathcal{E}^{C}}\right]^{2}\leq\mathbb{E}\left[|W_{t}|^{2}\right]\mathbb{P}\left[\mathcal{E}^{C}\right]\leq\mathbb{E}\left[|\widetilde{W}_{t}|^{2}\right]\mathbb{P}\left[\mathcal{E}^{C}\right]\leq
e^{\kappa_{1}t}\times\dfrac{\beta_{1}}{L},$

where we have used $|\mathsf{Piv}\left(t\right)|\leq|W_{t}|$ for the first
inequality, the Cauchy-Schwarz inequality for the second inequality, Lemma 20
for the third inequality, and Lemma 19 for the last inequality. Therefore,

$\displaystyle\mathbb{E}\left[|\mathsf{Piv}\left(t\right)|\right]\leq\psi(t)+e^{\kappa_{1}t/2}\times\dfrac{\sqrt{\beta_{1}}}{\sqrt{L}}.$

We can take $\kappa=\kappa_{1}/2$ and $\beta=\sqrt{\beta_{1}}$ to finish the
proof. ∎

##  6 Upper bound

In this section, we prove the upper bound in Theorem 1. For simplicity, we
only give the proof for $d=1$, but we will comment on how to adapt the proof
for $d=2$.

Recall that in the decomposition of the local flip-rate function $c(\cdot)$ in
Proposition 7, $f_{1}\equiv 1,\,f_{2}\equiv-1$, so the updates given by
$f_{1}$, $f_{2}$ are called _oblivious_ because they can be carried out
without looking at the spins of any sites. These updates will play a special
role in our proof. It is more convenient to merge two oblivious deterministic
updates into a single oblivious random update as follows. We allow the
collection of marks to contain another type of mark: the refresh marks, which
can appear on the sites in the space-time slab. We adapt the definition of the
collection of marks accordingly. From now on, let

$\bar{\rho}:=\dfrac{\lambda_{1}-\lambda_{2}}{\lambda_{1}+\lambda_{2}}.$ (62)

We will use the following graphical construction.

#### Graphical Construction 2.

Let the background process $\Xi$ be defined as follows.

$\Xi=\left((\Xi^{\text{exclusion}}_{u})_{0\leq u\leq
L-1},(\Xi^{\text{refresh}}_{u})_{0\leq u\leq
L-1},(\Xi^{\text{Glauber}}_{i,u})_{3\leq i\leq q,\,0\leq u\leq L-1}\right),$
(63)

where $(\Xi^{\text{exclusion}}_{u})_{0\leq u\leq
L-1},(\Xi^{\text{refresh}}_{u})_{0\leq u\leq
L-1},(\Xi^{\text{Glauber}}_{i,u})_{3\leq i\leq q,\,0\leq u\leq L-1}$ are
independent homogeneous Poisson processes, and

  * •

$\Xi^{\text{exclusion}}_{u}$ is of intensity $L^{2}$, $0\leq u\leq L-1$,

  * •

$\Xi^{\text{refresh}}_{u}$ is of intensity $\lambda_{1}+\lambda_{2}$, $0\leq
u\leq L-1$,

  * •

$\Xi^{\text{Glauber}}_{i,u}$ is of intensity $\lambda_{i}$, $3\leq i\leq
q,\,0\leq u\leq L-1$.

The process $\Xi$ naturally defines a collection of marks $\mathcal{C}$
consisting of exclusion, refresh, and Glauber marks as follows.

  * •

Whenever $\Xi^{\text{exclusion}}_{u}$ jumps, we put an exclusion mark on the
edge $(u,u+1)$.

  * •

Whenever $\Xi^{\text{refresh}}_{u}$ jumps, we put a refresh mark at site $u$.

  * •

Whenever $\Xi^{\text{Glauber}}_{i,u}$ jumps, we put a Glauber mark of type $i$
at site $u$.

Given the collection of marks $\mathcal{C}$, an infinite sequence of
independent Rademacher variables $\mathfrak{R}_{\bar{\rho}}$, and an initial
condition $x_{0}\in\mathcal{X}$, we construct the process, which we always
denote by $(X^{x_{0}}_{t})_{t\geq 0}$, as follows.

  * •

The process $(X^{x_{0}}_{t})_{t\geq 0}$ is a piecewise constant process
starting from $x_{0}$ and can only jump when a mark appears.

  * •

When we see an exclusion mark, say on the edge $(u,u+1)$, we make the
transition $x\mapsto x^{u\leftrightarrow u+1}$ (exchange the spins of sites
$u$ and $u+1$).

  * •

When we see a refresh mark, say at site $u$, we replace the $u-th$ coordinate
of $X$ by an independent $\mathfrak{R}_{\bar{\rho}}$ (randomize the spin at
site $u$).

  * •

When we see a Glauber mark, say of type $i$ at site $u$, we make the
transition $x\mapsto x^{u,f_{i}(x_{u+\cdot})}$ (update site $u$ using the
function $f_{i}$).

Then, when the collection of marks $\mathcal{C}$ is generated by $\Xi$ and the
Rademacher variables $\mathfrak{R}_{\bar{\rho}}$ are independent of $\Xi$, the
process $(X^{x_{0}}_{t})_{t\geq 0}$ is a Markov process with the generator
$\mathcal{L}_{GE}$ starting from $x_{0}$.

#### The grand coupling.

We can use the same Poisson process $\Xi$ and the same independent Rademacher
variables $\mathfrak{R}_{\bar{\rho}}$ to construct the Glauber-Exclusion
process from any configuration $x\in\mathcal{X}$. It is not hard to see that
this coupling preserves order, i.e. , $x\leq x^{\prime}\Rightarrow\forall
t\geq 0,\,X^{x}_{t}\leq X^{x^{\prime}}_{t}$.

#### Correlation marks associated with Glauber marks.

It is convenient to associate each Glauber mark with a set of points as
follows. Whenever we see a Glauber mark at site $u$, we put on each site in
$B(u,m)$ a mark, and we call those marks the correlation marks associated to
the aforementioned Glauber mark. These correlation marks indicate that the
spins of those sites are (possibly) used in the corresponding Glauber updates.

We introduce another Markov process $Z$, which we call the _independent-site
process_ , constructed as follows.

#### Independent-site process.

From the collection of marks generated by $\Xi$, let $(Z_{t})_{t\geq 0}$ be
the process taking values in $\mathcal{Z}:=\\{0,1\\}^{\Lambda_{L}}$
constructed as follows. For any initial configuration $z_{0}\in\mathcal{Z}$,

  * •

The process $(Z^{z_{0}}_{t})_{t\geq 0}$ is a piecewise constant process
starting from $z_{0}$ and can only jump when a mark appears.

  * •

When we see an exclusion mark, say on the edge $(u,u+1)$, we make the
transition $z\mapsto z^{u\leftrightarrow u+1}$.

  * •

When we see a refresh mark, say at site $u$, we make the transition $z\mapsto
z^{u,1}$.

  * •

When we see a Glauber mark, say at site $u$, we remove all the sites with the
associated correlation marks from $Z$, i.e. we make the transition $z\mapsto
z\setminus B(u,m)$.

Here we have identified the elements of $\\{0,1\\}^{\Lambda_{L}}$ with the
subsets of $\Lambda_{L}$. As its name suggests, the process $Z$ tracks the
positions of independent sites in X conditionally on $\Xi$. Roughly speaking,
when a site is resampled according to an independent Rademacher
$\mathfrak{R}_{\bar{\rho}}$ due to a refresh mark, it becomes independent of
all other sites and hence is added to $Z$. On the other hand, when a site $u$
is updated due to a Glauber mark, it creates a correlation with its neighbors,
and all these neighbors are removed from $Z$. By abuse of notation, for any
$z\subset\Lambda_{L}$, we denote by $\mathbb{P}_{z}\left[\cdot\right]$ the
distribution of the process $Z$ starting from $z$.

#### The grand coupling for $Z$.

We can use the same background process $\Xi$ to construct the process $Z$ from
any configuration $z\subset\Lambda_{L}$.

######  Remark 10 (Attractiveness of the process $Z$).

For any $z_{1},z_{2}\subset\Lambda_{L}$ such that $z_{1}\leq z_{2}$, if
$Z^{z_{1}}$ and $Z^{z_{2}}$ are constructed using the grand coupling, then
almost surely,

$\displaystyle\forall t\geq 0,\;Z^{z_{1}}_{t}\leq Z^{z_{2}}_{t}.$

#### Red, blue, and green regions.

Let the red, blue, and green regions be defined as follows:

$\displaystyle\mathrm{Blue}(t)$ $\displaystyle=Z^{\O}_{t},$
$\displaystyle\mathrm{Red}(t)$
$\displaystyle=\\{u\in\Lambda_{L}|X^{+}_{t}(u)\neq X^{-}_{t}(u)\\},$
$\displaystyle\mathrm{Green}(t)$
$\displaystyle=\Lambda_{L}\setminus\left(\mathrm{Red}(t)\cup\mathrm{Blue}(t)\right).$

Here, $(X^{+}_{t})_{t\geq 0}$ and $(X^{-}_{t})_{t\geq 0}$ denote the process
$X$ starting from configurations all-plus and all-minus, respectively. To
lighten the notation, we write $X^{x}_{t}(\mathrm{Blue})$ for
$X^{x}_{t}(\mathrm{Blue}(t))$, and similarly for $\mathrm{Red}$, and
$\mathrm{Green}$. Let $\mathcal{F}_{t}$ be the $\sigma$-algebra generated by
$\Xi([0,t])$ and
$\left(X_{s}^{x}(\Lambda_{L}\setminus\mathrm{Blue}(s)\right)_{x\in\mathcal{X},0\leq
s\leq t}$. The interest in introducing red, blue, and green regions is the
following.

######  Observation 27.

By construction, almost surely, conditionally on $\mathcal{F}_{t}$, for any
$x\in\mathcal{X}$,
$X^{x}_{t}(\mathrm{Blue})\sim\mathfrak{R}_{\bar{\rho}}^{\otimes{\mathrm{Blue}(t)}}$
and is independent of $X^{x}_{t}(\mathrm{Red}\cup\mathrm{Green})$.

From now on, for any subset $E\subset\Lambda_{L}$, we will use the shorthand
notation $\nu_{E}:=\mathfrak{R}_{\bar{\rho}}^{\otimes E}$. The meanings of the
red, blue, and green regions above are as follows: the red region
$\mathrm{Red}$ is the set of disagreements of $X^{+}$ and $X^{-}$.
Conditionally on $\mathcal{F}_{t}$, the spins in the blue region
$\mathrm{Blue}$ have the product measure $\nu_{\mathrm{Blue}(t)}$, independent
of the spins on $\mathrm{Red}\cup\mathrm{Green}$, and the green region
$\mathrm{Green}$ is the set of sites whose spins may present correlation but
independent of the initial configuration. More precisely, notice that, as
$X^{+}_{t}(\mathrm{Green})=X^{-}_{t}(\mathrm{Green})$, by monotonicity, in the
grand coupling, the spins on $\mathrm{Green}$ do not depend on the initial
configuration anymore:

$X^{+}_{t}(\mathrm{Green})=X^{-}_{t}(\mathrm{Green})=X^{x}_{t}(\mathrm{Green}),\;\forall
x\in\mathcal{X}.$

Therefore, we can safely denote it by $X_{t}(\mathrm{Green})$. Note also that
$X_{t}(\mathrm{Green})$ and $X^{x}_{t}(\mathrm{Red}),\,x\in\mathcal{X}$, are
$\mathcal{F}_{t}$ measurable.

The following lemma summarizes the discussion above.

######  Lemma 28 (The conditional law).

Almost surely, for any $t\geq 0$, conditionally on $\mathcal{F}_{t}$,

$X^{x}_{t}\sim\nu_{\mathrm{Blue}(t)}\otimes\delta_{X^{x}_{t}(\mathrm{Red})}\otimes\delta_{X_{t}(\mathrm{Green})},$
(64)

where $\delta_{X_{t}(\mathrm{Green})}$ (resp.
$\delta_{X^{x}_{t}(\mathrm{Red})}$) is the Dirac measure on
$\\{-1,1\\}^{\mathrm{Green}(t)}$ (resp. $\\{-1,1\\}^{\mathrm{Red}(t)}$) which
gives all mass to $X_{t}(\mathrm{Green})$ (resp. $X^{x}_{t}(\mathrm{Red})$).

###  6.1 Proof of the upper bound

First, note that, by the definition of $\mathrm{Red}$,

$\displaystyle\left|\mathrm{Red}(t)\right|=\dfrac{1}{2}\mathbb{E}\left[\sum_{u\in\Lambda_{L}}\left(X^{+}_{t}(u)-X^{-}_{t}(u)\right)\Big{|}\mathcal{F}_{t}\right].$

This and Corollary 25 lead to the following lemma.

######  Lemma 29 (Decay of the red region).

$\displaystyle\mathbb{E}\left[\left|\mathrm{Red}(t)\right|\right]=\dfrac{L}{2}(\rho^{+}(t)-\rho^{-}(t))+\mathcal{O}\left(1\right).$
(65)

For any $\beta\in\mathbb{R}_{>0}$, let

$\mathrm{BAD}_{\beta}=\left\\{z\in\mathcal{Z}:\exists\,0\leq
l\leq\left\lceil\dfrac{L}{\left\lceil\beta\log
L\right\rceil}\right\rceil,\,z\cap\left\llbracket l\left\lceil\beta\log
L\right\rceil,(l+1)\left\lceil\beta\log
L\right\rceil-1\right\rrbracket=\O\right\\},$

where $\left\llbracket i,j\right\rrbracket:=[i,j]\cap\mathbb{Z}.$

We will need the following results.

######  Lemma 30 (BAD is rarely visited).

There exist constants $\beta_{2},\beta_{3}\in\mathbb{R}_{>0}$ such that, for
any $t>\beta_{3}$,

$\max_{z\in\mathcal{Z}}\mathbb{P}_{z}\left[\exists
s\in[t,t+1]:Z_{s}\in\mathrm{BAD}_{\beta_{2}}\right]=\mathcal{O}\left(1/L^{6}\right).$
(66)

######  Proposition 31 (“Adjacent” measures quickly become close).

Let $z\subset\Lambda_{L}$, let $u\in\Lambda_{L}\setminus z$, and let $y$ be an
arbitrary spin configuration on $\Lambda_{L}\setminus(z\cup\\{u\\})$. Let
$\pi_{1},\pi_{2}$ be the two probability distributions on $\mathcal{X}$ given
by

$\pi_{1}=\delta^{u}_{1}\otimes\nu_{z}\otimes\delta_{y};\;\pi_{2}=\nu_{z\cup\\{u\\}}\otimes\delta_{y},$
(67)

where $\delta^{u}_{1}$ denotes the Dirac measure on $\\{-1,1\\}^{\\{u\\}}$
which gives all mass to $1$. Let $\beta_{2}$ be as in Lemma 30. Then there
exists a constant $\kappa$ such that

$\left\|\mathbb{P}_{\pi_{1}}\left[X_{1}\in\cdot\right]-\mathbb{P}_{\pi_{2}}\left[X_{1}\in\cdot\right]\right\|_{TV}\leq\kappa\left(\dfrac{\log
L}{\sqrt{L}}+\sqrt{\mathbb{P}_{z\cup\\{u\\}}\left[\exists
s\in[0,1]:Z_{s}\in\mathrm{BAD}_{\beta_{2}}\right]}\right).$ (68)

The same result holds if we replace $\pi_{1}$ by
$\delta^{u}_{-1}\otimes\nu_{z}\otimes\delta_{y}$.

Proposition 31 says that very quickly, one cannot distinguish two processes
starting from the measure of the form $\pi_{2}$ and its perturbation
$\pi_{1}$: the total variation distance drop from $1$ to $\dfrac{\log
L}{\sqrt{L}}$ in a time $\mathcal{O}\left(1\right)$. This is the result whose
proof involves the information percolation framework and the idea from
excursion theory.

With Lemma 30 and Proposition 31, we can now prove the upper bound in Theorem
1.

###### Proof of the upper bound.

Let $t=t_{1}+1$, for some number $t_{1}$ that we will choose later. Let
$\beta_{2},\beta_{3}$ be as in Lemma 30. To lighten the notation, we write
$\mathrm{BAD}$ for $\mathrm{BAD}_{\beta_{2}}$. We will estimate directly
$\textrm{d}_{\textsc{tv}}\left(\mathbb{P}_{x_{1}}\left[X_{t}\in\cdot\right],\mathbb{P}_{x_{2}}\left[X_{t}\in\cdot\right]\right)$
for $x_{1},x_{2}$ arbitrary in $\mathcal{X}$, as we do not have an explicit
formula for the invariant measure $\pi$. We use the grand coupling to
construct the Glauber-Exclusion process from any initial configuration. To
lighten the notation, here we write $\mathrm{Red},\,\mathrm{Blue}$, and
$\mathrm{Green}$ for $\mathrm{Red}(t_{1}),\mathrm{Blue}(t_{1})$, and
$\mathrm{Green}(t_{1})$. Let $\left|\mathrm{Red}\right|=\mathcal{R}$ and