# Fast Consensus and Metastability
in a Highly Polarized Social Network

Antonio Galves  Kádmo de S. Laxa

(October 27, 2022)

###### Abstract

A polarized social network is modeled as a system of interacting marked point
processes with memory of variable length. Each point process indicates the
successive times in which a social actor express a "favorable" or "contrary"
opinion. After expressing an opinion, the social pressure on the actor is
reset to 0, waiting for the group’s reaction. The orientation and the rate at
which an actor express an opinion is influenced by the social pressure exerted
on them, modulated by a polarization coefficient. We prove that the network
reaches an instantaneous but metastable consensus, when the polarization
coefficient diverges.

Keywords: Interacting marked point processes with memory of variable length,
metastability, fast consensus, social networks.

AMS MSC: 60K35, 60G55, 91D30.

##  1 Introduction

Huge discrepancies between the results of electoral intentions carried out a
few days before the actual voting and the electoral poll results during the
first round of the 2018 and 2022 presidential elections in Brazil were
striking (see for instance Branco, (2018), Franco, (2018), Nicas, (2022) and
Phillips, (2022)).

It was conjectured that these discrepancies in the brazilian elections were
due to campaigns in support of a group of candidates launched on social media
a few days before the elections (see for instance Belli, (2018) and Mello,
(2018, 2019)). This conjecture rises a question: is social-media campaigning
enough to change in a quite short period of time the voting intention of a
significant portion of voters?

To address this question we introduce a new stochastic model that mimics some
important features of real world social networks. We prove that for this
model, in a situation of high polarization, the opinion expressed by the
majority of the actors can globally change in a very short interval of time.
In other words, a social network described by our model is a machine to
produce consensus in a very short time. The consensus so produced remains for
a long and unpredictable time. This is the message of this article.

The model we propose can be informally described as follows.

  1. 1.

The model is a system with interacting marked point processes.

  2. 2.

Each point process indicates the successive times in which a social actor
express either a “favorable” $(+1)$ or “contrary” $(-1)$ opinion on a certain
subject.

  3. 3.

When an actor express an opinion, social pressure on them is reset to 0, and
simultaneously, social pressures on all the other actors change by one unit in
the direction of the opinion that was just expressed.

  4. 4.

The orientation and the rate at which an actor express an opinion is
influenced by the social pressure exerted on them, modulated by a polarization
coefficient.

In this model, the actors listen to and influence each other. Thus, after
expressing an opinion, each actor waits for the group’s reaction to the
opinion they have just expressed. It is this reaction that will guide the
actor’s next manifestation. This is the content of the third point presented
above. The resetting of the social pressure on the actor every time they
express an opinion makes the point processes describing the activity of the
actor to have a memory of variable length.

Starting with the classical voter model, introduced by Holley and Liggett,
(1975), several articles addressed issues associated to opinion dynamics in a
social network. See Wasserman and Faust, (1994), Castellano et al., (2009) and
Aldous, (2013) for a general review on this subject. However, to the best of
our knowledge, a model with the features introduced here was not considered
yet in the literature of social networks.

The model introduced here belongs to the same class of systems of interacting
point process with memory of variable length that was introduced in discrete
time by Galves and Löcherbach, (2013) and in continuous time by De Masi et
al., (2014) to model system of spiking neurons. This class of systems was
since then studied in several articles, including Duarte et al., (2015),
Brochini et al., (2016), Duarte and Ost, (2016), Fournier and Löcherbach,
(2016), Galves and Löcherbach, (2016), Yaginuma, (2016), Ferrari et al.,
(2018), André, (2019), Baccelli and Taillefumier, (2019), Duarte et al.,
(2019), Galves et al., (2019), André and Planche, (2021), Baccelli and
Taillefumier, (2021), Nascimento, (2021), Santis et al., (2021), Baccelli et
al., (2022), Chariker and Lebowitz, (2022) and Löcherbach and Monmarché,
(2022).

Let us now informally present our results. The existence of the process and
the uniqueness of its invariant probability measure is the content of Theorem
1.

When the polarization coefficient diverges, the invariant probability measure
concentrates on the set of consensus lists and the time the system needs to
get there goes to zero. Here by a consensus list we mean any list in which all
the social pressures push in the same direction. This is the content of
Theorem 2.

In the social network, the consensus has a metastable behavior. This means
that the direction of the social pressures on the actors globally changes
after a long and unpredictable random time. This is the content of Theorem 3.

The notion of metastability considered here is inspired by the so called
pathwise approach to metastability introduced by Cassandro et al., (1984). For
more references and an introduction to the topic, we refer the reader to
Olivieri and Vares, (2005), Hollander, (2009) and Fernández et al., (2015).

This article is organized as follows. In Section 2 we introduce the model, the
notation and state the main results. In Section 3 we introduce some extra
notation and prove two auxiliary propositions. In Section 4, 5 and 6 we prove
Theorems 1, 2 and 3, respectively.

##  2 Definitions, notation and main results

Let $\mathcal{A}=\\{1,2,...,N\\}$ be the set of social actors, with $N\geq 3$,
and let $\mathcal{O}=\\{-1,+1\\}$ be the set of opinions that an actor can
express, where $+1$ (respectively, $-1$) represents a favorable opinion
(respectively, a contrary opinion).

Let $\beta\geq 0$ be the polarization coefficient of this network. The
polarization coefficient of the network parametrizes the tendency of each
actor $a\in\mathcal{A}$ to follow the social pressure that the actors
belonging to $\mathcal{A}\setminus\\{a\\}$ exert on $a$.

To describe the time evolution of the social network we introduce a family of
maps on the set of lists of social pressures. For any actor $a\in\mathcal{A}$,
for any opinion $\mathfrak{o}\in\mathcal{O}$ and for any list
$u=(u(a):a\in\mathcal{A})$, where $u(a)$ is a integer number, we define the
new list $\pi^{a,\mathfrak{o}}(u)$ as follows

$\pi^{a,\mathfrak{o}}(u)(b)=\begin{cases}u(b)+\mathfrak{o},\text{ if }b\neq
a,\\\ 0,\text{ if }b=a.\end{cases}$

The time evolution of the social network can be described as follows.

  * •

Assume that at time $0$, the list of social pressures exerted on the actors is
$u=(u(a):a\in\mathcal{A})$.

  * •

Independent exponential random times with parameters
$\exp\left(\beta\mathfrak{o}u(a)\right)$ are associated to each actor
$a\in\mathcal{A}$ and each opinion $\mathfrak{o}\in\mathcal{O}$.

  * •

Denote $(A_{1},O_{1})$ the pair (actor, opinion) associated to the exponential
random time that occurs first.

  * •

At this random time, the list of social pressures changes from $u$ to
$\pi^{A_{1},O_{1}}(u)$.

  * •

At the new list of social pressures $\pi^{A_{1},O_{1}}(u)$, independent
exponential random times with parameters
$\exp\left(\beta\mathfrak{o}\pi^{A_{1},O_{1}}(u)(a)\right)$ are associated to
each actor $a\in\mathcal{A}$ and opinion $\mathfrak{o}\in\mathcal{O}$.

  * •

Denote $(A_{2},O_{2})$ the pair (actor, opinion) associated to the exponential
random time that occurs first, and so on.

Let $((A_{n},O_{n}):n\geq 1)$ be the sequence of pairs (actor, opinion),
associated to the exponential random times realizing the successive minima and
let $(T_{n}:n\geq 1)$ be the successive random times associated to them. Let
also $(U_{t}^{\beta,u})_{t\in[0,+\infty)}$ be the time evolution defined as
follows

$U_{t}^{\beta,u}=u\text{, if }0\leq t<T_{1},$

and for any $t\geq T_{1}$,

$U_{t}^{\beta,u}=\pi^{A_{m},O_{m}}(U^{\beta,u}_{T_{m-1}})\text{, if }T_{m}\leq
t<T_{m+1},$

where $T_{0}=0$.

So defined, the system of social pressures
$(U_{t}^{\beta,u})_{t\in[0,+\infty)}$ evolves as a Markov jump process taking
values in the set

$\mathcal{S}=\\{u=(u(a):a\in\mathcal{A})\in\mathbb{Z}^{N}:\min\\{|u(a)|:a\in\mathcal{A}\\}=0\\}$

and with infinitesimal generator defined as follows

$\mathcal{G}f(u)=\sum_{\mathfrak{o}\in\mathcal{O}}\sum_{b\in\mathcal{A}}\exp\left(\beta\mathfrak{o}u(b)\right)\left[f(\pi^{b,\mathfrak{o}}(u))-f(u)\right],$
(1)

for any bounded function $f:\mathcal{S}\to\mathbb{R}$.

The opinion dynamics of the social network is described by the system of
interacting marked point process $((T_{n},(A_{n},O_{n})):n\geq 1)$ together
with the initial list of social pressures $U_{0}^{\beta,u}=u$.

Observes that the rates of these marked point processes have a variable length
dependency from the past. This comes from the fact that each time actor $a$
express an opinion, the social pressure on them is reset to $0$ and therefore,
the actor forgets the past.

In a more formal way, for any pair $(a,\mathfrak{o})$, with $a\in\mathcal{A}$,
$\mathfrak{o}\in\mathcal{O}$ and for any pair of real numbers $s<t$, let us
define the counting measure

$Z^{a,\mathfrak{o}}((s,t])=\sum_{n=1}^{+\infty}\mathbf{1}\\{s<T_{n}\leq
t,(A_{n},O_{n})=(a,\mathfrak{o})\\}.$

For any time $t>0$, let $L_{t}^{a}$ be the last expression time of actor $a$
before time $t$

$L_{t}^{a}=\sup\\{T_{n}\leq t:A_{n}=a\\}.$

We use the convention that $\sup\emptyset=0$. Then, the social pressure on
actor $a$ at time $t>0$ can be equivalently defined as

$U_{t}^{\beta,u}(a)=\begin{cases}u(a)+\displaystyle\sum_{\mathfrak{o}\in\mathcal{O}}\sum_{b\in\mathcal{A}\setminus\\{a\\}}\mathfrak{o}Z^{b,\mathfrak{o}}((0,t]),&\text{
if }L_{t}^{a}=0,\\\
\displaystyle\sum_{\mathfrak{o}\in\mathcal{O}}\sum_{b\in\mathcal{A}\setminus\\{a\\}}\mathfrak{o}Z^{b,\mathfrak{o}}((L_{t}^{a},t]),&\text{
if }L_{t}^{a}>0.\end{cases}$

A standard computation shows that the rate of the point process
$N^{a,\mathfrak{o}}$ at time $t$, conditioned in the past history, is given by

$\exp(\beta\mathfrak{o}U_{t}^{\beta,u}(a)),$

and this value depends on the history of the system in the interval
$(L_{t}^{a},t]$ which has a variable length.

The process $(U_{t}^{\beta,u})_{t\in[0,+\infty)}$ is well defined for any
$t\in[0,\sup\\{T_{m}:m\geq 1\\})$. The unique thing that must yet be clarified
is whether this process is defined for any positive time $t$, i.e. if
$\sup\\{T_{m}:m\geq 1\\}=+\infty$ or not. This is part of the content of the
first theorem.

######  Theorem 1.

For any $\beta\geq 0$ and for any starting list $u\in\mathcal{S}$, the
following holds.

  1. 1.

The sequence $(T_{m}:m\geq 1)$ of jumping times of the process
$(U_{t}^{\beta,u})_{t\in[0,+\infty)}$ satisfies

${\mathbb{P}}(\sup\\{T_{m}:m\geq 1\\}=+\infty)=1,$

which assures the existence of the process for all time $t\in[0,+\infty)$.

  2. 2.

The process $(U_{t}^{\beta,u})_{t\in[0,+\infty)}$ has a unique invariant
probability measure $\mu^{\beta}$.

We define the set of positive (respectively negative) consensus lists as

$\mathcal{C}^{+}=\\{u\in\mathcal{S}:u\neq\vec{0},u(a)\geq 0\text{, for all
}a\in\mathcal{A}\\}$

and

$\mathcal{C}^{-}=\\{u\in\mathcal{S}:u\neq\vec{0},u(a)\leq 0\text{, for all
}a\in\mathcal{A}\\},$

where $\vec{0}\in\mathcal{S}$ is the null list. Consider also the set of
positive and negative ladder lists

$\mathcal{L}^{+}=\\{u\in\mathcal{S}:\\{u(1),...,u(N)\\}=\\{0,1,...,N-1\\}\\}$

and

$\mathcal{L}^{-}=\\{u\in\mathcal{S}:\\{u(1),...,u(N)\\}=\\{0,-1,...,-(N-1)\\}\\}.$

The set of ladder lists is given by
$\mathcal{L}=\mathcal{L}^{+}\cup\mathcal{L}^{-}$.

To state the next theorems, we define for any $u\in\mathcal{S}$ and for any
$B\subset\mathcal{S}$, the reaching time $R^{\beta,u}(B)$ as follows

$R^{\beta,u}(B)=\inf\\{t>0:U_{t}^{\beta,u}\in B\\}.$

Theorem 2 states that the invariant measure gets concentrated in the set of
the ladder lists as the polarization coefficient diverges. Moreover, for any
non-null initial list, the time it takes for the process to reach the set of
ladder lists goes to $0$, as the polarization coefficient diverges.

######  Theorem 2.

  1.   2. 1.

There exists a constant $C>0$, such that for any $\beta\geq 0$ the invariant
probability measure $\mu^{\beta}$ satisfies

$\mu^{\beta}(\mathcal{L})\geq 1-Ce^{-\beta}.$

  3. 2.

For any fixed $\delta>0$

$\sup_{u\in\mathcal{S}\setminus\\{\vec{0}\\}}{\mathbb{P}}\Big{(}R^{\beta,u}(\mathcal{L})>e^{-\beta(1-\delta)}\Big{)}\to
0\text{, as }\beta\to+\infty.$

Theorem 3 states that a highly polarized social network has a metastable
behavior.

######  Theorem 3.

For any $v\in\mathcal{C}^{+}$,

$\frac{R^{\beta,v}(\mathcal{C}^{-})}{\mathbb{E}[R^{\beta,v}(\mathcal{C}^{-})]}\to\text{Exp}(1)\text{
in distribution, as }\beta\to+\infty,$

where $\text{Exp}(1)$ denotes the mean $1$ exponential distribution.

##  3 Auxiliary notation and results

In this section we will prove some auxiliary results that will be used to
prove Theorems 1, 2 and 3. To do this, we need to extend the notation
introduced before.

Extra notation

  * •

The Markov chain embedded in the process $(U_{t}^{\beta,u})_{t\in[0,+\infty)}$
will be denoted $(\tilde{U}^{\beta,u}_{n})_{n\geq 0}$. In other terms,

$\tilde{U}^{\beta,u}_{0}=u\ \text{ and }\
\tilde{U}^{\beta,u}_{n}=U_{T_{n}}^{\beta,u},\text{ for any $n\geq 1$. }$

  * •

The invariant probability measure of the Markov chain
$(\tilde{U}_{n}^{\beta,u})_{n\geq 0}$ will be denoted $\tilde{\mu}^{\beta}$.

  * •

For any list $u\in\mathcal{S}$, the first return time of the embedded Markov
chain $(\tilde{U}_{n}^{\beta,u})_{n\geq 0}$ to $u$ will be denoted

$\tilde{R}^{\beta,u}(u)=\inf\\{n\geq 1:\tilde{U}^{\beta,u}_{n}=u\\}.$

  * •

For any $u\in\mathcal{S}$, the opposite list $-u\in\mathcal{S}$ is given by

$(-u)(a)=-u(a)\text{, for all }a\in\mathcal{A}.$

  * •

Let $\sigma:\mathcal{A}\to\mathcal{A}$ be a bijective map. For any
$u\in\mathcal{S}$, the permuted list $\sigma(u)\in\mathcal{S}$ is given by

$\sigma(u)(a)=u(\sigma(a))\text{, for all }a\in\mathcal{A}.$

  * •

The following event will appear several times in what follows. For a fixed
$u\in\mathcal{S}$ and for any $n\geq 1$,

$M_{n}=\\{A_{n}\in\text{argmax}\\{|\tilde{U}_{n-1}^{\beta,u}(a)|:a\in\mathcal{A}\\}\text{\
\ and \ \ }O_{n}\tilde{U}_{n-1}^{\beta,u}(A_{n})\geq 0\\}.$

######  Proposition 4.

  1.   2. 1.

For any $m\geq 1$,

${\mathbb{P}}\left(\bigcap_{j=1}^{m}M_{j}\right)\geq\zeta_{\beta}^{m},$

where

$\zeta_{\beta}=\frac{e^{\beta}}{e^{\beta}+e^{-\beta}+2(N-1)}\to 1,\text{ as
}\beta\to+\infty.$

  3. 2.

For any $u\in\mathcal{S}$,

${\mathbb{P}}\left(\tilde{U}^{\beta,u}_{3(N-1)}\in\mathcal{L}\
\Big{|}\bigcap_{j=1}^{3(N-1)}M_{j}\right)=1.$

We will first prove Part $1$ of Proposition 4.

###### Proof.

To prove Part $1$ of Proposition 4, we first observe that by the Markov
property,

${\mathbb{P}}\left(\bigcap_{j=1}^{m}M_{j}\right)=\sum_{v\in\mathcal{S}}{\mathbb{P}}\left(\bigcap_{j=1}^{m-1}M_{j},\tilde{U}^{\beta,u}_{m-1}=v\right){\mathbb{P}}\left(M_{m}\Big{|}\tilde{U}^{\beta,u}_{m-1}=v\right).$
(2)

The smallest value for

${\mathbb{P}}\left(M_{m}\ |\ \tilde{U}_{m-1}^{\beta,u}=v\right)$

is obtained for any list in which all actors, except one, have null pressure,
and the unique actor with non-null pressure has pressure $+1$ or $-1$. This
implies that

$\inf\left\\{{\mathbb{P}}\left(M_{m}\ |\
\tilde{U}_{m-1}^{\beta,u}=v\right):v\in\mathcal{S}\right\\}={\mathbb{P}}(M_{m}\
|\ \tilde{U}_{m-1}^{\beta,u}=(1,0,...,0))=\zeta_{\beta}.$

Applying this lower bound $m$ times in Equation (2), we conclude the proof of
Part 1. ∎

The proof of Part 2 of Proposition 4 is based in two Lemmas. Before proving
Lemmas 5 and 6, let us introduce some extra notation.

$\mathcal{S}^{+}$ is the set of lists $u\in\mathcal{S}$ such that there exists
$n(u)\in\\{1,\ldots,N-1\\}$ and a sequence of $n(u)$ different actors
$a_{1}(u),\ldots,a_{n(u)}(u)$ satisfying

$u(a_{i}(u))=j,\text{ for }j=1,...,n(u)\quad\text{ and }\quad
u(a)\geq-(n(u)-1),\text{ for all }a\in\mathcal{A}.$

In the same way, we define $\mathcal{S}^{-}$ as set of lists $u\in\mathcal{S}$
such that there exists $n(u)\in\\{1,\ldots,N-1\\}$ and a sequence of $n(u)$
different actors $a_{1}(u),\ldots,a_{n(u)}(u)$ satisfying

$u(a_{i}(u))=-j,\text{ for }j=1,...,n(u)\quad\text{ and }\quad
u(a)\leq(n(u)-1),\text{ for all }a\in\mathcal{A}.$

Let

$\tau=\inf\\{n\geq
1:A_{n}\in\\{A_{1},...,A_{n-1}\\}\cup\\{a\in\mathcal{A}:u(a)=0\\}\\}.$

Note that the event $\\{\tau=1\\}\cap M_{1}$ is only possible if $u=\vec{0}$.
In this case, $\tilde{U}_{1}^{\beta,u}\in\mathcal{S}^{+}\cup\mathcal{S}^{-}$.

######  Lemma 5.

Assume that $\tau\geq 2$. For any initial list $u\in\mathcal{S}$, if the event
$M_{\tau}$ occurs, then

$\tilde{U}_{\tau}^{\beta,u}\in\mathcal{S}^{+}\cup\mathcal{S}^{-}.$

###### Proof.

At instant $\tau-1$, we have that $\tilde{U}_{\tau-1}^{\beta,u}(A_{\tau-1})=0$
and for any $j=1,\ldots,\tau-2$,

$\tilde{U}_{\tau-1}^{\beta,u}(A_{j})=O_{j+1}+\ldots+O_{\tau-1},$

which implies that

$|\tilde{U}^{\beta,u}_{\tau-1}(A_{j})-\tilde{U}^{\beta,u}_{\tau-1}(A_{j-1})|=|O_{j}|=1.$

Moreover, for any $a\in\mathcal{A}$ such that $u(a)=0$, we have that

$\tilde{U}_{\tau-1}^{\beta,u}(a)=O_{1}+\ldots+O_{\tau-1}.$

Let,

$m=\max\\{|O_{\tau-1}|,|O_{\tau-2}+O_{\tau-1}|,\ldots,|O_{1}+\ldots+O_{\tau-1}|\\}.$

It follows that there exists a set with $m+1$ actors

$\\{a_{0},a_{1},\ldots,a_{m}\\}\subset\\{A_{1},\ldots,A_{\tau-1}\\}\cup\\{a\in\mathcal{A}:u(a)=0\\},$

such that either

$\\{\tilde{U}_{\tau-1}^{\beta,u}(a_{j}):j=0,1,\ldots,m\\}=\\{0,1,\ldots,m\\}$

or

$\\{\tilde{U}_{\tau-1}^{\beta,u}(a_{j}):j=0,1,\ldots,m\\}=\\{0,-1,\ldots,-m\\}.$

By assumption,

$A_{\tau}\in\text{argmax}\\{|\tilde{U}_{\tau-1}^{\beta,u}(a)|:a\in\mathcal{A}\\}\text{\
\ and \ \ }O_{n}\tilde{U}_{\tau-1}^{\beta,u}(A_{n})\geq 0.$

This implies that

$|\tilde{U}_{\tau-1}^{\beta,u}(A_{\tau})|=m\geq|\tilde{U}_{\tau-1}^{\beta,u}(a)|\text{,
for all }a\in\mathcal{A},$

and therefore, either

$\\{\tilde{U}_{\tau}^{\beta,u}(a_{j}):j=0,1,\ldots,m-1\\}=\\{1,\ldots,m\\}\
\text{ and }\ \tilde{U}_{\tau}^{\beta,u}(a)\geq-(m-1),\text{ for all
}a\in\mathcal{A},$

or

$\\{\tilde{U}_{\tau}^{\beta,u}(a_{j}):j=0,1,\ldots,m-1\\}=\\{0,-1,\ldots,-m\\}\
\text{ and }\ \tilde{U}_{\tau}^{\beta,u}(a)\leq m-1,\text{ for all
}a\in\mathcal{A}.$

This concludes the proof of Lemma 5.

∎

######  Lemma 6.

For any $u\in\mathcal{S}^{+},$ if the event
$\displaystyle\bigcap_{j=1}^{n(u)-1}M_{j}$ occurs, then
$\tilde{U}_{n(u)-1}^{\beta,u}\in\mathcal{C}^{+}$.

###### Proof.

If $M_{1}$ occurs, then

$\tilde{U}_{0}^{\beta,u}(A_{1})=\max\\{|u(a)|:a\in\mathcal{A}\\}\text{ and
}O_{1}=+1.$

Moreover, by assumption, for any $j=0,\ldots,n(u)-1$ there exists an actor
$a_{j}(u)\in\mathcal{A}$ such that
$\tilde{U}_{0}^{\beta,u}(a_{j}(u))=u(a_{j}(u))=j$. Therefore,

$\tilde{U}_{1}^{\beta,u}(a_{j}(u))=u(a_{j}(u))+1=j+1.$

As a consequence,

$\\{1,\ldots,n(u)\\}\subset\\{\tilde{U}_{1}^{\beta,u}(a):a\in\mathcal{A}\\}\quad\text{
and }\quad\tilde{U}_{1}^{\beta,u}(a)\geq-(n(u)-2),\text{ for all
}a\in\mathcal{A}.$

In general, for any $k=1,\ldots,n(u)-1$, if $\bigcap_{j=1}^{k}M_{j}$ occurs,
then there exists a sequence of actors $a_{0}(\tilde{U}^{\beta,u}_{k-1})$,
$\ldots$, $a_{n(u)}(\tilde{U}^{\beta,u}_{k-1})$ such that for any
$j=0,\ldots,n(u)-1$, we have that

$\tilde{U}_{k}^{\beta,u}(a_{j}(\tilde{U}^{\beta,u}_{k-1}))=j+1,$

and therefore,

$\\{1,\ldots,n(u)\\}\subset\\{\tilde{U}_{k}^{\beta,u}(a):a\in\mathcal{A}\\}\quad\text{
and }\quad\tilde{U}_{k}^{\beta,u}(a)\geq-(n(u)-(k+1)),\text{ for all
}a\in\mathcal{A}.$

This concluded the proof of Lemma 6. ∎

In the proof of Part $2$ of Proposition 4 that follows, we will indicate the
initial list $u\in\mathcal{S}$ when referring to the event $M_{n}$. In other
words, for a fixed $u\in\mathcal{S}$ and for any $n\geq 1$, we will denote

$M_{n}^{u}=\\{A_{n}\in\text{argmax}\\{|\tilde{U}_{n-1}^{\beta,u}(a)|:a\in\mathcal{A}\\}\text{\
\ and \ \ }O_{n}\tilde{U}_{n-1}^{\beta,u}(A_{n})\geq 0\\}.$

###### Proof.

We will now prove Part 2 of Proposition 4.

Lemma 5 and the fact that $\tau\leq N$ for any initial list $u\in\mathcal{S}$,
imply that if $\bigcap_{k=1}^{N}M_{k}^{u}$ occurs, then

$\tilde{U}^{\beta,u}_{N}\in\mathcal{S}^{+}\cup\mathcal{S}^{-}.$

Lemma 6 and the fact that $n(v)\leq N-1$ for any
$v\in\mathcal{S}^{+}\cup\mathcal{S}^{-}$, imply that if
$\bigcap_{k=1}^{N-2}M_{k}^{v}$ occurs, then

$\tilde{U}^{\beta,v}_{N-2}\in\mathcal{C}^{+}\cup\mathcal{C}^{-}.$

We also have that for any $v^{\prime}\in\mathcal{C}^{+}\cup\mathcal{C}^{-}$,
if $\bigcap_{k=1}^{N-1}M_{k}^{v^{\prime}}$ occurs, then

$\tilde{U}^{\beta,v^{\prime}}_{N-1}\in\mathcal{L}.$

We conclude the proof by putting all this together and using the Markov
property. ∎

For any fixed $l\in\mathcal{L}^{+}$, let $c_{\beta,l}$ be the positive real
number such that

${\mathbb{P}}(R^{\beta,l}(\mathcal{L}^{-})>c_{\beta,l})=e^{-1}.$ (3)

Due to the symmetric properties of the process, it is clear that
$c_{\beta,l}=c_{\beta,l^{\prime}}$, for any pair of lists $l$ and $l^{\prime}$
belonging to $\mathcal{L}^{+}$. Therefore, in what follows we will omit to
indicate $l$ in the notation of $c_{\beta}.$ The next proposition gives an
lower bound to $c_{\beta}$.

######  Proposition 7.

There exists $C_{1}>0$ such that for any $\beta\geq 0$,

$c_{\beta}\geq C_{1}e^{\beta}.$

###### Proof.

For any fixed $l\in\mathcal{L}^{+}$, let

$\tau_{-}^{(1)}=\inf\\{T_{n}:O_{n}=-1,U_{T_{n-1}}^{\beta,l}(A_{n})>0\\}.$

Consider also

$\tau_{-}^{(2)}=\inf\\{T_{n}:E_{n}^{-}\cap(E_{n-1}^{-}\cup E_{n-2}^{-})\\},$

where $E_{n}^{-}=\\{O_{n}=-1,U_{T_{n-1}}^{\beta,l}(A_{n})=0\\}$. By
definition,

$R^{\beta,l}(\mathcal{S}\setminus\mathcal{C}^{+})\geq\min\\{\tau_{-}^{(1)},\tau_{-}^{(2)}\\}.$

For any $t>0$,

${\mathbb{P}}(\tau_{-}^{(1)}>t)\geq{\mathbb{P}}(\text{Exp}((N-1)e^{-\beta})>t),$

where $\text{Exp}((N-1)e^{-\beta})$ is a random variable exponentially
distributed with mean $e^{\beta}/(N-1).$

Let $n_{0}^{-}=0$ and for $j\geq 1$,

$n_{j}^{-}=\inf\\{n>n_{j-1}^{-}:O_{n}=-1,U_{T_{n-1}}^{\beta,l}(A_{n})=0\\}.$

We have that

$\tau^{(2)}_{-}=\sum_{j=1}^{J}(T_{n_{j}^{-}}-T_{n_{j-1}^{-}}),$

where $J=\inf\\{j\geq 1:n_{j}^{-}=n_{j-1}^{-}+1\text{ or
}n_{j}^{-}=n_{j-1}^{-}+2\\}.$

Therefore, for any $t>0$,

${\mathbb{P}}(\tau_{-}^{(2)}>t|\tau_{-}^{(1)}>t)\geq{\mathbb{P}}\left(\sum_{j=1}^{G}E_{j}>t\right),$

where $(E_{j})_{j\geq 1}$ is a sequence of i.i.d. random variables
exponentially distributed with mean $1$, and $G$ is random variable
independent from $(E_{j})_{j\geq 1}$ with Geometric distribution assuming
values in $\\{1,2,...\\}$ with parameter

$\lambda_{\beta}=2\times\frac{(N-1)}{(N-1)+e^{\beta}}.$

This implies that for any $t>0$,

${\mathbb{P}}(\tau_{-}^{(2)}>t|\tau_{-}^{(1)}>t)\geq{\mathbb{P}}\left(\text{Exp}(\lambda_{\beta})>t\right),$

where $\text{Exp}(\lambda_{\beta})$ is a random variable exponentially
distributed with mean $1/\lambda_{\beta}$.

Therefore, for any $t>0$,

${\mathbb{P}}(R^{\beta,l}(\mathcal{S}\setminus\mathcal{C}^{+})>t)\geq{\mathbb{P}}[\text{Exp}((N-1)e^{-\beta})>t]{\mathbb{P}}[\text{Exp}(\lambda_{\beta})>t].$

This implies that

$\displaystyle
e^{-1}={\mathbb{P}}(R^{\beta,l}(\mathcal{L}^{-})>c_{\beta})\geq{\mathbb{P}}(R^{\beta,l}(\mathcal{S}\setminus\mathcal{C}^{+})>c_{\beta})\geq
e^{-c_{\beta}\left((N-1)e^{-\beta}+\lambda_{\beta}\right)},$

and therefore

$c_{\beta}\geq\left((N-1)e^{-\beta}+\lambda_{\beta}\right)^{-1}.$

With this we concluded the proof of Proposition 7.

∎

##  4 Proof of Theorem 1

To prove Part $1$ of Theorem 1 we first need to prove the following Lemma.

######  Lemma 8.

For any list $u\in\mathcal{S}$,

$\inf\left\\{n\geq 1:|U^{\beta,u}_{T_{n-1}}(A_{n})|<N\right\\}\leq N.$

###### Proof.

The initial list of social pressures $u$ belongs to $\mathcal{S}$. Therefore,
there exists $a_{0}\in\mathcal{A}$ such that $u(a_{0})=0$. By definition, for
any $m\geq 1$,

$|U^{\beta,u}_{T_{m}}(a_{0})|\leq\left|\sum_{j=1}^{m}O_{j}\right|\leq\sum_{j=1}^{m}\left|O_{j}\right|=m.$

More generally, if $A_{n}=a$ then for any $m\geq 1$,

$|U^{\beta,u}_{T_{n+m}}(a)|\leq\left|\sum_{j=1}^{m}O_{n+j}\right|\leq\sum_{j=1}^{m}\left|O_{n+j}\right|=m.$

Therefore, if

$|\\{a_{0},A_{1},...,A_{N-1}\\}|=N,$

then

$\inf\left\\{n\geq 1:|U^{\beta,u}_{T_{n-1}}(A_{n})|<N\right\\}\leq N.$

Now, if

$|\\{a_{0},A_{1},...,A_{N-1}\\}|\leq N-1,$

there exists $m\in\\{1,...,N-1\\}$ such that

$A_{m}\in\\{a_{0},A_{1},...,A_{m-1}\\}.$

This implies that

$|U^{\beta,u}_{T_{m-1}}(A_{m})|\leq m\leq N-1,$

and therefore,

$\inf\left\\{n\geq 1:|U^{\beta,u}_{T_{n-1}}(A_{n})|<N\right\\}\leq m<N.$

∎

Define $T_{0}^{<}=T_{0}^{>}=0$ and for any $k\geq 1$,

$T_{k}^{<}=\inf\\{T_{n}>T_{k-1}^{<}:|U^{\beta,u}_{T_{n-1}}(A_{n})|<N\\},$
$T_{k}^{>}=\inf\\{T_{n}>T_{k-1}^{>}:|U^{\beta,u}_{T_{n-1}}(A_{n})|\geq N\\}.$

Lemma 8 implies that $T_{k}^{<}$ is well defined for any $k\geq 1$. Now we can
prove Part 1 of Theorem 1.

###### Proof.

To prove Part 1 of Theorem 1, we will construct the process
$(U_{t}^{\beta,u})_{t\in[0,+\infty)}$ with jump times $\\{T_{n}:n\geq 1\\}$ as
the superposition of two process with jump times $\\{T_{k}^{<}:k\geq 1\\}$ and
$\\{T_{k}^{>}:k\geq 1\\}$ in the following way. For any $v\in\mathcal{S}$, the
jump rates of these two processes are

$q^{<}(v)=\sum_{a\in A}\mathbf{1}\\{v(a)<N\\}(e^{\beta v(a)}+e^{-\beta v(a)})$

and

$q^{>}(v)=\sum_{a\in A}\mathbf{1}\\{v(a)\geq N\\}(e^{\beta v(a)}+e^{-\beta
v(a)}).$

For any list $v\in\mathcal{S}$, we have that

$q^{<}(v)\leq\lambda,$

with

$\lambda=N(e^{\beta(N-1)}+e^{-\beta(N-1)}).$

For any $v\in\mathcal{S}$, define

$\Phi^{<,-}_{v}(0)=\Phi^{<,+}_{v}(0)=0\text{ and
}\Phi^{>,-}_{v}(0)=\Phi^{>,+}_{v}(0)=\lambda.$

For any $a\in\mathcal{A}$, define

$\Phi^{<,-}_{v}(a)=\Phi^{<,+}_{v}(a-1)+\mathbf{1}\\{|v(a)|<N\\}e^{-\beta
v(a)},$ $\Phi^{<,+}_{v}(a)=\Phi^{<,-}_{v}(a)+\mathbf{1}\\{|v(a)|<N\\}e^{+\beta
v(a)},$ $\Phi^{>,-}_{v}(a)=\Phi^{>,+}_{v}(a-1)+\mathbf{1}\\{|v(a)|\geq
N\\}e^{-\beta v(a)},$
$\Phi^{>,+}_{v}(a)=\Phi^{>,-}_{v}(a)+\mathbf{1}\\{|v(a)|\geq N\\}e^{+\beta
v(a)}.$

Using this partitions indexed by $\mathcal{S}$ (see Figure 1), we now
construct the process $(U_{t}^{\beta,u})_{t\in[0,+\infty)}$ as follows.
Consider an homogeneous rate $1$ Poisson point process in the plane
$[0,+\infty)^{2}$. Call $\mathcal{N}$ the counting measure of this process.
Given the initial list $u\in\mathcal{S}$, define

$T_{1}=\inf\left\\{t>0:\mathcal{N}\Big{(}(0,t]\times\left\\{[0,q^{<}(u))\cup[\lambda,\lambda+q^{>}(u))\right\\}\Big{)}=1\right\\}.$

Denoting $R_{1}$ as the second coordinate of the mark $(T_{1},R_{1})$ of
$\mathcal{N}$, we have

$\text{$A_{1}=b$, $O_{1}=-1$, \quad if }\quad
R_{1}\in[\Phi^{<,+}_{u}(b-1),\Phi^{<,-}_{u}(b))\cup[\Phi^{>,+}_{u}(b-1),\Phi^{>,-}_{u}(b)),$
$\text{ $A_{1}=b$, $O_{1}=+1$, \quad if}\quad
R_{1}\in[\Phi^{<,-}_{u}(b),\Phi^{<,+}_{u}(b))\cup[\Phi^{>,-}_{u}(b),\Phi^{>,+}_{u}(b)).$

At time $T_{1}$, we have $U_{T_{1}}^{\beta,u}=\pi^{A_{1},O_{1}}(u)$.

$\lambda$$\Phi_{u}^{>,+}(5)$$\Phi_{u}^{>,-}(2)$$\Phi_{u}^{>,+}(2)$$\Phi_{u}^{>,-}(4)$$\Phi_{u}^{>,+}(4)$$\Phi_{u}^{>,-}(5)$$\Phi_{u}^{<,-}(1)$$\Phi_{u}^{<,+}(1)$$\Phi_{u}^{<,-}(3)$$\Phi_{u}^{<,+}(3)$$\Phi_{u}^{<,-}(6)$$\Phi_{u}^{<,+}(6)$$T_{1}$
Figure 1: The regions of plane $[0,T_{1}]\times[0,\lambda+q^{>}(u)]$
considered on the construction $T_{1}$. In this example, $N=6$ and the list
$u\in\mathcal{S}$, satisfies $u(a)<N$ for $a=1,3,6$ and $u(a)\geq N$ for
$a=2,4,5$.

More generally, for $n\geq 1$, we have

$T_{n}=\inf\left\\{t>T_{n-1}:\mathcal{N}\Big{(}(T_{n-1},t]\times\left\\{[0,q^{<}(U_{T_{n-1}}^{\beta,u}))\cup[\lambda,\lambda+q^{>}(U_{T_{n-1}}^{\beta,u}))\right\\}\Big{)}=1\right\\}.$

Denoting $R_{n}$ as the second coordinate of the mark $(T_{n},R_{n})$ of
$\mathcal{N}$, we have

$\text{$A_{n}=b$, $O_{n}=-1$, \quad if }\quad
R_{n}\in\left[\Phi^{<,+}_{U_{T_{n-1}}^{\beta,u}}(b-1),\Phi^{<,-}_{U_{T_{n-1}}^{\beta,u}}(b)\right)\cup\left[\Phi^{>,+}_{U_{T_{n-1}}^{\beta,u}}(b-1),\Phi^{>,-}_{U_{T_{n-1}}^{\beta,u}}(b)\right),$
$\text{$A_{n}=b$, $O_{n}=+1$, \quad if}\quad
R_{n}\in\left[\Phi^{<,-}_{U_{T_{n-1}}^{\beta,u}}(b),\Phi^{<,+}_{U_{T_{n-1}}^{\beta,u}}(b)\right)\cup\left[\Phi^{>,-}_{U_{T_{n-1}}^{\beta,u}}(b),\Phi^{>,+}_{U_{T_{n-1}}^{\beta,u}}(b)\right).$

At time $T_{n}$, we have
$U_{T_{n}}^{\beta,u}=\pi^{A_{n},O_{n}}(U_{T_{n-1}}^{\beta,u})$ (see Figure 2).

$\lambda$$\lambda+q^{>}(u)$$q^{<}(u)$$\lambda+q^{>}(U_{T_{1}}^{\beta,u})$$q^{<}(U_{T_{1}}^{\beta,u})$$\lambda+q^{>}(U_{T_{2}}^{\beta,u})$$q^{<}(U_{T_{2}}^{\beta,u})$$T_{1}$$T_{2}$$\ldots$
Figure 2: The regions of the plane
$(T_{n},T_{n+1})\times[0,q^{<}(U_{T_{n}}^{\beta,u})]$ and
$(T_{n},T_{n+1})\times[\lambda,\lambda+q^{>}(U_{T_{n}}^{\beta,u})]$, for
$n=0,1,2,\ldots$.

For any $n\geq 1$, we have that

$T_{n}^{<}=\inf\left\\{T_{m}>T_{n-1}^{<}:R_{m}\in\big{[}0,q^{<}(U_{T_{m-1}}^{\beta,u})\big{)}\right\\},$
$T_{n}^{>}=\inf\left\\{T_{m}>T_{n-1}^{>}:R_{m}\in\big{[}\lambda,\lambda+q^{>}(U_{T_{m-1}}^{\beta,u})\big{)}\right\\}.$

Define $T_{0}^{\lambda}=0$ and for any $n\geq 1$,

$T_{n}^{\lambda}=\inf\left\\{t>T_{n-1}^{\lambda}:\mathcal{N}\Big{(}(T_{n-1}^{\lambda},t]\times[0,\lambda)\Big{)}=1\right\\}.$

By construction, $\\{T_{n}^{<}:n\geq 1\\}\subset\\{T_{n}^{\lambda}:n\geq
1\\}$. Also, $\\{T_{n}^{\lambda}:n\geq 1\\}$ are the marks of a homogeneous
Poisson point process with rate $\lambda$. Since $\sup\\{T_{n}^{\lambda}:n\geq
1\\}=+\infty$, we have that $\sup\\{T_{n}^{<}:n\geq 1\\}=+\infty$. By Lemma 8,
the event $\sup\\{T_{n}^{<}:n\geq 1\\}=+\infty$ implies that
$\sup\\{T_{n}^{>}:n\geq 1\\}=+\infty$. $\\{T_{n}:n\geq 1\\}$ is the
superposition of $\\{T_{n}^{<}:n\geq 1\\}$ and $\\{T_{n}^{>}:n\geq 1\\}$, and
then, we conclude that $\sup\\{T_{n}:n\geq 1\\}=+\infty$. ∎

To conclude the proof of Theorem 1, we will prove that
$(U_{t}^{\beta,u})_{t\in[0,+\infty]}$ is an ergodic Markov process.

###### Proof.

To prove Part $2$ of Theorem 1, let $l\in\mathcal{L}^{+}$ satisfies
$l(a)=a-1$, for all $a\in\mathcal{A}$. For any $u\in\mathcal{S}$, we have that

$l=\pi^{1,+1}\circ\pi^{2,+1}\circ\ldots\circ\pi^{N,+1}(u),$

and then, if the event $\bigcap_{j=1}^{N}\\{A_{j}=N-j+1,O_{j}=+1\\}$ occurs,
then $\tilde{U}^{\beta,u}_{N}=l.$

For any $u^{\prime}\in\mathcal{S}$,

$\displaystyle{\mathbb{P}}\Big{(}\tilde{U}^{\beta,u}_{n+2N}=l\ \Big{|}\
\tilde{U}^{\beta,u}_{n}=u^{\prime}\Big{)}\geq$
$\displaystyle{\mathbb{P}}\left(\bigcap_{a\in\mathcal{A}}\\{|\tilde{U}_{n+N}^{\beta,u}(a)|<N\\}\Big{|}\
\tilde{U}_{n}^{\beta,u}=u^{\prime}\right){\mathbb{P}}\left(\tilde{U}_{n+2N}^{\beta,u}=l\
\Big{|}\
\bigcap_{a\in\mathcal{A}}\\{|\tilde{U}_{n+N}^{\beta,u}(a)|<N\\}\right).$

For any $u\in\mathcal{S}$, using Proposition 4 we have that

${\mathbb{P}}\left(\bigcap_{a\in\mathcal{A}}\\{|\tilde{U}_{N}^{\beta,u}(a)|<N\\}\right)\geq{\mathbb{P}}\left(\bigcap_{j=1}^{N}M_{j}\right)\geq\zeta_{\beta}^{N}.$

Also, there exists $\epsilon^{*}>0$ such that

${\mathbb{P}}\left(\tilde{U}_{n+2N}^{\beta,u}=l\ \Big{|}\
\bigcap_{a\in\mathcal{A}}\\{|\tilde{U}_{n+N}^{\beta,u}(a)|<N\\}\right)\geq$
$\min\left\\{{\mathbb{P}}\left(\tilde{U}_{n+2N}^{\beta,u}=l\ \Big{|}\
\tilde{U}_{n+N}^{\beta,u}=v\right):v\in\mathcal{S},\bigcap_{a\in\mathcal{A}}\\{|v(a)|<N\\}\right\\}=\epsilon^{*}.$

Therefore, for any $u^{\prime}\in\mathcal{S}$,

${\mathbb{P}}\Big{(}\tilde{U}^{\beta,u}_{n+2N}=l\ \Big{|}\
\tilde{U}^{\beta}_{n}=u^{\prime}\Big{)}\geq\zeta_{\beta}^{N}\epsilon^{*}.$

Recall that $\tilde{R}^{\beta,l}(l)=\inf\\{n\geq
1:\tilde{U}^{\beta,l}_{n}=l\\}.$ The last inequality implies that for any
$t>0$,

${\mathbb{P}}(\tilde{R}^{\beta,l}(l)>t)\leq{\mathbb{P}}(2N\times\text{Geom}(\zeta_{\beta}^{N}\epsilon^{*})>t),$

where $\text{Geom}(r)$ denotes a random variable with geometric distribution
assuming values in $\\{1,2,\ldots\\}$ and with parameter $r\in(0,1)$. This
implies that $\mathbb{E}(\tilde{R}^{\beta,l}(l))<+\infty$ and then,
$(\tilde{U}^{\beta,u}_{n})_{n\geq 0}$ is a positive-recurrent Markov chain.

The jump rate of the process $(U_{t}^{\beta,u})_{t\in[0,+\infty)}$ satisfies

$\sum_{a\in\mathcal{A}}(e^{\beta v(a)}+e^{-\beta v(a)})\geq 2N,$

for any $v\in\mathcal{S}$. Putting all this together we conclude that
$(U_{t}^{\beta,u})_{t\in[0,+\infty)}$ is ergodic.

∎

##  5 Proof of Theorem 2

To prove Part $1$ of Theorem 2 we will need the following Proposition.

######  Proposition 9.

For any $\beta>0$ and $u\notin\mathcal{L}$ such that
$\max\\{|u(a)|:a\in\mathcal{A}\\}<N$ we have that

$\tilde{\mu}^{\beta}(u)\leq C^{\prime}e^{-\beta(N-1)},$

where $C^{\prime}>0$.

###### Proof.

First, we have that

$\mathbb{E}(\tilde{R}^{\beta,u}(u))\geq\sum_{m=3(N-1)}^{+\infty}{\mathbb{P}}(\tilde{R}^{\beta,u}(u)\geq
m).$ (4)

For any $u\notin\mathcal{L}$ such that $\max\\{|u(a)|:a\in\mathcal{A}\\}<N$,
considering Proposition 4 we have that

${\mathbb{P}}\left(\\{\tilde{U}_{n}^{\beta,u}\neq u,\text{ for
}n=1,2,...,3(N-1)\\}\cap\\{\tilde{U}_{3(N-1)}^{\beta,u}\in\mathcal{L}\\}\right)\geq{\mathbb{P}}\left(\bigcap_{j=1}^{3(N-1)}M_{j}\right).$

Also, for any $l\in\mathcal{L}$ and for any $m\geq 1$, we have that

${\mathbb{P}}\left(\\{\tilde{U}_{n}^{\beta,l}\neq u,\text{ for
}n=1,2,...,m\\}\right)\geq{\mathbb{P}}\left(\bigcap_{j=1}^{m}\\{O_{j}\tilde{U}_{j-1}^{\beta,l}(A_{j})>0\\}\right).$

Therefore, for any $m\geq 1$ we have that

${\mathbb{P}}(\tilde{R}^{\beta,u}(u)\geq
3(N-1)+m)\geq{\mathbb{P}}\left(\bigcap_{j=1}^{3(N-1)}M_{j}\right){\mathbb{P}}\left(\
\bigcap_{j=1}^{m}\\{O_{j}\tilde{U}_{j-1}^{\beta,l}(A_{j})>0\\}\right).$ (5)

For any $l\in\mathcal{L}$, we have that

${\mathbb{P}}\left(\bigcap_{j=1}^{m}\\{O_{j}\tilde{U}_{j-1}^{\beta,l}(A_{j})>0\\}\right)\geq\eta^{m},$
(6)

where

$\eta=\frac{\displaystyle\sum_{j=1}^{N-1}e^{\beta
j}}{\displaystyle\sum_{j=0}^{N-1}(e^{\beta j}+e^{-\beta j})}.$

Therefore, by Proposition 4 and Equations (4), (5) and (6), it follows that

$\mathbb{E}(\tilde{R}^{\beta,u}(u))\geq{\mathbb{P}}\left(\bigcap_{j=1}^{3(N-1)}M_{j}\right)\left(1+\sum_{m=1}^{\infty}\eta^{m}\right)\geq\frac{(2N)^{-3(N-1)}}{1-\eta}.$

From the classical Kac’s Lemma (see Kac, (1947)), we conclude that

$\tilde{\mu}^{\beta}(u)\leq C^{\prime}e^{-\beta(N-1)},$

where $C^{\prime}=(N+1)(2N)^{3(N-1)}$. ∎

######  Corollary 10.

$\tilde{\mu}^{\beta}(\vec{0})<C^{\prime}_{1}e^{-N\beta}$ for all $\beta>0$,
where $C^{\prime}_{1}>0$.

###### Proof.

Let $v\in\mathcal{S}$ be the list in which $v(1)=0$ and $v(a)=1,\text{ for
}a\in\mathcal{A}\setminus\\{1\\}$. We have that

$\\{u\in\mathcal{S}:{\mathbb{P}}(\tilde{U}^{\beta,u}_{1}=\vec{0})>0\\}=\\{\sigma(v),-\sigma(v)\in\mathcal{S}\text{
such that }\sigma:\mathcal{A}\to\mathcal{A}\text{ is a bijective map}\\}.$

Therefore,

$\tilde{\mu}^{\beta}(\vec{0})=\sum_{u\in\mathcal{S}}{\mathbb{P}}(\tilde{U}^{\beta,u}_{1}=\vec{0})\tilde{\mu}^{\beta}(u)=2N{\mathbb{P}}(\tilde{U}^{\beta,v}_{1}=\vec{0})\tilde{\mu}^{\beta}(v).$

Since

${\mathbb{P}}(\tilde{U}^{\beta,v}_{1}=\vec{0})=\frac{1}{2+(N-1)(e^{+\beta}+e^{-\beta})}\leq
e^{-\beta},$

from Proposition 9 we conclude that

$\tilde{\mu}^{\beta}(\vec{0})\leq(N+1)(2N)^{3(N-1)}e^{-\beta(N-1)}(2N)e^{-\beta}=C^{\prime}_{1}e^{-\beta
N}.$

∎

Now we can prove the Part 1 of Theorem 2.

###### Proof.

To prove Part 1 of Theorem 2, first note that for any $u\in\mathcal{S}$, the
invariant measure $\mu^{\beta}$ satisfies

$\mu^{\beta}(u)=\frac{\tilde{\mu}^{\beta}(u)}{q_{\beta}(u)}\left(\sum_{u^{\prime}\in\mathcal{S}}\frac{\tilde{\mu}^{\beta}(u^{\prime})}{q_{\beta}(u^{\prime})}\right)^{-1},$

where for any $v\in\mathcal{S}$,

$q_{\beta}(v)=\sum_{a\in\mathcal{A}}(e^{\beta v(a)}+e^{-\beta v(a)})$

is the jump rate of $(U_{t}^{\beta,u})_{t\in[0,+\infty)}$ at list $v$.
Therefore,

$\mu^{\beta}(\mathcal{L})=\sum_{u\in\mathcal{L}}\frac{\tilde{\mu}^{\beta}(u)}{q_{\beta}(u)}\left(\sum_{u^{\prime}\in\mathcal{S}}\frac{\tilde{\mu}^{\beta}(u^{\prime})}{q_{\beta}(u^{\prime})}\right)^{-1}=\frac{1}{1+\displaystyle\frac{\sum_{j=0}^{N-1}(e^{\beta
j}+e^{-\beta
j})}{\tilde{\mu}^{\beta}(\mathcal{L})}\sum_{u^{\prime}\not\in\mathcal{L}}\frac{\tilde{\mu}^{\beta}(u^{\prime})}{q_{\beta}(u^{\prime})}}.$

By using Proposition 4 to bound $\tilde{\mu}^{\beta}(\mathcal{L})$, we have
that

$\mu^{\beta}(\mathcal{L})\geq\frac{1}{1+\displaystyle(2N)^{3(N-1)+1}e^{\beta(N-1)}\sum_{u^{\prime}\not\in\mathcal{L}}\frac{\tilde{\mu}^{\beta}(u^{\prime})}{q_{\beta}(u^{\prime})}}$

For any $u\in\mathcal{S}$ such that $\max\\{|u(a)|:a\in\mathcal{A}\\}\geq N$,
we have that $q_{\beta}(u)\geq e^{\beta N}$, and then

$\frac{\tilde{\mu}^{\beta}(u)}{q_{\beta}(u)}\leq\tilde{\mu}^{\beta}(u)e^{-\beta
N}.$

Observe that for any $u\neq\vec{0}$, $q_{\beta}(u)\geq e^{\beta}$. Therefore,
by Proposition 9 and Corollary 10, it follows that for any
$u\in\mathcal{L}^{c}$ such that $\max\\{|u(a)|:a\in\mathcal{A}\\}<N$, we have
that

$\frac{\tilde{\mu}^{\beta}(u)}{q_{\beta}(u)}\leq C^{\prime}_{1}e^{-\beta N}.$

Since $|\\{u\in\mathcal{L}^{c}:\max\\{|u(a)|:a\in\mathcal{A}\\}<N\\}|\leq
N(2N-1)^{N-1}$, we conclude that

$\mu^{\beta}(\mathcal{L})\geq\frac{1}{1+Ce^{-\beta}}\geq 1-Ce^{-\beta},$

where

$C=(2N)^{3(N-1)+1}[1+C^{\prime}_{1}N(2N-1)^{N-1}].$

∎

Now we can prove Part 2 of Theorem 2.

###### Proof.

To prove Part 2 of Theorem 2, we first note that for any
$u\in\mathcal{S}\setminus\\{\vec{0}\\}$, we have that

${\mathbb{P}}\Big{(}R^{\beta,u}(\mathcal{L})>t\Big{)}\leq{\mathbb{P}}\left(R^{\beta,u}(\mathcal{L})>t\
,\
\bigcap_{j=1}^{3(N-1)}M_{j}\right)+{\mathbb{P}}\left(\bigcup_{j=1}^{3(N-1)}M_{j}^{c}\right).$
(7)

Part $2$ of Proposition 4 implies that the right-hand side of Equation (7) is
bounded above by

${\mathbb{P}}\left(T_{3(N-1)}>t\ ,\
\bigcap_{j=1}^{3(N-1)}M_{j}\right)+{\mathbb{P}}\left(\bigcup_{j=1}^{3(N-1)}M_{j}^{c}\right).$

By Part $1$ of Proposition 4, we have that

${\mathbb{P}}\left(\bigcup_{j=1}^{3(N-1)}M_{j}^{c}\right)\leq
1-\left(\frac{e^{\beta}}{e^{\beta}+e^{-\beta}+2(N-1)}\right)^{3(N-1)}.$ (8)

Note that the exit rate of any list different from the null list is always
bigger than $e^{\beta}$. Therefore,

${\mathbb{P}}\left(T_{3(N-1)}>t\ ,\
\bigcap_{j=1}^{3(N-1)}M_{j}\right)\leq{\mathbb{P}}\left(\sum_{n=1}^{3(N-1)}E_{n}>t\right),$
(9)

where $(E_{n})_{n\geq 1}$ is an i.i.d. sequence of exponentially distributed
random variables with mean $1/e^{\beta}$. It follows that

${\mathbb{P}}\left(\sum_{n=1}^{3(N-1)}E_{n}>t\right)\leq{\mathbb{P}}\left(\bigcup_{n=1}^{3(N-1)}\left\\{E_{n}>\frac{t}{3(N-1)}\right\\}\right)\leq
3(N-1)e^{-e^{\beta}t/(3(N-1))}.$

We conclude the proof by considering $t=e^{-\beta(1-\delta)}$ and noting that
the bounds in (8) and (9) does not depend on $u$. ∎

The next corollary follows from Theorem 2.

######  Corollary 11.

For any fixed $\delta>0$,

${\mathbb{P}}\Big{(}R^{\beta,\vec{0}}(\mathcal{L})>\tau+e^{-\beta(1-\delta)}\Big{)}\to
0\text{ as }\beta\to\infty,$

where $\tau$ exponentially distributed random time with mean $1/2N$
independent from $(U_{t}^{\beta,\vec{0}})_{t\in[0,+\infty)}$.

###### Proof.

Corollary 11 follows directly from Part $2$ of Theorem 2 and the fact that
given the initial list $\vec{0}$, the first jump time $T_{1}$ is an
exponentially distributed random time with mean $1/2N$. ∎

##  6 Proof of Theorem 3

Recall that in Section 3, for any $l\in\mathcal{L}^{+}$ and for any $\beta\geq
0$ we considered $c_{\beta}$ as the positive real number such that

${\mathbb{P}}(R^{\beta,l}(\mathcal{L}^{-})>c_{\beta})=e^{-1}.$

To prove Theorem 3, we prove Proposition 12 which is interesting by itself.

######  Proposition 12.

For any $l\in\mathcal{L}^{+}$

$\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}\to\text{Exp}(1),\text{ as
}\beta\to+\infty,$

where $\text{Exp}(1)$ is a random variable exponentially distributed with mean
$1$.

To prove Proposition 12, we will first prove the following lemma.

######  Lemma 13.

For any $\beta\geq 0$, for any $l\in\mathcal{L}$ and for any $s>0$,

${\mathbb{P}}(U_{s}^{\beta,l}\in\mathcal{S}\setminus\mathcal{L})\leq\frac{1-\mu^{\beta}(\mathcal{L})}{\mu^{\beta}(\mathcal{L})}.$

###### Proof.

For any $s>0$,

$\mu^{\beta}(\mathcal{L})=\sum_{u\in\mathcal{L}}\mu^{\beta}(u){\mathbb{P}}(U^{\beta,u}_{s}\in\mathcal{L})+\sum_{u\in\mathcal{S}\setminus\mathcal{L}}\mu^{\beta}(u){\mathbb{P}}(U^{\beta,u}_{s}\in\mathcal{L}).$

By the symmetric properties of the process, it follows that for any
$l,l^{\prime}\in\mathcal{L}$,

${\mathbb{P}}(U^{\beta,l}_{s}\in\mathcal{L})={\mathbb{P}}(U^{\beta,l^{\prime}}_{s}\in\mathcal{L}).$

Therefore,

$\mu^{\beta}(\mathcal{L})\leq\mu^{\beta}(\mathcal{L}){\mathbb{P}}(U^{\beta,l}_{s}\in\mathcal{L})+(1-\mu^{\beta}(\mathcal{L})).$

We conclude the proof of Lemma 13 by rearranging the terms of this last
inequality.

∎

###### Proof.

We will now prove Proposition 12. First of all, we will prove that for any
$l\in\mathcal{L}^{+}$ and for any pair of positive real numbers $s,t\geq 0$,
the following holds

$\lim_{\beta\to+\infty}\left|{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>s+t\right)-{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>s\right){\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>t\right)\right|=0.$
(10)

To simplify the presentation of the proof, we will use the shorthand notation
$R^{\beta,u}$ instead of $R^{\beta,u}(\mathcal{L}^{-})$, for any
$u\notin\mathcal{L}^{-}$.

Indeed, for a fixed $l\in\mathcal{L}^{+}$,

$\left|{\mathbb{P}}\left(\frac{R^{\beta,l}}{c_{\beta}}>s+t\right)-{\mathbb{P}}\left(\frac{R^{\beta,l}}{c_{\beta}}>s\right){\mathbb{P}}\left(\frac{R^{\beta,l}}{c_{\beta}}>t\right)\right|\leq$
$\sum_{u\in\mathcal{S}\setminus\mathcal{L}^{-}}{\mathbb{P}}\left(U_{c_{\beta}s}^{\beta,l}=u,\frac{R^{\beta,l}}{c_{\beta}}>s\right)\left|{\mathbb{P}}\left(\frac{R^{\beta,u}}{c_{\beta}}>t\right)-{\mathbb{P}}\left(\frac{R^{\beta,l}}{c_{\beta}}>t\right)\right|.$
(11)

By the symmetric properties of the process, the left-hand Equation (11) is
bounded above by

$\sum_{u\in\mathcal{S}\setminus\mathcal{L}}{\mathbb{P}}\left(U_{c_{\beta}s}^{\beta,l}=u,\frac{R^{\beta,l}}{c_{\beta}}>s\right)\left|{\mathbb{P}}\left(\frac{R^{\beta,u}}{c_{\beta}}>t\right)-{\mathbb{P}}\left(\frac{R^{\beta,l}}{c_{\beta}}>t\right)\right|\leq{\mathbb{P}}\left(U_{c_{\beta}s}^{\beta,l}\in\mathcal{S}\setminus\mathcal{L}\right).$
(12)

By Lemma 13, Equation (12) and Theorem 2 implies (10).

By definition,

${\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>1\right)=e^{-1}.$

Iterating (10) with $t=s=2^{-n}$, for $n=1,2,\ldots$, we have that

${\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>2^{-n}\right)\to
e^{-2^{-n}},\text{ as }\beta\to+\infty.$

More generally, we have that for any

$t\in\left\\{\sum_{n=1}^{m}b(n)2^{-n}:b(n)\in\\{0,1\\},n=1,...,m,m\geq
1\right\\}$

is valid that

${\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>t\right)\to
e^{-t},\text{ as }\beta\to+\infty.$ (13)

Any real number $r\in(0,1)$ has a binary representation

$r=\sum_{n=1}^{+\infty}b(n)2^{-n},$

where for any $n\geq 1$, $b(n)\in\\{0,1\\}$. Therefore, the monotonicity of

$t\to{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>t\right)$

implies that the convergence in (13) is valid for any $t\in(0,1)$. Moreover,
for any positive integer $n\geq 1$, Equation (10) implies that

${\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>n\right)\to
e^{-n},\text{ as }\beta\to+\infty.$

We conclude that (13) is valid for any $t>0$. ∎

######  Remark 14.

For any $l\in\mathcal{L}^{+}$ and for any $\beta\geq 0$, the function
$f_{\beta}:[0,+\infty)\to[0,1]$ given by

$f_{\beta}(t)={\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>t\right)$

is monotonic. Also, by Proposition 12, it converges pointwise as
$\beta\to+\infty$ to a continuous function. Therefore, given
$\epsilon_{\beta}>0$ such that
$\displaystyle\lim_{\beta\to+\infty}\epsilon_{\beta}=0$, for any $t>0$ we have
that

$\lim_{\beta\to+\infty}{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>t+\epsilon_{\beta}\right)=\lim_{\beta\to+\infty}{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>t-\epsilon_{\beta}\right)=e^{-t}.$

To prove Theorem 3, we will first prove the two following lemmas.

######  Lemma 15.

For any $u\in\mathcal{S}$, let

$\rho_{u}={\mathbb{P}}\left(R^{\beta,u}(\mathcal{L}^{+})<R^{\beta,u}(\mathcal{L}^{-})\Big{|}\bigcap_{j=1}^{3(N-1)}M_{j}\right).$

Then, for any $t>0$,

$\lim_{\beta\to+\infty}\sup_{u\in\mathcal{S}\setminus\mathcal{L}^{-}}\left|{\mathbb{P}}\left(\frac{R^{\beta,u}(\mathcal{L}^{-})}{c_{\beta}}>t\right)-e^{-t}\rho_{u}\right|=0.$

###### Proof.

Denoting
$E_{\beta,u}=\\{R^{\beta,u}(\mathcal{L})<e^{\beta/2},\bigcap_{j=1}^{3(N-1)}M_{j}\\}$,
we have that for any $u\not\in\mathcal{L}$,

${\mathbb{P}}\left(\frac{R^{\beta,u}(\mathcal{L}^{-})}{c_{\beta}}>t\right)={\mathbb{P}}\left(\frac{R^{\beta,u}(\mathcal{L}^{-})}{c_{\beta}}>t,E_{\beta,u},R^{\beta,u}(\mathcal{L}^{+})<R^{\beta,u}(\mathcal{L}^{-})\right)+$
(14)
${\mathbb{P}}\left(\frac{R^{\beta,u}(\mathcal{L}^{-})}{c_{\beta}}>t,E_{\beta,u},R^{\beta,u}(\mathcal{L}^{-})<R^{\beta,u}(\mathcal{L}^{+})\right)+{\mathbb{P}}\left(\frac{R^{\beta,u}(\mathcal{L}^{-})}{c_{\beta}}>t,E_{\beta,u}^{c}\right).$

By Proposition 7, there exists $\beta_{t}>0$ such that for any
$\beta>\beta_{t}$, $c_{\beta}t>e^{\beta/2}$. This implies that, for any
$\beta>\beta_{t}$,

${\mathbb{P}}\left(\frac{R^{\beta,u}(\mathcal{L}^{-})}{c_{\beta}}>t,E_{\beta,u},R^{\beta,u}(\mathcal{L}^{-})<R^{\beta,u}(\mathcal{L}^{+})\right)=0.$

Considering $l\in\mathcal{L}^{+}$, for any
$u\in\mathcal{S}\setminus\mathcal{L}^{-}$ and for all $\beta>\beta_{t}$, the
left-hand side of Equation (14) is bounded bellow by

${\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>t\right)\rho_{u}[1-{\mathbb{P}}(E_{\beta,u}^{c})]$
(15)

and bounded above by

${\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>t-\frac{e^{\beta/2}}{c_{\beta}}\right)\rho_{u}+{\mathbb{P}}(E_{\beta,u}^{c}).$
(16)

By Theorem 2 and Corollary 11,

$\lim_{\beta\to+\infty}\sup_{u\in\mathcal{S}\setminus\mathcal{L}^{-}}{\mathbb{P}}\left(E_{\beta,u}^{c}\right)=0.$

By Proposition 7, it follows that
$\displaystyle\lim_{\beta\to+\infty}e^{\beta/2}/c_{\beta}=0$. Therefore, by
Remark 14 we have that

$\lim_{\beta\to+\infty}{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>t\right)=\lim_{\beta\to+\infty}{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>t-\frac{e^{\beta/2}}{c_{\beta}}\right)=e^{-t}.$

We conclude the proof by noting that the limits in the last equation do not
depend on $u$.

∎

######  Lemma 16.

There exists $\alpha\in(0,1)$ and $\beta_{\alpha}>0$ such that for any
$\beta>\beta_{\alpha}$ and any $l\in\mathcal{L}^{+}$, the following upperbound
holds

${\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>n\right)\leq\alpha^{n},$

for any positive integer $n\geq 1$.

###### Proof.

By Lemma 15, for any fixed $\alpha\in(e^{-1},1)$, there exists
$\beta_{\alpha}$ such that for all $\beta>\beta_{\alpha}$ and for any
$u\notin\mathcal{L}^{-}$,

${\mathbb{P}}\left(\frac{R^{\beta,u}(\mathcal{L}^{-})}{c_{\beta}}>1\right)\leq\alpha<1.$
(17)

For any $l\in\mathcal{L}^{+}$ and for any $n\in\\{2,3,\ldots\\}$,

${\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>n\right)=\sum_{u\in\mathcal{S}\setminus\mathcal{L}^{-}}{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>n-1,U_{c_{\beta}(n-1)}^{\beta,l}=u\right){\mathbb{P}}\left(\frac{R^{\beta,u}(\mathcal{L}^{-})}{c_{\beta}}>1\right).$

Equation (17) implies that for any $\beta>\beta_{\alpha}$,

${\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>n\right)\leq\alpha{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}>n-1\right).$
(18)

We finish the proof by iterating (18). ∎

###### Proof.

We will now prove Theorem 3.

First of all, we will prove that for any $l\in\mathcal{L}^{+}$, the following
holds

$\frac{R^{\beta,l}(\mathcal{L}^{-})}{\mathbb{E}[R^{\beta,l}(\mathcal{L}^{-})]}\to\text{Exp}(1)\text{
in distribution, as }\beta\to+\infty.$ (19)

Considering Proposition 12, we only need to show that

$\lim_{\beta\to+\infty}\frac{\mathbb{E}[R^{\beta,l}(\mathcal{L}^{-})]}{c_{\beta}}=1.$

Actually,

$\lim_{\beta\to+\infty}\frac{\mathbb{E}[R^{\beta,l}(\mathcal{L}^{-})]}{c_{\beta}}=\lim_{\beta\to+\infty}\int_{0}^{+\infty}{\mathbb{P}}(R^{\beta,l}(\mathcal{L}^{-})>c_{\beta}s)ds.$

Lemma 16 and the Dominated Convergence Theorem, allow us to put the limit
inside the integral in the last term

$\lim_{\beta\to+\infty}\int_{0}^{+\infty}{\mathbb{P}}(R^{\beta,l}(\mathcal{L}^{-})>c_{\beta}s)ds=\int_{0}^{+\infty}\lim_{\beta\to+\infty}{\mathbb{P}}(R^{\beta,l}(\mathcal{L}^{-})>c_{\beta}s)ds=\int_{0}^{+\infty}e^{-s}ds=1.$

This and Proposition 12 imply (19).

For any $\beta>0$, for any $u\in\mathcal{C}^{+}$ and for any $s>0$,

${\mathbb{P}}(R^{\beta,u}(\mathcal{C}^{-})\geq
c_{\beta}s)={\mathbb{P}}(R^{\beta,u}(\mathcal{C}^{-})\geq
c_{\beta}s,E_{\beta,u})+{\mathbb{P}}(R^{\beta,u}(\mathcal{C}^{-})\geq
c_{\beta}s,E_{\beta,u}^{c}),$

where

$E_{\beta,u}=\\{R^{\beta,u}(\mathcal{L}^{+})<\min\\{1,R^{\beta,u}(\mathcal{C}^{-})\\},R^{\beta,u}(\mathcal{L}^{-})<R^{\beta,u}(\mathcal{C}^{-})+1\\}.$

For any $l\in\mathcal{L}^{+}$, we have

${\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}\geq
s+\frac{1}{c_{\beta}}\right){\mathbb{P}}\left(E_{\beta,u}\right)\leq{\mathbb{P}}(R^{\beta,u}(\mathcal{C}^{-})\geq
c_{\beta}s,E_{\beta,u})\leq{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}\geq
s-\frac{1}{c_{\beta}}\right){\mathbb{P}}\left(E_{\beta,u}\right).$ (20)

By Theorem 2, for any $u\in\mathcal{C}^{+}$,

$\lim_{\beta\to+\infty}{\mathbb{P}}(E_{\beta,u})=1,$

and then,

$\lim_{\beta\to+\infty}{\mathbb{P}}(R^{\beta,u}(\mathcal{C}^{-})\geq
c_{\beta}s,E_{\beta,u})=\lim_{\beta\to+\infty}{\mathbb{P}}(R^{\beta,u}(\mathcal{C}^{-})\geq
c_{\beta}s).$

By Proposition 7, $\displaystyle\lim_{\beta\to+\infty}c_{\beta}^{-1}=0.$
Therefore, Remark 14 implies that

$\lim_{\beta\to+\infty}{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}\geq
s+\frac{1}{c_{\beta}}\right)=\lim_{\beta\to+\infty}{\mathbb{P}}\left(\frac{R^{\beta,l}(\mathcal{L}^{-})}{c_{\beta}}\geq
s-\frac{1}{c_{\beta}}\right)=e^{-s}.$

The conclusion follows from Equation (20) and by observing that the Dominated
Convergence Theorem allow us to replace $c_{\beta}$ by
$\mathbb{E}[R^{\beta,u}(\mathcal{C}^{-})]$ as we did to prove that Equation
(19) holds.

∎

## Acknowledgments

This work is part of USP project Mathematics, computation, language and the
brain and FAPESP project Research, Innovation and Dissemination Center for
Neuromathematics (grant 2013/07699-0). AG is partially supported by CNPq
fellowship (grant 314836/2021-7). KL was successively supported by Capes
fellowship (grant 88887.340896/2019-00), CNPq fellowship (grant 164080/2021-0)
and and FAPESP fellowship (grant 2022/07386-0).

We thank M. Cassandro, M. Galves, J. Peschanski and R. Prôa for many
interesting discussions.

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Antonio Galves
Instituto de Matemática e Estatística
Universidade de São Paulo
Rua do Matão, 1010
São Paulo-SP, 05508-090
Brazil
e-mail address<EMAIL_ADDRESS>
Kádmo de Souza Laxa
Instituto de Matemática e Estatística
Universidade de São Paulo
Rua do Matão, 1010
São Paulo-SP, 05508-090
Brazil
e-mail address<EMAIL_ADDRESS>