# The mean field stubborn voter model

Lisa Hartung1 1 Institut für Mathematik, Johannes Gutenberg-Universität Mainz
<EMAIL_ADDRESS>,  Christian Mönch2 2 Institut für Mathematik, Johannes
Gutenberg-Universität Mainz<EMAIL_ADDRESS>and  Florian Völlering3 3
Mathematisches Institut, Universität Leipzig<EMAIL_ADDRESS>leipzig.de

###### Abstract.

We analyse the effect of a fat-tailed waiting time distribution in the voter
model on the complete graph with $N$ vertices. Our main result is the
existence of a limiting infinite voter model on the slowest updating sites. We
further derive explicitly the consensus probabilities in the limit model. To
prove this, we study properties of the coalescing system of random walks that
forms the dual of the limit voter model and proof, among others, coming down
from infinity.

AMS-MSC 2020: 82C22 (primary); 91D30, 60K35 (secondary)

Key Words: voter model, coalescing random walk, random process in random
environment

This collaborative research was initiated within the scientific network
_Stochastic processes on evolving networks_ funded by Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – grant no.
412848929. CM’s research is funded by Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – grant no. 443916008 (SPP 2265).

##  1\. Introduction & main results

We investigate the following variant of the voter model: voters are arrayed on
the complete graph with vertices $[N]=\\{1,\dots,N\\}.$ A voter in site
$x\in[N]$ is initially assigned a type (or opinion)
$\smash{\eta^{(N)}_{0}(x)\in\\{0,1\\}}$ and a clock that rings at i.i.d.
$\smash{\operatorname{Exp}(1/w_{x}^{(N)})}$ times, where the mean waiting
times $\smash{w_{1}^{(N)}\geq w_{2}^{(N)}\geq\dots\geq w_{N}^{(N)}\geq 1}$ are
given by the order statistics of $N$ i.i.d. standard
$\operatorname{Pareto}(\alpha)$ random variables with $\alpha\in(0,1).$
Whenever any clock rings, the corresponding voter copies the opinion of a
uniformly chosen neighbour, which gives rise to a Markov process
$\smash{(\eta_{t}^{(N)}(x))_{t\geq 0}}$ with right continuous trajectories in
$\\{0,1\\}^{[N]}$. We couple realisations of $w^{(N)}$ in the obvious way and
consider the triangular scheme $\smash{\\{w^{(N)}_{x},1\leq x\leq
N,N\in\mathbb{N}\\}}$ as derived from a single infinite i.i.d. sequence $w$.

Due to the heavy tail decay of the weights $w$, voters with low indices change
their opinion very slowly. We hence refer to this model as the _stubborn voter
model_. In this article, we are interested in the influence that the stubborn
voters exert on the fixation of the system for large $N$ and in the existence
of an infinite limiting model that captures the dynamics among the most
stubborn voters. As we will see below, even if only one voter $x$ starts with
opinion $1$ and all other voters start with opinion $0$, the probability that
the system fixates in state $1\in\\{0,1\\}^{[N]}$ remains bounded away from
$0$, as long as the initial position $x$ of the single $1$-voter does not grow
with $[N]$.

The homogeneous voter model on $\mathbb{Z}^{d}$ has been extensively studied
in the 1970s and 1980s, see [1, 8, 15, 2] for some important early results and
Liggett’s monograph [12] for an overview of the classical theory in the
general context of stochastic interacting particle systems. With the advent of
Network Science in the last 20 years, interest in models for consensus
dynamics has renewed, particularly for agents that are networked in an
inhomogeneous fashion as opposed to the classical translation invariant
setting on $\mathbb{Z}^{d}$. Furthermore, the influence of ‘stubborn’ or
‘zealous’ agents on opinion dynamics has received considerable attention in
the (socio-)physics literature, see for instance [13, 18, 16, 17].
Mathematically rigorous results for zealous/stubborn voters were obtained in
[9] and [14]. However, in all these references, stubbornness is modelled by
the presence of a few selected opinions that _never_ change, whereas we assume
the presence of an inherent heavy-tailed stubbornness distribution that leads
to some voters changing their opinion more slowly than others.

We now provide an overview of our main results. We call the sites with the
largest weights _slow sites_ as they take, on average, the longest to update
their opinion. Since the distribution function $F$ of the weights satisfies
$1-F(x)\sim x^{-\alpha}$ with $\alpha\in(0,1)$, the weights of the slow sites
are of order $N^{1/\alpha}$. More precisely, let $(\chi_{i})_{i\in\mathbb{N}}$
be i.i.d. $\operatorname{Exp}(1)$-distributed and set

$\xi_{i}=\Big{(}\sum_{j\in[i]}\chi_{i}\Big{)}^{-\frac{1}{\alpha}},\quad
i\in\mathbb{N},$

then it is well known that

(1)
$N^{-\frac{1}{\alpha}}\left(\big{(}w_{i}^{(N)}\big{)}_{i=1}^{N},\sum_{i\in[N]}w_{i}^{(N)}\right)\overset{d}{\longrightarrow}\left((\xi_{i})_{i=1}^{\infty},\sum_{i\in\mathbb{N}}\xi_{i}\right)\quad\text{
as }N\to\infty,$

see [7, Theorem 2.33] for this formulation and [11, Section 2.3] for the
general framework. Here and throughout $\overset{d}{\to}$ denotes convergence
in distribution. The first main result covers the existence of a limiting
voter model as $N\to\infty$, when speeding up time by $N^{\frac{1}{\alpha}}$.

######  Theorem 1.1.

Let $w^{(N)}$ and $\xi$ be as in (1) and let
$(\eta_{0}^{(N)})_{N\in\mathbb{N}}$ denote a sequence of initial conditions
satisfying

$\eta_{0}^{(N)}\overset{d}{\longrightarrow}\eta_{0}\text{ as }N\to\infty,$

for some random or deterministic $\eta_{0}\in\\{0,1\\}^{\mathbb{N}}$. Then the
corresponding Markov processes $(\eta_{t}^{(N)})_{t\geq 0}$ satisfy

$\eta^{(N)}_{N^{\frac{1}{\alpha}}(\cdot)}\overset{d}{\longrightarrow}\eta_{(\cdot)}\text{
as }N\to\infty,$

where the limit process $\eta$ has initial distribution $\eta_{0}$ and
transition rates governed by $\xi$, i.e. conditionally on $\xi$,
$(\eta_{t})_{t\geq 0}$ is a voter model with update rates
$\xi_{x},x\in\mathbb{N}.$

Note that the existence of the limit process is highly non-trivial, since on
the complete graph with vertex set $\mathbb{N}$, there is no such thing as a
‘uniformly chosen neighbour’ of a vertex. In fact, the assumption of heavy
tailed waiting times is essential for the limit process to exist. We always
consider the voter models under a fixed environment (quenched), i.e. under the
distributions

$\mathbb{P}=\mathbb{P}^{\xi}=\mathcal{L}(\eta|\xi)$

or

$\mathbb{P}=\mathbb{P}^{w}=\mathcal{L}(\eta^{(N)}|w).$

Next, we obtain the consensus probabilities in the limit model.

######  Theorem 1.2.

Let $\eta_{0}\in\\{0,1\\}^{\mathbb{N}}$ be any initial distribution and let

$\tau=\inf_{t>0}\\{\eta_{t}\equiv 0\text{ or }\eta_{t}\equiv 1\\}$

denote the consensus time. Then,

$\mathbb{E}[\tau]<\infty\text{ for almost every realisation of }\xi.$

Moreover,

(2) $\mathbb{P}(\eta_{\tau}\equiv
1)=\frac{\sum_{x\in\mathbb{N}}\eta_{0}(x)\xi_{x}}{\sum_{x\in\mathbb{N}}\xi_{x}}.$

For completeness, we also state the corresponding formula for the consensus
probability in the finite systems.

######  Theorem 1.3.

Let $\eta_{0}^{(N)}$ be any initial distribution and let

$\tau^{(N)}=\inf_{t>0}\\{\eta_{t}^{(N)}\equiv 0\text{ or }\eta_{t}\equiv 1\\}$

denote the consensus time. Then,

$\sup_{N}N^{-\frac{1}{\alpha}}\mathbb{E}[\tau^{(N)}]<\infty\text{ for almost
every realisation of }w.$

Moreover,

(3) $\mathbb{P}(\eta_{\tau}^{(N)}\equiv
1)=\frac{\sum_{x\in\mathbb{N}}\eta^{(N)}_{0}(x)w_{x}^{(N)}}{\sum_{x\in\mathbb{N}}w^{(N)}_{x}}.$

Note that the consensus time for $\eta^{[N]}$ is trivially finite, because the
system has only finitely many states and two of them are absorbing. However,
due to the divergence of timescales, the expected consensus times for
$\eta^{(N)}$ are not uniformly bounded without rescaling. Below, we prove
Theorem 1.2 directly but an alternative way to derive the asymptotic consensus
probabilities is to combine (3) with Theorem 1.1 and (1).

  * Remark (countably many opinions).

It will be clear from the proofs that Theorems 1.1–1.3 remain valid mutatis
mutandis, if opinions are drawn from a countable alphabet instead of
$\\{0,1\\}$. In particular, we may initialise the infinite system in such a
way that each voter in $\mathbb{N}$ carries a distinct opinion. The expected
consensus time in this case is still finite and the probability that the whole
system ends up with the initial opinion of vertex $x\in\mathbb{N}$ is
$\xi_{x}/\sum_{z\in\mathbb{N}}\xi_{z}$.

### Structure of the paper

In Section 2, we introduce the system of coalescing random walks that forms
the dual of the voter model. In Sections 3-5, we work exclusively with the
dual: In Section 3, we construct the Markov process $X$ that describes the
motion of a single walker among slow sites indexed by $\mathbb{N}$ and derive
some of its properties that are of independent interest. In Section 4, we show
that the walk $Y^{(N)}$ on $[N]$ with jump rates governed by $w^{(N)}$
converges to $X$ and that the limiting procedure can be extended to
corresponding coalescing systems of finitely many walks. Finally, in Section
5, we show that the limit system of coalescing walks comes down from infinity
in Theorem 5.1. This allows us to treat infinitely many lineages in the voter
model simultaneously. Finally, we prove Theorem 1.1, Theorems 1.2 in Section
6.

##  2\. The dual system: coalescing random walks with heavy tailed rates

The following duality relation for the voter model is well known, see e.g.
[12, Chapter V]: Given $w^{(N)}$, consider the continuous time random walk
$Y^{(N)}$ on $[N]$ with transition rates

$q^{(N)}(x,y)=\frac{1}{Nw^{(N)}_{x}},\quad x,y\in[N].$

Provided with some initial condition $\gamma_{0}^{(N)}\in\\{0,1\\}^{[N]}$,
which we interpret as the positions of $\sum_{x\in[N]}\gamma^{(N)}_{0}(x)$
particles, we let each of the particles move independently according to the
walk $Y^{(N)}$ and coalesce particles which are located in the same site. This
leads to a Markovian system $\big{(}\gamma_{t}^{(N)}\big{)}_{t\geq 0}$ of
coalescing random walks that is dual to the voter model in the sense that for
all $t\geq 0,A\subset[N]$,

(4) $\mathbb{P}\big{(}\eta^{(N)}_{t}(x)\equiv 1\text{ on
}A\big{|}\eta^{(N)}_{0}\big{)}=\mathbb{P}\big{(}\eta^{(N)}_{0}(x)\equiv
1\text{ on
}\operatorname{supp}\gamma^{(N)}_{t}\big{|}\operatorname{supp}\gamma^{(N)}_{0}=A\big{)},$

where $\operatorname{supp}\gamma_{t}=\\{x:\gamma_{t}(x)=1\\},t\geq 0.$ Note
that the dual dynamics can be reconstructed from the primal dynamics and vice
versa. We make use of this fact in Section 6 when proving Theorem 1.1 through
duality.

##  3\. Limiting walk on the slow sites

### Construction and properties of the limit process

The convergence in (1) indicates that a natural choice of metric on the space
of environments is the $\ell^{1}$-distance.

######  Lemma 3.1.

Let $\xi^{(n)},\xi\in\ell_{1},n\geq 1,$ be non-negative then
$\lim_{n\to\infty}\xi^{(n)}_{x}=\xi_{x}$ for all $x\in\mathbb{N}$ and

$\lim_{n\to\infty}\sum_{x\in\mathbb{N}}\xi^{(n)}_{x}=\sum_{x\in\mathbb{N}}\xi_{x}$

if and only if $\xi^{(n)}\to\xi$ in $\ell^{1}$.

###### Proof.

The second assertion obviously implies the first one. For the converse
implication, fix $N\in\mathbb{N}$ and write $Z_{\geq N}(\xi)$ for $\sum_{x\geq
N}\xi_{x}$, then

$|\xi-\xi^{(n)}|_{\ell^{1}}\leq\sum_{x<N}|\xi_{x}^{(n)}-\xi_{x}|+Z_{\geq
N}(\xi)+Z_{\geq N}(\xi^{(n)}).$

Fix $\epsilon>0$, then $N$ so large that $Z_{\geq N}(\xi)\leq\epsilon$. Fix
$L$ so large that $|Z_{\geq N}(\xi)-Z_{\geq N}(\xi^{(n)})|\leq\epsilon$
whenever $n\geq L$. We obtain for $n\geq L$

$|\xi-\xi^{(n)}|_{\ell^{1}}\leq\sum_{x<N}|\xi_{x}^{(n)}-\xi_{x}|+3\epsilon,$

and taking the limit $n\to\infty$ yields
$|\xi-\xi^{(n)}|_{\ell^{1}}<3\epsilon$ due to coordinate-wise convergence,
which concludes the argument since $\epsilon$ is chosen arbitrarily. ∎

Let $\mathbf{P}$ denote the distribution of the limiting environment $\xi$.
Note that the strong law of large numbers yields that $\mathbf{P}$-almost
surely

$i/2<\sum_{j\in[i]}\eta_{i}<2i,\text{ for all but finitely many
}i\in\mathbb{N},$

and consequently we have

(5) $\xi_{i}\asymp i^{-1/\alpha}\quad\mathbf{P}\text{-almost surely}.$

Here, $f(i)\asymp g(i)$ means that $f(i)/g(i)$ is bounded away from $0$ and
$\infty$ uniformly in $i\in\mathbb{N}$. In particular, by (5), the right-hand
side of (1) is a well-defined random variable on $\ell_{1}\times(0,\infty)$.
We have that

$\xi([N])=\sum_{x\in[N]}\xi_{x}\overset{N\to\infty}{\longrightarrow}\xi(\mathbb{N})=\sum_{x\in\mathbb{N}}\xi_{x}<\infty,\quad\mathbf{P}\text{-almost
surely},$

and that

$Z(\lambda)=\sum_{x\in\mathbb{N}}\frac{\lambda\xi_{x}}{1+\lambda\xi_{x}}\in\mathbb{R},\quad\text{
exists }\mathbf{P}\text{-almost surely for any
}\lambda\in\mathbb{R}\setminus\\{-\xi_{x}^{-1},x\in\mathbb{N}\\}.$

Eventually, our goal is to construct a limiting coalescing system that
captures the behaviour of the coalescing random walk amongst the slow sites,
thereby extending (1) from the environments to the processes. Our first goal
is thus to construct a random walk on $\mathbb{N}$ whose transitions are
governed by the limit environment $\xi=(\xi_{i})_{i\in\mathbb{N}}$ described
prior to (1). As the holding times outside the slow sites tend to zero under
the rescaling, this construction must be done carefully. We start the
construction with finitely many sites. Let $X^{(N)}=(X_{t}^{(N)})_{t\geq 0}$
be a random walk on $[N]$ with uniform jump distribution and waiting times
governed by $\xi$. We assume that the starting point $X_{0}^{(N)}\in[N]$ is
chosen according to some distribution $\nu_{N}$ with
$\operatorname{supp}(\nu_{N})\subset[N]$. For technical convenience, we
consider $X^{(N)}$ as a Markov process on $\mathbb{N}$ via the canonical
embedding of $[N]$ into $\mathbb{N}$, which leads to the transition rates

$r^{(N)}(x,y)=\begin{cases}\frac{1}{N\xi_{x}},&\text{ if }x,y\in[N],\\\
0,&\text{ otherwise.}\end{cases}$

  * Remark.

Note that at every jump time $X^{(N)}$ may remain at its current position with
probability $1/N$, i.e. it is a lazy version of the random walk on $[N]$ with
holding times governed by $\xi$. Defining $X^{(N)}$ in this manner leads to
slightly simpler calculations. It is easy to see that the limit process (see
Theorem 3.8 below) is the same for both the lazy version $X^{(N)}$ and the
non-lazy version.

We mostly work under the quenched measure $\mathbb{P}=\mathbb{P}^{\xi}$
conditional on the environment $\xi$. We use the shorthand
$\mathbb{P}_{x}(\,\cdot\,)$ for $\mathbb{P}(\,\cdot\,|X^{[N]}_{0}=x),$ where
it is tacitly assumed that $N\geq x$. Similarly we define
$\mathbb{P}_{\nu^{N}}=\sum_{x\in[N]}\nu^{N}(x)\mathbb{P}_{x}$, and write
$\mathbb{P}_{\nu}$ instead of $\mathbb{P}_{\nu^{N}}$ when there is no
ambiguity regarding the initial distribution. We frequently initiate $X^{(N)}$
in a uniformly chosen site and denote the corresponding distributions by
$\mathbb{P}_{\mathbf{u}}=N^{-1}\sum_{x\in[N]}\mathbb{P}_{x}.$

Note that $X^{(N)}$ only depends on the initial segment of $\xi$. If we keep
one realisation of $\xi$ fixed and choose $M>N$, then $X^{(N)}$ can be viewed
as a projection of $X^{(M)}$ up to a time change. To make this precise, let
$W=(W_{t})_{t\geq 0}$ be any Markov process on $\mathbb{N}$ under $\mathbb{P}$
with stationary distribution $\nu$. Let $A\subset\mathbb{N}$ be such that
$\nu(A)>0$, thereby ensuring that, $\mathbb{P}$-almost surely,

$W^{-1}(A)=\\{t:W_{t}\in A\\}$

is unbounded and $T_{A}=\inf\\{t>0:W_{t}\in A\\}$ is finite. Let now
$\psi^{W}_{A}:[0,\infty)\to W^{-1}(A)$ be given by

$\psi^{W}_{A}(t)=\int_{0}^{t}\mathbbm{1}_{\\{W_{t}\in A\\}}\,\textup{d}t.$

Since $\psi_{A}^{W}$ is non-decreasing, it admits an increasing right-
continuous inverse $\phi_{A}^{W}$. We call the time-changed process
$\textup{tr}_{A}(W)$ defined via

$\textup{tr}_{A}(W)_{t}=W_{\phi^{W}_{A}(t)},\quad t\geq 0,$

the _trace_ of $W$ on $A.$

######  Lemma 3.2.

Let $M>N\in\mathbb{N}$. For any $x\in[N]$, the processes
$\textup{tr}_{[N]}\big{(}X^{(M)}\big{)}$ and $X^{(N)}$ have the same
distribution under $\mathbb{P}_{x}$. The same assertion remains true, if both
$X^{(M)}$ and $X^{(N)}$ are started in a uniformly chosen site in $[M]$ and
$[N]$, respectively.

###### Proof.

For any given realisation of $\xi$, this follows immediately from the
definition of $\textup{tr}_{[N]}\big{(}X^{(M)}\big{)}$ and the uniformity of
the jump distributions of $X^{(N)}$ and $X^{(M)}$. ∎

The next lemma establishes that the time spent outside the slow sites is
asymptotically negligible.

######  Lemma 3.3.

For $M>N\geq 1$ let $\phi_{M,N}:[0,\infty)\to[0,\infty)$ be the associated
time change of the trace process $\textup{tr}_{[N]}\big{(}X^{(M)}\big{)}$ of
$X^{(M)}$ started in a fixed initial distribution $\nu^{M}$ on $[M]$. For any
sequence $(\epsilon_{N})_{N\in\mathbb{N}}$ in $(0,1)$, with

$\lim_{N\to\infty}\epsilon_{N}N^{\frac{1}{\alpha}-1}=\infty,$

it holds that

$\lim_{N\to\infty}\sup_{M>N}\mathbb{P}_{\nu^{M}}(|\phi_{M,N}(t)-t|\geq\epsilon_{N})=0,\quad
t>0.$

Throughout the proof of Lemma 3.3 and the remainder of this section, we
frequently decompose the path of $X^{(M)}$ into excursions. To prepare the
arguments, we establish a few auxiliary results pertaining to the excursion
structure of $X^{(M)}$ first. For a given proper non-empty subset $A$ of
$[M]$, denote by $T_{0}^{A}$ the first hitting time of $A$ by $X^{(M)}$ and by
$\mathcal{E}^{A}_{1},\mathcal{E}^{A}_{2},\dots$ the successive excursion
intervals of $X^{(M)}$ from $A$ after time $T^{A}_{0}$. Note that
$T^{A}_{j}<\infty$ for all $j\in\mathbb{N}_{0}$ $\mathbb{P}$-almost surely and
that $T^{A}_{0}=0$ precisely if $X_{0}\in A.$ Clearly, the set of times
$[T^{A}_{0},\infty)\setminus\bigcup_{n=1}^{\infty}\mathcal{E}^{A}_{n}$ can
also be identified with a sequence
$\mathcal{I}^{A}_{1},\mathcal{I}^{A}_{2},\dots$ of non-adjacent
$\mathbb{P}$-almost surely finite intervals. Let $E^{A}$ have the distribution
of $|\mathcal{E}^{A}_{1}|$ and let $I^{A}$ have the distribution of
$|\mathcal{I}^{A}_{1}|$. By stationarity of the transition rates, uniformity
of the jump distribution and the Markov property, we have that

$({E}^{A}_{n})_{n\in\mathbb{N}}:=(|\mathcal{E}^{A}_{n}|)_{n\in\mathbb{N}}\text{
and }({I}^{A}_{n})_{n\in\mathbb{N}}:=(|\mathcal{I}^{A}_{n}|)_{n\in\mathbb{N}}$

are two independent sequences of i.i.d. copies of $E^{A}$ and $I^{A}$. We
refer to $E^{A}$ and $I^{A}$ as the generic _excursion length_ and the generic
_incursion length_ of $A$, respectively. Observe also, that
$T_{0}\overset{d}{=}E^{A}$ if and only if $X_{0}$ is distributed uniformly on
$[M]\setminus A$. We further define $H^{A}$ to be
$\operatorname{Exp}(\xi_{U}^{-1})$-distributed conditionally on $U$, where $U$
is uniform on $A$. Hence, $H^{A}$ is a _typical holding time_ of $X^{(M)}$
conditioned to stay in $A$. The next two lemmas collect some properties of
holding times and excursion/incursion lengths.

######  Lemma 3.4.

Let $A\subset[M]$ be such that $\max_{x\in A}\xi_{x}\leq 1/2$ and let
$H^{A}_{1},H^{A}_{2},\dots$ be i.i.d. typical $A$-holding times. Set

$\bar{\xi}(A)=\frac{\sum_{x\in A}\xi_{x}}{|A|}.$

For any $l\in\mathbb{N}$ and $\lambda>0$, we have

$\mathbb{P}\left(\sum_{j=1}^{l}H^{A}_{i}>l\lambda\right)\leq\textup{e}^{-l(\lambda-2\bar{\xi}(A)+O(\bar{\xi}(A)^{2}))}.$

###### Proof.

Markov’s inequality together with the fact that
$\mathbb{E}\textup{e}^{H^{A}_{1}}=|A|^{-1}\sum_{x\in A}\frac{1}{1-\xi_{x}}$
yields

$\mathbb{P}\left(\sum_{j=1}^{l}H^{A}_{i}>l\lambda\right)=\mathbb{P}\left(\textup{e}^{\sum_{j=1}^{l}H^{A}_{i}}>\textup{e}^{l\lambda}\right)\leq\textup{e}^{-l\lambda+l\log(|A|^{-1}\sum_{x\in
A}\frac{1}{1-\xi_{x}})}.$

Now the assertion follows by a Taylor expansion of $\log(\cdot)$ around $1$,
since

(6) $\frac{1}{|A|}\sum_{x\in A}\frac{1}{1-\xi_{x}}=1+\frac{1}{|A|}\sum_{x\in
A}\frac{\xi_{x}}{1-\xi_{x}}\leq 1+2\bar{\xi}(A)\leq 2,$

by assumption on $(\xi_{x})_{x\in A}$. ∎

######  Lemma 3.5.

Fix an i.i.d. sequence of typical $A$-holding times
$(H^{A}_{i})_{i\in\mathbb{N}}$ and let $G$ denote a
$\operatorname{Geo}(1-|A|/M)$ random variable independent of
$(H^{A}_{i})_{i\in\mathbb{N}}$. Then,

(7) $I^{A}\overset{\textup{d}}{=}\sum_{i=1}^{G}H^{A}_{i},$

and furthermore

(8) $\mathbb{E}\Big{[}\textup{e}^{-\lambda
I^{A}}\Big{]}=\frac{(M-|A|)\,\frac{1}{|A|}\sum_{x\in
A}\frac{1}{1+\lambda\xi_{x}}}{M-\sum_{x\in
A}\frac{1}{1+\lambda\xi_{x}}},\quad\lambda\in\mathbb{R}\setminus\\{-\xi_{x}^{-1},x\in
A\\},$

as well as

(9) $\mathbb{E}I^{A}=\frac{M}{M-|A|}\,\frac{1}{|A|}\sum_{x\in A}\xi_{x}.$

In particular, for $A=[N]$ with $N<M$, it follows that $\mathbf{P}$-almost
surely for $E^{[N]}=I^{[M]\setminus[N]}$

(10) $\sup_{M>N}\mathbb{E}\big{[}E^{[N]}\big{]}\in
O\big{(}N^{-\frac{1}{\alpha}}\big{)}.$

###### Proof.

The representation (7) follows from two simple observations. Firstly, upon
entering $A$, $X^{(M)}$ sits in a uniformly distributed site and has a site-
independent probability of $1-\frac{|A|}{M}$ at each jump to leave $A$ and
each jump within $A$ leads with probability $|A|^{-1}$ to any given site $x$.
Secondly, at each site $x$, the holding time is an independent
$\operatorname{Exp}(1/\xi_{x})$-random variable. (8) can then be obtained from
(7) by elementary calculations using

(11) $\mathbb{E}\textup{e}^{-\lambda H^{A}_{1}}=|A|^{-1}\sum_{x\in
A}\frac{1}{1+\lambda\xi_{x}},$

and the independece of $G$ and $(H^{A}_{i})_{i\in\mathbb{N}}$. (9) follows
directly from (8). (10) is a straightforward consequence of (9) and (5). ∎

Next, we prove Lemma 3.3.

###### Proof of Lemma 3.3.

We consider the excursions of $X^{(M)}$ from $[N]$. Set $T_{j}=T_{j}^{[N]}$,
$j\in\mathbb{N}_{0}$. Let, for fixed $t>0$,

$\ell(t)=\max\\{n\geq 0:T_{n}\leq\phi_{M,N}(t)\\}.$

Note that for any $t\geq 0$,

$\phi_{M,N}(t)-t=T_{0}+\sum_{j=1}^{\ell(t)}E_{j}\geq 0,$

with $E_{j},j\in\mathbb{N}$ i.i.d. copies of $E^{[N]}$. In particular, for any
fixed $\epsilon>0$

(12)
$\mathbb{P}_{\nu}(\phi_{M,N}(t)-t>2\epsilon)\leq\mathbb{P}_{\nu}(T_{0}\geq\epsilon)+\mathbb{P}\Big{(}\sum_{j=1}^{\ell(t)}E_{j}>\epsilon\Big{)}.$

For the second term on the right hand side of (12), we first note that it does
not depend on the initial condition (8). Moreover,

(13)
$\mathbb{P}\Big{(}\sum_{j=1}^{\ell(t)}E_{j}>\epsilon\Big{)}\leq\mathbb{P}\Big{(}\sum_{j=1}^{L}E_{j}>\epsilon\Big{)}+\mathbb{P}(\ell(t)>L),\quad
L\in\mathbb{N}.$

We denote the incursion lengths inside $[N]$ by $I_{1},I_{2},\dots$. Then, by
Chebyshev’s inequality

$\mathbb{P}(\ell(t)>L)\leq\mathbb{P}\Big{(}\sum_{j=1}^{L}I_{j}<t\Big{)}\leq\textup{e}^{\lambda
t}\mathbb{E}\Big{[}\textup{e}^{-\lambda
I_{1}}\Big{]}^{L}\leq\textup{e}^{-q_{N}(\lambda)L+\lambda t},\quad
L\in\mathbb{N},\lambda>0,$

where we used that by (8),

(14) $\mathbb{E}\Big{[}\textup{e}^{-\lambda
I_{1}}\Big{]}=\frac{(M-N)(1-q_{N}(\lambda))}{M-N(1-q_{N}(\lambda))}\leq
1-{q_{N}(\lambda)},\quad\lambda>0,$

with $q_{N}(\lambda)\equiv
1-N^{-1}\sum_{x\in\mathbb{N}}\frac{1}{1+\lambda\xi_{x}}$. Fixing
$K_{0}\in\mathbb{N}$, we obtain for any choice of $L=L(N,M)$ satisfying
$L>K_{0}N$

(15)
$\lim_{N\to\infty}\sup_{M>N}\mathbb{P}(\ell(t)>L)\leq\textup{e}^{-K_{0}Z(\lambda)+\lambda
t},\quad\lambda>0,$

since
$Nq_{N}(\lambda)=\sum_{y\in[N]}\frac{\lambda\xi_{y}}{1+\lambda\xi_{y}}\to
Z(\lambda)$ as $N\to\infty.$ On the other hand, for $L=2K_{0}N$, we bound the
second term in (13) from above by

(16)
$\mathbb{P}\Big{(}\sum_{j=1}^{L}E_{j}>\epsilon\Big{)}\leq\frac{2K_{0}N\mathbb{E}E_{1}}{\epsilon}\leq\frac{CK_{0}}{\epsilon}N^{1-1/\alpha}$

for some $C$ depending only on $\xi$, by Markov’s inequality and (10). We may
thus conclude from (13) and (15) that

(17)
$\lim_{N\to\infty}\sup_{M>N}\mathbb{P}\Big{(}\sum_{j=1}^{\ell(t)}E_{j}>\epsilon\Big{)}=0,$

since $K_{0}$ can be made arbitrarily large. It remains to show that
$\mathbb{P}_{\nu}(T_{0}>\epsilon)$ vanishes. If $X_{0}\in[N]$, then $T_{0}=0$.
Otherwise, the first jump of $X$, say at time $\sigma_{0}$, either sends $X$
to $[N]$ in which case $\sigma_{0}=T_{0}$ or to a uniformly distributed point
in $[M]\setminus[N]$ in which case the remaining time to enter $[N]$ has
precisely the distribution of $E_{1}$. We conclude

$\mathbb{P}_{\nu}(T_{0}>\epsilon)\leq\sum_{x\in[M]\setminus[N]}\mathbb{P}_{x}(\sigma_{0}>\epsilon/2)\nu(x)+\mathbb{P}(E_{1}>\epsilon/2)$

and thus

(18)
$\lim_{N\to\infty}\sup_{M>N}\mathbb{P}_{\nu}(T_{0}>\epsilon)\leq\lim_{N\to\infty}\sup_{M>N}\big{(}\mathbb{P}(\tau_{N+1}>\epsilon/2)+\mathbb{P}(E_{1}>\epsilon/2)\big{)}=0,$

where we have used that $\sigma_{0}$ is dominated by an
$\operatorname{Exp}(1/\xi_{N+1})$ random variable $\tau_{N+1}$ uniformly in
the starting point $x\in[M]\setminus[N]$ and that
$\sup_{M>N}\mathbb{P}(E_{1}>\epsilon/2)\in O(\epsilon^{-1}N^{-1/\alpha})$ by
(10) and Markov’s inequality. Combining (17) and (18) according to (12)
concludes the proof for fixed $\epsilon>0$. The assertion for decaying
$\epsilon_{N}$ follows from the observation that (15) holds independently of
the choice of $\epsilon$ and that in the remaining estimates (16) and (18) the
term on the right hand side of (16) dominates. ∎

We next describe the limiting process associated with the sequence
$(X^{N})_{N\in\mathbb{N}}$. For $N\in\mathbb{N}$, let
$P^{(N)}=(P^{(N)}_{t})_{t\geq 0}$ denote the semigroup of $X^{(N)}$ given
$\xi.$

######  Proposition 3.6.

Let $(\nu^{M})_{M\in\mathbb{N}}$ be any sequence of distributions such that
$\operatorname{supp}\nu^{M}\subset[M]$ for each $M\in\mathbb{N}$ and such that
$\lim_{M\to\infty}\nu^{M}([N])=0$ for any $N\in\mathbb{N}$. There exists a
family of probability laws $(Q_{t})_{t>0}$ on $\mathbb{N}$ which does not
depend on the choice of $(\nu^{M})_{M\in\mathbb{N}}$ and satisfies

$Q_{t}=\lim_{M\to\infty}\nu^{M}P^{(M)}_{t},\;t>0,\quad\mathbf{P}\text{-almost
surely.}$

Before we give the proof, we formulate an auxiliary result that is also useful
later on.

######  Lemma 3.7.

For any $M>N>2$ and $t>0$, we have that

$\displaystyle\mathbb{P}_{\mathbf{u}}(X^{(M)}_{t}\in\mathbb{N}\setminus[N]$
$\displaystyle|X_{0}^{(M)}\in[M]\setminus[N])$
$\displaystyle\leq\frac{2M-N}{M-N}\left(\textup{e}^{-s_{N}t}+\frac{N}{s_{N}}\left(\frac{t}{\xi([N])}+C\right)\right),$

where $C$ is a universal constant and

$s_{N}=s_{N}(\xi)=\frac{1}{2}\left(1/\xi_{N+1}\wedge
N/2\xi(\mathbb{N}\setminus[N])\right).$

###### Proof.

Let $I=I^{[N]}$ and $E=E^{[N]}$ be the generic incursion and excursion lengths
of $X^{(M)}$ from $[N]$. For any $M>N$, we set

$Z_{t}^{(M)}=\mathbbm{1}_{\\{X^{(M)}\in[M]\setminus[N]\\}},\quad t\geq 0.$

Conditionally on $X_{0}^{(M)}\in[M]\setminus[N]$, the starting point
$X_{0}^{(M)}$ is uniformly distributed on $[M]\setminus[N]$ and thus the
conditional distribution of $(Z_{t}^{(M)})_{t\geq 0}$ is that of a simple two
state renewal process, with transition function
$p^{M}(r)=\mathbb{P}_{\mathbf{u}}(Z_{r}^{(M)}=1|X_{0}^{(M)}\in[M]\setminus[N])$
given by

(19)
$p^{M}(r)=\bar{F}_{E}(r)+\int_{0}^{r}\bar{F}_{E}(r-s)\,\textup{d}\ell^{M}(s),r\geq
0.$

Here, $\bar{F}_{E}(r)=\mathbb{P}(E>r)$ and $\ell^{M}$ is the renewal measure
associated with the generic cycle length $E+I$. Let us first provide bounds on
$\bar{F}_{E}$. Let $G$ denote the number of stays inside $[M]\setminus[N]$
that $X^{(M)}$ accumulates during a generic excursion $E$, and let
$H_{1},H_{2},\dots$ denote independent generic $[M]\setminus[N]$-holding
times. We have, by Chebyshev’s inequality,

$\displaystyle\mathbb{P}(E>t)$
$\displaystyle=\mathbb{P}\left(\sum_{j=1}^{G}H_{j}>t\right)=\sum_{j=1}^{\infty}\frac{N}{M}(1-N/M)^{j-1}\mathbb{P}\left(\sum_{i=1}^{j}H_{i}>t\right)$
$\displaystyle\leq\sum_{j=1}^{\infty}\frac{N}{M}(1-N/M)^{j-1}\textup{e}^{-st}(\mathbb{E}\textup{e}^{sH_{1}})^{j}.$

Using (11) and taking $s$ equal to

$s_{N}=s_{N}(\xi)=\frac{1}{2}\left(1/\xi_{N+1}\wedge
N/2\xi(\mathbb{N}\setminus[N])\right),$

which is positive $\mathbf{P}$-almost surely for all $N>2$, we get

$\displaystyle\mathbb{P}(E>t)$
$\displaystyle\leq\frac{N}{M-N}\textup{e}^{-s_{N}t}\sum_{j=1}^{\infty}\left(\frac{1}{M}\sum_{x=N+1}^{M}\frac{1}{1-s_{N}\xi_{x}}\right)^{j}$
$\displaystyle\leq\frac{N}{M-N}\textup{e}^{-s_{N}t}\sum_{j=1}^{\infty}\left(\frac{M-N+2s_{N}\xi(\mathbb{N}\setminus[N])}{M}\right)^{j}$
$\displaystyle\leq\textup{e}^{-s_{N}t}\frac{N}{M-N}\frac{2M-N}{N}=\textup{e}^{-s_{N}t}\frac{2M-N}{M-N}.$

The second inequality uses a bound similar to (6). The third inequality is
also true by the choice $s_{N}$. Inserting this estimate into (19) yields

(20)
$p^{M}(t)\leq\frac{2M-N}{M-N}\left(\textup{e}^{-s_{N}t}+\int_{0}^{t}\textup{e}^{-s_{N}(t-s)}\,\textup{d}\ell^{M}(s)\right).$

Note that $\ell^{M}$ is dominated by the renewal measure $r^{N}$ associated
with the typical holding time $H=H^{[N]}$ inside $[N]$, because each cycle
contains at least one stay inside $[N]$. We obtain

$\int_{0}^{t}\textup{e}^{-s_{N}(t-s)}\,\textup{d}\ell^{M}(s)\leq\frac{r^{N}(t)}{s_{N}}\int_{0}^{t}s_{N}\textup{e}^{-s_{N}(t-s)}\frac{1}{r^{N}(t)}\,\textup{d}r^{N}(s)\leq\frac{r^{N}(t)}{s_{N}},$

where the last inequality follows from the fact that the integral can be
viewed as the distribution function at $t$ of the sum of two independent
random variables. We conclude that

(21)
$p^{M}(t)\leq\frac{2M-N}{M-N}\textup{e}^{-s_{N}t}+\frac{2M-N}{M-N}\frac{r^{N}(t)}{s_{N}},$

and the renewal function $r^{N}(t)$ satisfies the bound

$r^{N}(t)\leq\frac{t}{\mathbb{E}H}+C\frac{\mathbb{E}H^{2}}{(\mathbb{E}H)^{2}}=\frac{tN}{\xi([N])}+C\frac{2N\sum_{x=1}^{N}\xi_{x}^{2}}{\xi([N])^{2}},$

where $C$ is some universal constant, see e.g. [3]. Combining this bound with
(21) concludes the proof. ∎

###### Proof of Proposition 3.6.

We set

$R_{t}^{M}(y)=\frac{1}{M}\sum_{x\in[M]}\delta_{x}P^{(M)}_{t}(y)=\mathbb{P}_{\mathbf{u}}(X^{(M)}_{t}=y),\quad
y\in[M],\,M\in\mathbb{N},$

and divide the proof into three parts: first we establish tightness of
$(R_{t}^{(M)})_{M\in\mathbb{N}}$, then we show the existence of a unique
accumulation point, and finally we argue that the uniform distributions in the
definition of $R^{(M)}_{t}$ may be replaced by any sequence
$(\nu^{M})_{M\in\mathbb{N}}$ that puts vanishing mass on the slow sites.
Part (i) – tightness: Let $\epsilon>0$ be given and fix $N\in\mathbb{N}$. Let
now $M_{0}>N/\epsilon$. We obtain

(22)
$R_{t}^{M}(\mathbb{N}\setminus[N])\leq\epsilon+\frac{M-N}{M}\mathbb{P}_{\mathbf{u}}(X^{(M)}_{t}\in\mathbb{N}\setminus[N]|X_{0}^{(M)}\in[M]\setminus[N]),\quad
M>M_{0},t>0.$

Thus, Lemma 3.7 implies that for some constant $C$ depending only on $\xi$,
and all sufficiently large $M$,

(23) $R_{t}^{M}(\mathbb{N}\setminus[N])\leq\epsilon+CtN^{1-1/\alpha},$

since $1/s_{N}=O(N^{-\frac{1}{\alpha}})$ $\mathbb{P}$-alomst surely. It
follows that $(R_{t}^{M})_{M\in\mathbb{N}}$ is indeed tight.
Part (ii) – uniqueness: We use a renewal theoretic argument to obtain the
uniqueness of the accumulation point. Fix $y\in\mathbb{N}$ and let

$W_{s}^{(M)}=\mathbbm{1}_{\\{X^{(M)}_{s}=y\\}},\quad s\geq 0.$

Since, for each $y\in\mathbb{N}$,
$\mathbb{P}_{\mathbf{u}}(X^{(M)}_{s}=y)=\mathbb{P}(W_{s}^{(M)}=1)$, it
suffices to show that the sequence $(W^{(M)})_{M\geq 1}$ of renewal processes
converges in distribution, since this identifies the limits of
$(R_{t}^{(M)}(y))_{M\in\mathbb{N}},y\in\mathbb{N}$. Furthermore, on the event
$\\{X_{0}^{(M)}\neq y\\}$, each $W^{(M)}$ is entirely characterised by the
sums

$\sum_{j=1}^{n}E_{j}\mathbbm{1}_{j\text{ odd}}+I_{j}\mathbbm{1}_{j\text{
even}},\quad n\in\mathbb{N},$

where $E_{j},j\in\mathbb{N}$ are i.i.d. copies of
$E(M)=I^{[M]\setminus\\{y\\}}$ and $I_{j},j\in\mathbb{N}$ are i.i.d. copies of
$I(M):=I^{\\{y\\}}$, respectively. Since
$\lim_{M\to\infty}\mathbb{P}_{\mathbf{u}}(X_{0}^{(M)}\neq y)=1$, it is thus
easy to see that convergence in distribution of $E(M),I(M)$ as $M\to\infty$
implies the convergence of $(W^{(M)})_{M\in\mathbb{N}}$. By (7), $I(M)$ is the
independent sum of $G\sim\operatorname{Geo}(1-1/M)$ copies of
$\operatorname{Exp}(1/\xi_{y})$, thus its convergence to
$\operatorname{Exp}(1/\xi_{y})$ is immediate. To obtain the limit of $E(M)$,
we use the corresponding characteristic function

$\psi_{M}(\lambda)=\mathbb{E}\left[\textup{e}^{i\lambda
E(M)}\right]=\frac{\frac{1}{M-1}\sum_{x\in[M]\setminus\\{y\\}}\frac{1}{1-i\lambda\xi_{x}}}{M-\sum_{x\in[M]\setminus\\{y\\}}\frac{1}{1-i\lambda\xi_{x}}},\quad\lambda\in\mathbb{R},$

which is obtained in the same manner as (8). The pointwise limit of
$(\psi_{M})_{M\in\mathbb{N}}$ is simply

(24)
$\psi(\lambda)=\frac{1}{1-\sum_{x\in\mathbb{N}\setminus\\{y\\}}\frac{i\lambda\xi_{x}}{1-i\lambda\xi_{x}}},\quad\lambda\in\mathbb{R},$

which is a well-defined point on the complex unit circle since
$|1-i\lambda\xi_{x}|\geq 1$ and thus
$\sum_{x\in\mathbb{N}\setminus\\{y\\}}\left|\frac{i\lambda\xi_{x}}{1-i\lambda\xi_{x}}\right|\leq\lambda\xi(\mathbb{N})<\infty$,
for any $\lambda\in\mathbb{R}.$ Lévy’s continuity theorem now yields that
$\psi$ is the characteristic function of a probability distribution if it is
continuous in $0$, which follows from

$\left|\frac{1}{1-\sum_{x\in\mathbb{N}\setminus\\{y\\}}\frac{i\lambda\xi_{x}}{1-i\lambda\xi_{x}}}-1\right|\leq\frac{\left|\sum_{x\in\mathbb{N}\setminus\\{y\\}}\frac{i\lambda\xi_{x}}{1-i\lambda\xi_{x}}\right|}{\left|1-\sum_{x\in\mathbb{N}\setminus\\{y\\}}\frac{i\lambda\xi_{x}}{1-i\lambda\xi_{x}}\right|}\leq\lambda\xi(\mathbb{N})|\psi(\lambda)|=\lambda\xi(\mathbb{N}).$

We conclude, that $E((M))_{M\in\mathbb{N}}$ and $(I(M))_{M\in\mathbb{N}}$ both
have well-defined distributional limits, whence $(W^{(M)})_{M\in\mathbb{N}}$
converges in distribution to a well-defined limiting renewal process.
Part (iii) – independence of initial distribution: Let $\tau$ denote the first
time that ${X}^{(M)}$ jumps, then for any $y\in[M]$

$\mathbb{P}_{\nu^{M}}({X}^{(M)}_{\tau+s}=y)=\mathbb{P}_{\mathbf{u}}({X}^{(M)}_{s}=y)=R_{s}^{M}(y),\quad
s>0,$

and, using the Markov property,

(25) $\displaystyle\mathbb{P}_{\nu^{M}}({X}^{(M)}_{t}=y,t>\tau)$
$\displaystyle=\mathbb{P}_{\nu^{M}}({X}^{(M)}_{\tau+(t-\tau)}=y,t>\tau)$ (26)
$\displaystyle=\sum_{x\in[M]}\nu^{M}(x)\int_{0}^{t}R_{t-s}^{M}(y)\frac{\textup{e}^{-s/\xi_{x}}}{\xi_{x}}\,\textup{d}s.$

By the assumption that $\nu^{M}([N])$ vanishes for any fixed $N$, we obtain
that

$\mathbb{P}(\tau>t)\leq\nu^{M}([N])+\int_{t}^{\infty}\frac{\textup{e}^{-s/\xi_{N}}}{\xi_{N}},$

which can be made arbitrarily small by choosing $N$ large and then increasing
$M$. Hence $\lim_{M\to\infty}\mathbb{P}_{\nu^{M}}(\tau<t)=1$. On the other
hand, we have that

$\displaystyle\sum_{x\in[M]\setminus[N]}\nu^{M}(x)\int_{0}^{t}$ $\displaystyle
R_{t-s}^{M}(y)\frac{\textup{e}^{-s/\xi_{x}}}{\xi_{x}}\,\textup{d}s\leq\sum_{x\in[M]}\nu^{M}(x)\int_{0}^{t}R_{t-s}^{M}(y)\frac{\textup{e}^{-s/\xi_{x}}}{\xi_{x}}\,\textup{d}s$
$\displaystyle\leq\sum_{x\in[M]\setminus[N]}\nu^{M}(x)\int_{0}^{t}R_{t-s}^{M}(y)\frac{\textup{e}^{-s/\xi_{x}}}{\xi_{x}}\,\textup{d}s+\nu^{M}([N]),$

and the weak convergence of $\operatorname{Exp}(\lambda)$ to $\delta_{0}$ as
$\lambda\to\infty$ implies that both the first and the last term can be
brought arbitrarily close to $R_{t}^{M}(y)$ by choosing first $N$ and then $M$
large. It thus follows from (25) that

$\lim_{M\to\infty}\mathbb{P}_{\nu^{M}}(X_{t}=y)=\lim_{M\to\infty}R_{t}^{M}(y)=Q_{t}(y),\quad
y\in\mathbb{N},t>0.$

∎

The laws $(Q_{t})_{t>0}$ describe the limiting distribution of the processes
$X^{(N)},N\in\mathbb{N}$ upon (re-)entering the slow sites. To define the
dynamics of the limiting process, let $\tau_{x},x\in\mathbb{N}$, be
$\operatorname{Exp}(1/\xi_{x})$ distributed and set, for
$f\in\mathcal{B}(\mathbb{N})=\\{g:\mathbb{N}\to\mathbb{R}:g\text{ bounded
}\\}$,

(27) $\displaystyle
P_{t}f(x)=f(x)\mathbb{P}(\tau_{x}>t)+\mathbb{E}(Q_{t-\tau_{x}}(f)\mathbbm{1}_{\\{\tau_{x}<t\\}}),\quad
x\in\mathbb{N},t\geq 0.$

######  Theorem 3.8.

The following assertions hold $\mathbf{P}$-almost surely:

  1. (a)

We have $\lim_{N\to\infty}P_{t}^{(N)}=P_{t}$ for every $t>0.$

  2. (b)

$P=(P_{t})_{t\geq 0}$ is a Markov semigroup inducing a process
$X=(X_{t})_{t>0}$ on $\mathbb{N}$ with waiting times governed by $\xi$.

  3. (c)

$(Q_{t})_{t\geq 0}$ is an entrance law of $X$, i.e. satisfies
$Q_{t}P_{s}=Q_{t+s}$, for all $s,t>0.$

  4. (d)

The trace process $\textup{tr}_{[N]}(X)$ coincides with $X^{(N)}$ in
distribution.

###### Proof.

We first prove (a). Fix any bounded function $f$ on $\mathbb{N}$ and any
$t>0$. We have

$P_{t}^{(N)}f(x)=f(x)\mathbb{P}(\tau_{x}>t)+\mathbb{E}(R^{N}_{t-\tau_{x}}(f)\mathbbm{1}_{\\{\tau_{x}<t\\}}),\quad
x\in[N],$

and it thus follows from Proposition 3.6 and dominated convergence that

(28) $\lim_{N\to\infty}P_{t}^{(N)}f(x)=P_{t}f(x),$

for any $x\in\mathbb{N}.$ To show that the identity in (c) holds, we observe
that

$R^{N}_{t}P^{(N)}_{s}=R^{N}_{t+s},\quad t,s\geq 0.$

(28) together with Proposition 3.6 allows us to take weak limits and establish
the identity

(29) $Q_{t}P_{s}=Q_{t+s},\quad t,s\geq 0.$

Next, we show the semigroup property needed for (b). (c) implies, for any
$x\in\mathbb{N}$,

$\displaystyle P_{t+s}f(x)=\,$
$\displaystyle\mathbb{P}(\tau_{x}>t+s)+\mathbb{E}(Q_{t-\tau_{x}}(P_{s}f)\mathbbm{1}_{\\{\tau_{x}<t\\}})+\mathbb{E}(Q_{t+s-\tau_{x}}(f)\mathbbm{1}_{\\{\tau_{x}\in[t,t+s]\\}})$
$\displaystyle=\,$
$\displaystyle\mathbb{P}(\tau_{x}>t)\mathbb{P}(\tau_{x}>s)+\mathbb{E}(Q_{t-\tau_{x}}(P_{s}f)\mathbbm{1}_{\\{\tau_{x}<t\\}})$
$\displaystyle+\textup{e}^{-t/\xi_{x}}\mathbb{E}(Q_{s-\tau_{x}}(f)\mathbbm{1}_{\\{\tau_{x}\in[0,s]\\}})$
$\displaystyle=\,$
$\displaystyle\mathbb{P}(\tau_{x}>t)\big{(}\mathbb{P}(\tau_{x}>s)+\mathbb{E}(Q_{s-\tau_{x}}(f)\mathbbm{1}_{\\{\tau_{x}\in[0,s]\\}})\big{)}$
$\displaystyle+\mathbb{E}(Q_{t-\tau_{x}}(P_{s}f)\mathbbm{1}_{\\{\tau_{x}<t\\}})$
$\displaystyle=\,$ $\displaystyle P_{t}P_{s}f(x),$

which establishes the semigroup property of $P$. Finally, the distributional
identity in (d) is inherited from the $X^{(N)}$, because
$\textup{tr}_{[N]}(\cdot)$ is continuous w.r.t. convergence in distribution. ∎

We interpret the environments $\xi$ as elements of the space
$\mathcal{S}^{\alpha}_{+}\subset\ell^{1}$ given by

$\mathcal{S}^{\alpha}=\\{\xi:\sup_{x\in\mathbb{N}}|\xi_{x}|x^{1/\alpha}<\infty\\},\;\mathcal{S}^{\alpha}_{+}=\mathcal{S}^{\alpha}\cap\\{\xi\in\ell^{1}:\xi\geq
0\\}.$

Furthermore, let $\mathcal{M}(\mathbb{N})$ denote the space of probability
measures on $\mathbb{N}$ equipped with $\ell^{1}$-metric (which in our
discrete setting is equivalent to the total variation metric) and let
$(\mathcal{B}(\mathbb{N}),\|\cdot\|)$ be the space of bounded linear operators
on $\mathbb{N}$ with $\|\cdot\|$ denoting the canonical operator norm
$\|L\|=\sup_{f:|f|_{\ell^{\infty}}\leq 1}|Lf|_{\ell^{\infty}}$. The next lemma
addresses continuity of $P$ and $Q$ in the environment variable $\xi$.

######  Lemma 3.9.

Let $\alpha$ be fixed. For every $t>0$, the maps
$Q_{t}:(\mathcal{S}^{\alpha}_{+},\ell^{1})\to(\mathcal{M}(\mathbb{N}),\ell^{1})$
and
$P_{t}:(\mathcal{S}^{\alpha}_{+},\ell^{1})\to(\mathcal{B}(\mathbb{N}),\|\cdot\|)$
are continuous.

###### Proof.

We begin by demonstrating the continuity of $Q_{t}$, for
$\xi\in\mathcal{S}^{\alpha}_{+}$ set $q(\xi)=Q_{t}(\xi)$ for brevity. First
note that the convergence of the characteristic functions established in Part
(ii) of the proof of 3.6 implies that if $\xi^{(n)}\to\xi$ in $\ell^{1}$ with
$\xi,\xi^{(n)}\in\mathcal{S}^{\alpha}_{+}$, then $q(\xi^{(n)})\to q(\xi)$
coordinate-wise, since the expression $\psi(\lambda)$ for the limiting
characteristic function is continuous in $\xi$ w.r.t. the $\ell^{1}$-distance.
It remains to extend the continuity to the stronger topology of $\ell^{1}$
convergence on $\mathcal{M}(\mathbb{N}).$

To this end, let $r_{M}(\xi)$ denote the approximating measures $R^{M,\xi}$
used in the proof of Proposition 3.6. Fix $N$ and observe that the constants
$C(\xi)$ in (23) can be chosen as a continuous function of the coordinate
$\xi_{N+1}$ and the sums $\xi(\mathbb{N}\setminus[N]),\xi([N])$. Note further
that $N$ in (23) is chosen independently of $\xi$; consequently the bound
there holds uniformly (with $C(\xi)$ replaced by some $C^{\prime}$) in some
small neighbourhood of $\xi$ w.r.t. the $\ell_{1}$-metric, i.e. for any
$\epsilon>0,N\in\mathbb{N}$, we can find some $\delta>0$ with

$r_{M}(\xi^{\prime};\mathbb{N}\setminus[N])\leq\epsilon+C^{\prime}(\delta,\xi)tN^{1-1/\alpha},\quad
M>\frac{N}{\epsilon},\xi^{\prime}\in\\{\eta:|\xi-\eta|_{\ell_{1}}<\delta\\}\equiv
U(\delta,\xi).$

In the limit $M\to\infty$, the dependence of $M$ on $\epsilon$ vanishes (mind
that in the calculation leading to (23), $\epsilon$ was needed to deal with
the initial condition in the finite system), which yields the uniform bound

(30) $q(\xi^{\prime};\mathbb{N}\setminus[N])\leq
C^{\prime}(\delta,\xi)tN^{1-1/\alpha},\xi^{\prime}\in U(\delta,\xi).$

We conclude that

(31)
$|q(\xi)-q(\xi^{\prime})|_{\ell^{1}}\leq\sum_{x=1}^{N}|q(\xi,\\{x\\})-q(\xi^{\prime},\\{x\\})|+C^{\prime}(\xi,\delta)tN^{1-1/\alpha},\quad\xi^{\prime}\in
U(\delta,\xi),$

from which continuity of $q(\cdot)$ follows.

To address continuity for $P_{t}$ in $\xi$, we write
$P_{t}=P_{t}(\xi),P^{\prime}_{t}=P_{t}(\xi^{\prime})$ and bound

(32) $\displaystyle|P_{t}f-P^{\prime}_{t}f|_{\infty}$
$\displaystyle\leq\sup_{x}|f(x)\mathbb{P}(\tau_{x}>t)-f(x)\mathbb{P}(\tau^{\prime}_{x}>t)|$
$\displaystyle\phantom{=}+\sup_{x}\left|\int_{0}^{t}Q^{\prime}_{t-s}f\frac{1}{\xi^{\prime}_{x}}\textup{e}^{-s/\xi^{\prime}_{x}}-Q_{t-s}f\frac{1}{\xi_{x}}\textup{e}^{-s/\xi_{x}}\,\textup{d}s\right|.$

Since $|f(x)|\leq 1$, the first term is at most

$\sup_{x}|\textup{e}^{-t/\xi_{x}}-\textup{e}^{-t/\xi^{\prime}_{x}}|$

which vanishes as $\xi^{\prime}\to\xi$ in $\ell^{1}$. Similarly, the second
term is bounded by

(33)
$\displaystyle\sup_{x}\int_{0}^{t}|(Q^{\prime}_{t-s}-Q_{t-s})|\frac{1}{\xi^{\prime}_{x}}\textup{e}^{-s/\xi^{\prime}_{x}}\,\textup{d}s+\sup_{x}|F_{\tau^{\prime}(x)}(t)-F_{\tau(x)}(t)|,$

where $F_{\tau_{x}}$ denotes the distribution function of $\tau_{x}$. It is
straightforward to see, using that $C(\xi^{\prime},\delta)$ does not depend on
$t$ in (31), that both terms vanish for any $t>0$ as $\xi^{\prime}\to\xi$ in
$\ell^{1}$. ∎

The process $X$ enters $\mathbb{N}$ according to the entrance law
$(Q_{t})_{t\geq 0}$ at some site $x$ and the random walk dynamics correspond
to waiting an $\operatorname{Exp}(\xi_{x}^{-1})$-distributed time at site $x$,
then jumping to infinity and instantaneously re-entering $\mathbb{N}$ again
via $(Q_{t})_{t>0}$. We next determine the invariant distribution of $X$.

######  Lemma 3.10.

The invariant distribution of $X$ is given by
$\mu(x)=\frac{1}{\xi(\mathbb{N})}\xi_{x},x\in\mathbb{N}$.

###### Proof.

$X^{(N)}$ is clearly ergodic, hence has a unique invariant distribution
$\mu_{N}$. It is straightforward to deduce from the occupation time formula

$\lim_{t\to\infty}\frac{\int_{0}^{t}\mathbbm{1}_{\\{X^{(N)}_{s}=x\\}}\,\textup{d}s}{t}=\mu_{N}(x),\;\mathbb{P}\text{-almost
surely for }x\in\mathbb{N},$

that $\mu_{N}(x)=\xi_{x}/\xi([N])$, $x\in[N]$, is the invariant distribution
of $X^{N}$. According to Theorem 3.8 (c) the semigroups $P^{(N)}$ converge to
$P$ and we clearly have $\lim_{N\to\infty}\mu_{N}=\mu$, hence [5, Theorem
9.10] or a direct calculation yield that $\mu$ is invariant for $X$. ∎

######  Lemma 3.11.

Let $f,g:\mathbb{N}\to\mathbb{R}$ have finite support. Then

$\lim_{N\to\infty}\langle P_{t}^{(N)}f,g\rangle_{\mu_{N}}=\langle
P_{t}f,g\rangle_{\mu},$

and

$\langle P_{t}f,g\rangle_{\mu}=\langle f,P_{t}g\rangle_{\mu}.$

###### Proof.

The convergence result is a straightforward consequence of the convergence of
$\mu_{N}$ to $\mu$ and of $P^{(N)}$ to $P$. Self-adjointess of $P$ w.r.t.
$\mu$ is inherited from $P^{(N)}$, since $X^{(N)}$ is reversible. ∎

######  Lemma 3.12.

Let $p_{t}^{(N)}(x,y)=\mathbb{P}_{x}(X_{t}^{(N)}=y)$. Then, for $x\neq y$,

$\lim_{N\to\infty}\frac{\,\textup{d}}{\,\textup{d}t}p_{t}^{(N)}(x,y)|_{t=0}=0,$

whereas

$\lim_{N\to\infty}p_{t}^{(N)}(x,y)>0,\quad\text{for all }t>0.$

###### Proof.

The first statement is simply a reformulation of the fact that the transition
rates $r^{(N)}(x,y)$ vanish as $N\to\infty$. To prove the second statement we
rewrite

$p_{t}(x,y)=\delta_{x}P_{t}\mathbbm{1}_{\\{y\\}}=\mathbb{E}(Q_{t-\tau_{x}}(y)\mathbbm{1}_{\\{\tau_{x}<t\\}}).$

The claim follows once we show $\operatorname{supp}Q_{s}=\mathbb{N}$ for any
$s>0.$. Note that for every $x\in\mathbb{N},t>0$ there exists some $K>x$ such
that $Q_{t}([K])>0$, hence there is a positive probability that $X$ has
visited $[K]$ before time $t$. Upon entering $[K]$ at time $T_{0}<t$, the
position $X_{T_{0}}\in[K]$ is uniformly distributed by Theorem 3.8 (d), and
thus there is a positive probability that $X_{s}=x$ for all $s\in[T_{0},t]$,
hence $Q_{t}(x)>0.$ ∎

######  Lemma 3.13.

For any $x,y\in\mathbb{N}$ we have

(34)
$\displaystyle\left\|\,\mathbb{P}_{x}(X_{t}\in\cdot)-\mathbb{P}_{y}(X_{t}\in\cdot)\,\right\|_{\textup{TV}}\leq
e^{-\xi_{1}^{-1}t},$

and in particular

(35)
$\displaystyle\left\|\,\mathbb{P}_{x}(X_{t}\in\cdot)-\mu\,\right\|_{\textup{TV}}\leq
e^{-\xi_{1}^{-1}t}.$

###### Proof.

We construct a Markovian coupling of two versions of the limit random walk
$X_{t}$ and $Y_{t}$. If $X_{t}<Y_{t}$, then the exit rate $\xi_{Y_{t}}^{-1}$
from the current state of $Y_{t}$ is larger than the corrsponding rate for
$X_{t}$. Hence, we can couple the two exponentials governing the exit so that
when $X_{t}$ jumps, $Y_{t}$ jumps as well. In the case of a joint exit, we can
couple the random walks to move identically, as both movements are prescribed
by $(Q_{t})_{t>0}$. The additional jumps of $Y_{t}$ may cause $Y_{t}$ to jump
to the left of $X_{t}$, in which case the roles of $X_{t}$ and $Y_{t}$ reverse
so that it is always the random walk with the lower index, which determines
the coupling speed. Since $\xi_{1}$ is the largest weight, the exit time for a
random walk started in the slowest site dominates this coupling argument,
which proves that the coupling time in this coupling is at most an exponential
with parameter $\xi_{1}^{-1}$.∎

Note that the coupling used in the proof of Lemma 3.13 is not optimal, as it
ignores the possibility that $Y_{t}$ jumps on top of $X_{t}$ for an earlier
coupling. We proceed by stating two direct implications of the lemma.

######  Corollary 3.14.

We have $\mathbf{P}$-almost surely that

  1. (a)

$\left\|\,Q_{t}-\mu\,\right\|_{\textup{TV}}\leq e^{-\xi_{1}^{-1}t};$

  2. (b)

$(P_{t})$ has a spectral gap of at least $\xi_{1}^{-1}$.

To construct a system of infinitely many coalescing versions of $X$ in Section
4, it is technically convenient to extend the trajectory space.

######  Lemma 3.15.

Let $\bar{\mathbb{N}}$ denote the one-point compactification of $\mathbb{N}$
with respect to the discrete topology. Then $X$ can be extended to a Feller
process on $\bar{\mathbb{N}}$.

###### Proof.

We do not distinguish notationally between $X$ and its extension. We use the
probabilistic characterisation of the Feller property given in [10, Lemma
17.3]. This amounts to establishing the two conditions

(36) $\mathbb{P}_{x}(X_{t}\in\cdot)\to\mathbb{P}_{y}(X_{t}\in\cdot)\text{ for
any }t\geq 0\text{ as }x\to y\text{ in }\bar{\mathbb{N}},$

and

(37) $X_{t}\overset{\mathbb{P}}{\to}X_{0}\text{ as }t\to 0.$

If $y\in\mathbb{N}$, then (36) holds trivially. If $y=\infty$, then as $x\to
y$ the time of the first jump converges to $0$ in probability and thus
$P_{t}\delta_{x}$ converges to $Q_{t}$ as required for (36). Similarly, item
(37) is trivial if the process is initiated in $\mathbb{N}$. If the process is
initiated via the entrance law, i.e. it starts at $\infty$, then $X_{t}$ must
leave any finite subset of $\mathbb{N}$ as $t\to 0$. But this is equivalent to
$X_{t}\to\infty$ in $\bar{\mathbb{N}}$, which yields (37). ∎

### Multiple walks.

We now consider systems of finitely many independent limit random walks. Let
$\mathbb{P}_{\nu_{1}\otimes\nu_{2}}$ denote the distribution of two
independent copies $(X_{t})_{t\geq 0}$ and $(Y_{t})_{t\geq 0}$ of the limiting
walk $X$, started in distributions $\nu_{1}$ and $\nu_{2}$, respectively. If
the starting positions are fixed $\nu_{1}=\delta_{x},\nu_{2}=\delta_{y}$, then
we write $\mathbb{P}_{(x,y)}$ instead. We say that $X$ and $Y$ _meet_ at time
$t$, if $X_{t}=Y_{t}$.

######  Corollary 3.16.

The probability that two independent random walks with fixed starting points
meet at time $t$ satisfies

(38)
$\displaystyle\mathbb{P}_{(x,y)}(X_{t}=Y_{t})\geq\sup_{\alpha\in(0,1)}(1-\alpha)^{2}\sum_{z\leq
h_{\alpha}(t)}\mu(z)^{2},$

where
$h_{\alpha}(t)=\sup\\{x\in\mathbb{N}:\mu(x)\geq\alpha^{-1}e^{\xi_{1}^{-1}t}\\}$.

###### Proof.

Fix $\alpha\in(0,1)$, then we have

(39) $\displaystyle\mathbb{P}_{(x,y)}(X_{t}=Y_{t})$
$\displaystyle=\sum_{z}p_{t}(x,z)p_{t}(y,z)$ (40) $\displaystyle=\sum_{z\leq
h_{\alpha}(t)}\big{(}\mu(z)+(p_{t}(x,z)-\mu(z))\big{)}\big{(}\mu(z)+(p_{t}(y,z)-\mu(z))\big{)}$
(41) $\displaystyle\geq\sum_{z\leq
h_{\alpha}(t)}(\mu(z)-\left\|\,p_{t}(x,\cdot)-\mu\,\right\|_{\textup{TV}})(\mu(z)-\left\|\,p_{t}(y,\cdot)-\mu\,\right\|_{\textup{TV}})$
(42) $\displaystyle\geq\sum_{z\leq
h_{\alpha}(t)}(\mu(z)-e^{-\xi_{1}^{-1}t})(\mu(z)-e^{-\xi_{1}^{-1}t}),$

where we used Corollary 3.14 in the last inequality and the fact that
$\mu(z)>\left\|\,p_{t}(x,\cdot)-\mu\,\right\|_{\textup{TV}}>\sup_{w\in\mathbb{N}}|p_{t}(x,w)-\mu(w)|$,
for $z\in h_{\alpha}(t)$ in the second inequality. Note, that by definition of
$h_{\alpha}(t)$, the last expression is at least $(1-\alpha)^{2}\sum_{z\leq
h_{\alpha}(t)}\mu(z)^{2}$. The claim now follows by taking the supremum in
$\alpha$. ∎

It follows that for $t$ of order $\xi_{1}^{-1}$ there is a realistic chance
that two independent random walks have met. This time-scale is also the best
possible if we make no assumptions on the starting points, since on time
scales much smaller than $\xi_{1}^{-1}$, two random walks starting in $1$ and
$2$ have not jumped with high probability, hence they cannot have met. We next
show that if we start both walks in the stationary distribution, they
typically do not meet in high index sites.

######  Lemma 3.17.

There is some constant $C<\infty$ (independent of $\xi$) such that for any
$A\subset\mathbb{N}$ and $T>0$, $\mathbf{P}$-almost surely

$\mathbb{P}_{\mu\otimes\mu}(X\text{ and }Y\text{ meet inside }A\text{ before
time }T)\leq C\Big{(}T\,\frac{\mu(A)}{\xi(\mathbb{N})}+\sum_{z\in
A}\mu(z)^{2}\Big{)}.$

###### Proof.

Let

$\tau_{z}=\inf\\{t:X_{t}=Y_{t}=z\\},z\in\mathbb{N}.$

Since $X$ and $Y$ individually leave $z$ at rate $\xi_{z}^{-1}$, the joint
leaving rate is $2\xi_{z}^{-1}$. Let $\sigma_{z}$ be the waiting time for the
next jump of either walker after $\tau_{z}$. Note that $\sigma_{z}$ is
independent of $\tau_{z}$ by the Markov property. Then

(43) $\displaystyle\mathbb{P}_{\mu\otimes\mu}\left(\tau_{z}\leq
T\right)=\mathbb{P}_{\mu\otimes\mu}\left(\tau_{z}\leq
T,\sigma_{z}\geq\epsilon\xi_{z}\right)+\mathbb{P}_{\mu\otimes\mu}\left(\tau_{z}\leq
T,\sigma_{z}<\epsilon\xi_{z}\right)$ (44)
$\displaystyle\leq\mathbb{P}_{\mu\otimes\mu}\left(\int_{0}^{T+\epsilon\xi_{z}}\mathbbm{1}_{\\{X_{t}=z\\}}\mathbbm{1}_{\\{Y_{t}=z\\}}\,\textup{d}t\geq\epsilon\xi_{z}\right)+\mathbb{P}_{\mu\otimes\mu}\left(\tau_{z}\leq
T\right)\mathbb{P}\left(\sigma_{z}<\epsilon\xi_{z}\right).$

Note that the increase in the time interval from $[0,T]$ to
$[0,T+\epsilon\xi_{z}]$ accommodates for the event that
$\tau_{z}\in(T-\epsilon\xi_{z},T]$. Since
$\mathbb{P}\left(\sigma_{z}<\epsilon\xi_{z}\right)=1-e^{-2\epsilon}$, we
obtain by regrouping terms that

$\mathbb{P}_{\mu\otimes\mu}\left(\tau_{z}\leq T\right)\leq
e^{2\epsilon}\mathbb{P}_{\mu\otimes\mu}\left(\int_{0}^{T+\epsilon\xi_{z}}\mathbbm{1}_{\\{X_{t}=z\\}}\mathbbm{1}_{\\{Y_{t}=z\\}}\,\textup{d}t\geq\epsilon\xi_{z}\right).$

By Markov’s inequality,

(45)
$\displaystyle\mathbb{P}_{\mu\otimes\mu}\left(\int_{0}^{T+\epsilon\xi_{z}}\mathbbm{1}_{\\{X_{t}=z\\}}\mathbbm{1}_{\\{Y_{t}=z\\}}\,\textup{d}t\geq\epsilon\xi_{z}\right)$
$\displaystyle\leq\frac{\mathbb{E}_{\mu\otimes\mu}\int_{0}^{T+\epsilon\xi_{z}}\mathbbm{1}_{\\{X_{t}=z\\}}\mathbbm{1}_{\\{Y_{t}=z\\}}\,\textup{d}t}{\epsilon\xi_{z}}$
(46)
$\displaystyle=\frac{(T+\epsilon\xi_{z})\mu(z)^{2}}{\epsilon\xi_{z}}=\Big{(}\xi_{z}+\frac{T}{\epsilon}\Big{)}\frac{\xi_{z}}{\xi(\mathbb{N})^{2}}.$

Therefore,

(47) $\displaystyle\mathbb{P}_{\mu\otimes\mu}\left(\tau_{z}\leq T\right)\leq
e^{2\epsilon}\Big{(}\xi_{z}+\frac{T}{\epsilon}\Big{)}\frac{\xi_{z}}{\xi(\mathbb{N})^{2}}$

Choosing $\epsilon$ of order one and applying a union bound over all $z\in A$
gives the desired result. ∎

For arbitrary starting positions, we obtain the following corollary by using
the simple fact that
$\mathbb{P}_{(x,y)}(E)\leq\frac{1}{\mu(x)\mu(y)}{\mathbb{P}_{\mu\otimes\mu}}(E)$
for any event $E$.

######  Corollary 3.18.

Let $x<y$ and $A\subset\mathbb{N}$, then

$\mathbb{P}_{(x,y)}(X\text{ and }Y\text{ meet in }A\text{ before time
}T)\leq\frac{C}{\xi_{x}\xi_{y}}\big{(}T\,\xi(A)+{\sum_{z\in
A}\xi_{z}^{2}}\big{)}.$

##  4\. The limit coalescing walk

We now establish the convergence of the finite systems of coalescing walks on
$[N]$ to finite systems of limiting walks as constructed in the previous
section.

######  Proposition 4.1.

Let $w^{(N)}_{1}\geq w^{(N)}_{2}\geq\dots w^{(N)}_{N}$ denote the order
statistics of $N$ i.i.d. Pareto($\alpha$) random variables and let $Y^{(N)}$
denote the random walk in the environment $(w^{(N)}_{i})_{i\in[N]}$ started in
position $y_{0}\leq M$ with uniform jump distribution. Then

$Y^{(N)}_{N^{\frac{1}{\alpha}}(\cdot)}\overset{d}{\longrightarrow}X_{(\cdot)},\quad\mbox{as
}N\to\infty.$

###### Proof.

It suffices to show that
$\big{(}Y^{(N)}_{N^{\frac{1}{\alpha}}t}\big{)}_{t\in[0,T]}$ can be perfectly
coupled to $X^{(N)}$ (which coincides with the trace of $X$ on $[N]$ by
Theorem 3.8(d)) up to some error probability that vanishes as $N\to\infty$.
Note that given the initial positions $Y_{0}^{(N)},\;X_{0}$ the processes can
be reconstructed from the environment of exponential clocks. Hence, it
suffices to show that the Poisson field of clock processes corresponding to
$w^{(N)}$ at time scale $N^{\frac{1}{\alpha}}$ converges weakly to the Poisson
field of clock processes parametrised by $\xi_{1},\dots,\xi_{N}$. This follows
from expressing (1) through pointwise convergence of characteristic functions
and the fact that a sequence of Poisson measure converges weakly to some
limiting Poisson measure if and only if the corresponding characteristic
functionals converge, see e.g.[4, Proposition 11.1.VIII]. ∎

Let us now formally define the coalescence operator
$\operatorname{coal}(\cdot)$ for an arbitrary collection
$(X^{1},X^{2},\cdots)$ of copies of $X$ with distinct initial conditions. We
assume $X^{1}_{0}=1,X^{2}_{0}=2,\dots$. This induces a natural order on the
paths: $X^{i}<X^{j}$ if and only if $X^{i}_{0}<X^{j}_{0}$. Note that this
convention imposes no additional restriction since an arbitrary collection of
paths can always be ordered according to starting positions and viewed as a
projection of the full configuration. For any $i$ denote by

$\sigma_{i}=\inf\\{t:X_{t}^{i}=X_{t}^{j}\text{ for any }j<i\\}$

the _coalescence time_ of the $i$-th walker. Note that each $\sigma_{i}$ is
measurable with respect to the $\sigma$-field $\sigma(X^{j},j\leq i)$ which is
generated by only finitely many walkers and that $\sigma_{i}>0$ almost surely
for each $i$. In particular the coalescence time of any walker is well-defined
and there are only countably many coalescence times. Define now

(48)
$\bar{N}\operatorname{coal}\big{(}(X^{1},X^{2},\dots)\big{)}_{t}=\Big{(}\sum_{j:\sigma_{j}>t}\delta_{X^{j}_{t}}(x)\Big{)}\in\\{0,1\\}^{\mathbb{N}},\quad
t\geq 0.$

The existence of this infinite system of coalescing walks is deduced from
standard results in the next proposition.

######  Proposition 4.2.

For each finite $M\subset\mathbb{N}$, let $\mathbb{Q}_{M}$ denote the
distribution of $\operatorname{coal}\big{(}(X^{j},j\in M)\big{)}$. Then

$\mathbb{Q}=\mathcal{L}\big{(}\operatorname{coal}\big{(}(X^{1},X^{2},\dots)\big{)}\big{)}$

is the projective limit of $\\{\mathbb{Q}_{M},M\subset\mathbb{N}\text{
finite}\\}.$

###### Proof.

It is clear, that if the projective limit exists, it must coincide with
$\mathbb{Q}$ and this would also establish the well-definedness of (48). To
this end, we invoke a version of the Kolmogorov extension theorem, namely [10,
Theorem 8.23]. Recall that by Lemma 3.15 we may view the particle trajectories
as paths in $\bar{\mathbb{N}}$, which is Borel isomorphic to
$\\{1/n,n\in\mathbb{N}\\}\cup\\{\infty\\}$ and thus a Borel space. Hence, we
only need to show the consistency of the family
$\\{\mathbb{Q}_{M},M\subset\mathbb{N}\text{ finite}\\}$. The latter is an
immediate consequence of the definition of the coalescence times
$\sigma_{j},j\in M$, if for each particle, one keeps track of the indices of
the particles that have already merged with it111 Formally, this requires to
extend the probability spaces involved. However, we only use the additional
structure in this proof and we found that the corresponding notation only
obscures the argument: Arguing inductively, it suffices to show that
$\mathbb{Q}_{M^{\prime}}$ coincides with the projection of $\mathbb{Q}_{M}$
onto the particles with indices in $M^{\prime}$ for $|M^{\prime}|=|M|-1$.
Denote the index of the additional (_spare_) particle in $M$ by $s$. If no
other particle coalesces with $s$ before $\sigma_{s}$, then there is nothing
to show. Assume this is not the case, then there exist coalescence times
$\sigma_{j_{1}}<\sigma_{j_{2}}<\dots<\sigma_{j_{k}}<\sigma_{s}$ at which
particles with indices in $M^{\prime}$ have merged with the spare particle.
Since these _virtual_ coalescences do not occur in the smaller system, we need
to define the trajectories of the non-spare particles involved after the
virtual coalescence events. By coupling the $M^{\prime}$ to the $M$ system, we
may declare the trajectory of the particle $j_{1}$ involved in the first
virtual coalescence to be that of the spare particle until the next
coalescence event involving $j_{1}$. By construction, the later will occur at
time $\sigma_{j_{2}}$. Depending on whether $j_{1}<j_{2}$ or $j_{1}>j_{2}$,
one of the particles survives and we set it once again to follow the
trajectory of the spare particle. We continue this procedure and thus
inductively define a path for the $M^{\prime}$-coalescing random walk. It
follows from the Markov property of the underlying walks that this path has
indeed the distribution $\mathbb{Q}_{M^{\prime}}$. ∎

Note that the law $\mathbb{Q}$ just constructed formally corresponds to a
coalescing system with trajectories in
$\\{0,1\\}^{\mathbb{N}}\times\mathbb{N}$, since by convention we do not
coalesce particles that meet at $\infty$. That this is justified follows from
Corollary 3.18: the probability that particles meet ‘near’ the boundary point
$\infty$ vanishes in the finite system, which is also crucially used in the
proof of Proposition 4.3 below. However, we may also consider the process
$\gamma$ derived from propagating an initial condition $\gamma_{0}$ according
to (projections of) $\mathbb{Q}$ as a process on $\\{0,1\\}^{\mathbb{N}}$,
just as in Sections 1 and 2.

######  Proposition 4.3.

Let $\gamma_{0}^{(N)}\overset{d}{\to}\gamma_{0}\in\\{0,1\\}^{\mathbb{N}}$ as
$N\to\infty$, where $\operatorname{supp}\gamma_{0}^{(N)}\subset[K]$ for some
$K$, uniformly in $N\in\mathbb{N}$. Then the corresponding coalescing random
walk systems

$\big{(}\gamma_{N^{\frac{1}{\alpha}}t}^{(N)}\big{)}_{t\geq 0}$

admit a distributional limit $\gamma=(\gamma_{t})_{t\geq 0}$ as $N\to\infty$,
in which individual particle dynamics are given by i.i.d. copies of $X$ and
particles coalesce whenever they share the same location.

###### Proof.

We define the following approximations to the coalescence operator

  * •

$\operatorname{coal}_{M}(\cdot)$ denotes the coalescence operator allowing
coalescences only at sites in $[M]$;

  * •

$\operatorname{coal}_{M,\epsilon}(\cdot)$ denotes the coalescence operator
allowing coalescences only at sites in $[M]$ and at times in
$\epsilon\mathbb{N}$.

By Proposition 4.1, we have that for any
$M\in\mathbb{N},\epsilon>0,\vec{t}=(t_{1},\dots,t_{d})\in[0,T]^{d}$ and
$\vec{x}\in\bigtimes_{j=1}^{d}\\{0,1\\}^{\mathbb{N}}$

(49) $\displaystyle\mathbb{P}\left(\operatorname{coal}_{M,\epsilon
N^{\frac{1}{\alpha}}}(Y^{(1,N)},\dots,Y^{(k,N)})_{N^{\frac{1}{\alpha}}\vec{t}}=\vec{x}\right)$
$\displaystyle\overset{N\to\infty}{\longrightarrow}\mathbb{P}\Big{(}$
$\displaystyle\operatorname{coal}_{M,\epsilon}(X^{(1)},\dots,X^{(k)})_{\vec{t}}=\vec{x}\Big{)}.$

Now,

(50) $\displaystyle\mathbb{P}\Big{(}\operatorname{coal}_{M,\epsilon
N^{\frac{1}{\alpha}}}$
$\displaystyle(Y^{(1,N)},\dots,Y^{(k,N)})_{N^{\frac{1}{\alpha}}\vec{t}}\neq\operatorname{coal}_{M}(Y^{(1,N)},\dots,Y^{(k,N)})_{N^{\frac{1}{\alpha}}\vec{t}}\Big{)}$
$\displaystyle\leq\mathbb{P}($ a jump occurs within $\displaystyle\epsilon
N^{\frac{1}{\alpha}}\text{ time units after a coalescence event at a site
}x\in[M]).$

By Markov’s inequality, the total number of jumps during
$[0,TN^{\frac{1}{\alpha}}]$ performed from sites in $[M]$ is at most of order
$O(TM^{\frac{1}{\alpha}+1})$ with high probability. Since every entry into
$[M]$ from $[N]\setminus[M]$ has to be preceded by an exit (recall that the
initial positions oft walkers are within $[M]$), the total number of entries
into sites in $[M]$ is of the same order. The latter number is an upper bound
on the number of coalescence events occurring in $[M]$. Furthermore, the
probability that a jump occurs at $M$ in the time interval $[0,\epsilon
N^{\frac{1}{\alpha}}]$ is of order $\epsilon M^{\frac{1}{\alpha}}$ and the
probability that this happens at any site in $[M]$ is of order $\epsilon
M^{1+\frac{1}{\alpha}}$. It follows that if
$\epsilon=o(M^{-2-\frac{2}{\alpha}})$, then the probability on the right hand
side of (50) vanishes as $N\to\infty$.

Furthermore,

$\displaystyle\mathbb{P}\big{(}\operatorname{coal}_{M}(Y^{(1,N)},\dots,Y^{(k,N)})_{N^{\frac{1}{\alpha}}\vec{t}}\neq\operatorname{coal}(Y^{(1,N)},\dots,Y^{(k,N)})_{N^{\frac{1}{\alpha}}\vec{t}}\big{)}$
$\displaystyle\leq{k\choose
2}\sup_{x,y\in[K]}\mathbb{P}_{(x,y)}\Big{(}Y^{1,N}\text{ and }Y^{2,N}\text{
meet before time }TN^{\frac{1}{\alpha}}\text{ in }[N]\setminus[M]\Big{)}.$

For any given bound $K\in\mathbb{N}$ on the initial positions of the walk, we
may choose $M$ so large, that the right hand side is $o(1)$, since

(51) $\displaystyle\mathbb{P}_{(x,y)}\left(Y^{1,N}\text{ and }Y^{2,N}\text{
meet before time }TN^{\frac{1}{\alpha}}\text{ in }[N]\setminus[M]\right)$ (52)
$\displaystyle=O\left(\frac{TN^{\frac{1}{\alpha}}\sum_{z=M+1}^{N}w^{(N)}_{z}}{w^{(N)}_{x}w^{(N)}_{y}}\right)$

by the same argumentation that led to Corollary 3.18. We conclude that we may
replace the approximate coalescence operator on the left hand side of (49) by
the true coalescence operator $\operatorname{coal}(\cdot)$ up to an error
probability that vanishes as $M,N\to\infty$. It remains to show that the same
substitution can be made on the right hand side of (49), which follows from
similar but slightly simpler calculations that we leave to the reader. ∎

##  5\. Coalescence time and coming down from infinity

In this section, we establish that $\gamma$ ‘comes down from infinity’.

######  Theorem 5.1.

Let $\gamma_{0}\equiv 1$. For almost every environment $\xi$, we have that

$\sum_{x\in\mathbb{N}}\gamma_{t}(x)<\infty\quad\text{ for any }t>0.$

Furthermore, $\gamma$ can be extended to a Feller process on
$\\{0,1\\}^{\mathbb{N}}\times 2^{\mathbb{N}}$ that enters
$\\{0,1\\}^{\mathbb{N}}$ continuously.

Theorem 5.1 is proved at the end of this section. The main ingredient for the
proof is the following auxiliary statement in which we control the expected
coalescence time $\mathbb{E}\tau_{M+1\to M}$. Here, $\tau_{k\rightarrow l}$
denotes the first time at which the process initiated with $k$ particles
enters a state with $l$ particles for $l<k$. In fact, it suffices to consider
the process $\gamma$ restricted to sites in $[N].$

######  Proposition 5.2.

There exists a constant $C(\xi)$ so that, for any $M$,

(53) $\displaystyle\mathbb{E}\big{[}\tau^{(N)}_{M+1\to
M}\big{|}\gamma_{0}^{(N)}\big{]}\leq C(\xi)M^{-\frac{1}{\alpha}},$

uniformly in $N$ and $\gamma_{0}^{(N)}\in\\{0,1\\}^{[N]}$.

### The proof of Proposition 5.2

The proof is a direct consequence of Lemmas 5.3–5.5 in this paragraph. Since a
coalescence can only happen when a particle jumps we need to control the total
number of jumps $J^{(N)}_{t}$ in a system of $M+1$ coalescing random walks on
$[N]$, which we do by means of a lower bound on $J_{t}^{(N)}$.

######  Lemma 5.3.

Assume there is a lower bound $J^{\prime}_{t}\leq J^{(N)}_{t}$ valid until
$\tau^{(N)}_{M+1\to M}$, i.e. $J^{\prime}_{t}\leq
J^{(N)}_{t}+\infty\mathbbm{1}_{t>\tau{(N)}_{M+1\to M}}$ uniformly in all
starting configurations $\gamma_{0}^{(N)}\in\\{0,1\\}^{[N]}$. Assume further
that there are constants $c_{1}=c_{1}(\xi)$, $c_{2}=c_{2}(\xi)$ satisfying

(54)
$\displaystyle\int_{0}^{t}\mathbb{P}\Big{(}J^{\prime}_{t}<c_{1}NM^{\frac{1}{\alpha}-1}t\Big{)}\,\textup{d}t\leq
c_{2}M^{-\frac{1}{\alpha}}.$

Then

(55) $\displaystyle\mathbb{E}\big{[}\tau^{(N)}_{M+1\to
M}\big{|}\gamma_{0}^{(N)}\big{]}\leq(c_{1}^{-1}+c_{2})M^{-\frac{1}{\alpha}}.$

###### Proof.

Each jump in the system has probability $\frac{M}{N}$ to land on top of
another particle, leading to a coalescence. Since these jumps are independent,
the number of jumps needed is a geometric random variable $Z$ with success
parameter $\frac{M}{N}$, and $\\{\tau^{(N)}_{M+1\to
M}>t\\}=\\{Z>J^{(N)}_{t}\\}$. Note that $Z$ and $J^{(N)}_{t}$ are not
independent. By assumption there is a coupling
$\widehat{\mathbb{P}}_{\mathbf{x}}$ with $J_{t}^{\prime}\leq J_{t}^{(N)}$ on
the event $\\{\tau^{(N)}_{M+1\to M}>t\\}$, where we set
$\mathbf{x}=\gamma_{0}^{(N)}$ for conciseness. Then, writing
$\tau=\tau^{(N)}_{M+1\to M}$, for any $a\in\mathbb{N}$,

(56)
$\displaystyle\mathbb{P}_{\mathbf{x}}\left(\tau>t\right)=\mathbb{P}_{\mathbf{x}}\left(\tau>t,Z\geq
a\right)+\mathbb{P}_{\mathbf{x}}\left(\tau>t,Z<a\right)$ (57)
$\displaystyle\leq\mathbb{P}_{\mathbf{x}}\left(Z\geq
a\right)+\mathbb{P}_{\mathbf{x}}\left(\tau>t,J^{(N)}_{t}<a\right)$ (58)
$\displaystyle\leq\mathbb{P}_{\mathbf{x}}\left(Z\geq
a\right)+\widehat{\mathbb{P}}_{\mathbf{x}}\left(\tau>t,J^{\prime}_{t}<a\right)$
(59) $\displaystyle\leq\mathbb{P}_{\mathbf{x}}\left(Z\geq
a\right)+\widehat{\mathbb{P}}_{\mathbf{x}}\left(J^{\prime}_{t}<a\right).$

With $a=\lfloor c_{1}NM^{\frac{1}{\alpha}-1}t\rfloor$,

(60) $\displaystyle\mathbb{P}_{\mathbf{x}}\left(Z\geq
a\right)=\left(1-\frac{M}{N}\right)^{a-1}\leq e^{-c_{1}M^{\frac{1}{\alpha}}t}$

and

(61) $\displaystyle\int_{0}^{\infty}\mathbb{P}_{\mathbf{x}}\left(Z\geq
a\right)\;dt\leq c_{1}^{-1}M^{-\frac{1}{\alpha}}.$

By assumption,

(62)
$\displaystyle\int_{0}^{t}\widehat{\mathbb{P}}_{\mathbf{x}}\left(J^{\prime}_{t}<a\right)\;dt\leq
c_{2}M^{-\frac{1}{\alpha}}$

and since $\mathbb{P}_{\mathbf{x}}(\tau^{(N)}_{M+1\to M}=t)=0$

(63) $\displaystyle\mathbb{E}_{\mathbf{x}}\tau^{(N)}_{M+1\to
M}=\int_{0}^{\infty}\mathbb{P}_{\mathbf{x}}\left(\tau^{(N)}_{M+1\to
M}>t\right)\;dt\leq(c_{1}^{-1}+c_{2})M^{-\frac{1}{\alpha}}.\qed$

We now need to construct a good lower bound $J_{t}^{\prime}$ on $J_{t}^{(N)}$.
Let $X_{t}^{\prime}$ be a random walk on $\\{M+1,...,N\\}$ started in $M+1$.
The mean waiting times in site $x$ are given by $\xi_{x}$ (so identical to
$X_{t}^{(N)}$), but it then jumps to $y$ with probability $\frac{1}{N}$ for
$y>M+1$ and $\frac{M+1}{N}$ for $y=M+1$. Let $J_{t}^{\prime}$ denote the
number of jumps this random walk performs until time $t$.

######  Lemma 5.4.

The process $J^{\prime}_{t}$ is a lower bound on $J^{(N)}_{t}$ until
$\tau^{(N)}_{M+1\to M}$, that is, $J^{\prime}_{t}\prec
J^{(N)}_{t}+\infty\mathbbm{1}_{t>\tau^{(N)}_{M+1\to M}}$ uniformly in all
starting configurations $\gamma_{0}^{(N)}$, where ‘$\prec$’ indicates
stochastic domination.

###### Proof.

Consider the system of coalescing random walks where we mark one particle,
with $X_{t}^{\prime\prime}$ the position of the marked particle. Then,
clearly, the number of jumps performed by $X^{\prime\prime}_{t}$ is a lower
bound on the total number of jumps (we do not care what happens with the
marked particle during a coalescence event, since the lower bound needs only
to be valid until the first coalescence). The mechanism for marking is as
follows: Initially, the particle with the largest position is marked, which is
automatically at a position larger than $M$. Then, when the marked particle
jumps to a site in $[M]$, we change the mark to the particle on the largest
position, which is larger than $M$ (again, assuming coalescence hasn’t
happened yet). Note that by construction, until the first coalescence,
$X^{\prime\prime}_{t}\geq M+1$.

We now couple $X^{\prime\prime}_{t}$ with $X^{\prime}_{t}$. If
$X^{\prime\prime}_{t}=X^{\prime}_{t}$, both particles move identically until
$X^{\prime\prime}_{t}$ jump into $[M]$. On this event, the marked particle
changes to the largest particle, so $X^{\prime\prime}_{t}\geq M+1$, with the
position random depending on the position of the other particles. On the other
hand, $X^{\prime}_{t}$ jumps to $M+1$. If $X^{\prime}_{t}=M+1$ and
$X^{\prime\prime}_{t}>M+1$ the jump rate of $X_{t}^{\prime\prime}$ is
$\xi_{X^{\prime\prime}_{t}}^{-1}$ and hence dominates the jump rate
$\xi_{M+1}^{-1}$ of $X_{t}^{\prime}$. Therefore we can couple the jump times
so that if $X_{t}^{\prime}$ jumps, $X^{\prime\prime}_{t}$ jumps as well and
both jump to the same target (unless the target is in $[M]$, in which case
$X_{t}^{\prime}$ jumps to $M+1$ (and effectively does not move, but we do
count this jump), and $X_{t}^{\prime\prime}$ jumps to some site $\geq M+1$.

From this construction we obtain that whenever $X_{t}^{\prime}$ jumps, so does
$X_{t}^{\prime\prime}$, and hence $J_{t}^{\prime}\leq J_{t}^{(N)}$ until the
first coalescence time. ∎

######  Lemma 5.5.

$J^{\prime}_{t}$ satisfies

(64)
$\displaystyle\int_{0}^{\infty}\mathbb{P}\Big{(}J^{\prime}_{t}<\frac{NM^{\frac{1}{\alpha}-1}}{4(1+C(\xi))}t\Big{)}\;dt\leq\frac{4}{C(\xi)}M^{-\frac{1}{\alpha}}.$

###### Proof.

Observe by Chebyshevs inequality

(65) $\mathbb{P}(J^{\prime}_{t}<a)=\mathbb{P}(\sum_{i=1}^{a}\sigma_{i}>t)\leq
e^{-\beta t}\mathbb{E}e^{\beta\sum_{i}\sigma_{i}},$

where $\sigma_{i}$ is the waiting time for the $i$-th jump. These are
$\operatorname{Exp}(\xi_{X^{\prime}_{i}}^{-1})$-distributed and independent
given $X^{\prime}_{i}$, $i=1,2,...$, where $X^{\prime}_{i}$ is the position of
$X^{\prime}_{t}$ before the $i$-th jump. We have $X^{\prime}_{1}=M+1$ and the
remaining positions are i.i.d. with
$\mathbb{P}(X^{\prime}_{i}=x)=\frac{1}{N}$, $x>M+1$ and
$\mathbb{P}(X^{\prime}_{i}=M+1)=\frac{M+1}{N}$. Therefore, for
$\beta\leq\frac{1}{2}\xi_{M+1}^{-1}$ (and hence $\beta\xi_{x}\leq\frac{1}{2}$
for all $x\geq M+1$),

(66) $\displaystyle\mathbb{E}e^{\beta\sum_{i}\sigma_{i}}$
$\displaystyle=\prod_{i=1}^{a}\mathbb{E}e^{\beta\sigma_{i}}=\prod_{i=1}^{a}\mathbb{E}\frac{\xi_{X^{\prime}_{i}}^{-1}}{\xi_{X^{\prime}_{i}}^{-1}-\beta}=\prod_{i=1}^{a}\mathbb{E}\frac{1}{1-\beta\xi_{X^{\prime}_{i}}}$
(67)
$\displaystyle=\prod_{i=1}^{a}\frac{1}{N}\sum_{x=1}^{N}\frac{1}{1-\beta\xi_{x\lor(M+1)}}$
(68)
$\displaystyle\leq\prod_{i=1}^{a}\frac{1}{N}\sum_{x=1}^{N}(1+2\beta\xi_{x\lor(M+1)})=\prod_{i=1}^{a}\big{(}1+\frac{2\beta}{N}\sum_{x=1}^{N}\xi_{x\lor(M+1)}\big{)}$

By (3) and (5),

(69) $\displaystyle\sum_{x=1}^{N}\xi_{x\lor(M+1)}\leq
M^{1-\frac{1}{\alpha}}+C(\xi)M^{1-\frac{1}{\alpha}}=(1+C(\xi))M^{1-\frac{1}{\alpha}}.$

Hence,

(70) $\displaystyle\mathbb{E}e^{\beta\sum_{i}\sigma_{i}}$
$\displaystyle\leq\prod_{i=1}^{a}(1+2\beta(1+C(\xi))M^{1-\frac{1}{\alpha}}N^{-1})\leq
e^{2\beta(1+C(\xi))aM^{1-\frac{1}{\alpha}}N^{-1}}.$

Assuming $t\geq 2(1+C(\xi))aM^{1-\frac{1}{\alpha}}N^{-1}$, we pick the maximal
possible value for $\beta$, namely $\beta=\frac{1}{2}\xi_{M+1}^{-1}$. Plugging
(70) into (65), we get

(71) $\displaystyle\mathbb{P}(J^{\prime}_{t}<a)\leq
e^{-\frac{1}{2}\xi_{M+1}^{-1}(t-2(1+C(\xi))aM^{1-\frac{1}{\alpha}}N^{-1})}.$

Choosing $a=\frac{NM^{\frac{1}{\alpha}-1}}{4(1+C(\xi))}t$ it follows that

(72)
$\displaystyle\mathbb{P}\Big{(}J^{\prime}_{t}<\frac{NM^{\frac{1}{\alpha}-1}}{4(1+C(\xi))}t\Big{)}\leq
e^{-\frac{1}{4}\xi_{M+1}^{-1}t}\leq
e^{-\frac{1}{4}C(\xi)M^{\frac{1}{\alpha}}t}.$

for all $t\geq 0$ and

(73)
$\displaystyle\int_{0}^{\infty}\mathbb{P}\Big{(}J^{\prime}_{t}<\frac{NM^{\frac{1}{\alpha}-1}}{4(1+C(\xi))}t\Big{)}\;\,\textup{d}t\leq\frac{4}{C(\xi)}M^{-\frac{1}{\alpha}}.$

∎

### The proof of Theorem 5.1

We conclude the section with the final derivation of the ‘coming down from
infinity’-property.

###### Proof of Theorem 5.1.

By Propsition 5.2, and Fatou’s Lemma

(74) $\mathbb{E}\tau_{\infty\to 1}=\sum_{M=2}^{\infty}\mathbb{E}\tau_{M+1\to
M}\leq\sum_{M=2}^{\infty}\liminf_{N\to\infty}\mathbb{E}\tau^{(N)}_{M+1\to
M}\leq C(\xi)\sum_{M=2}^{\infty}M^{-\frac{1}{\alpha}}<\infty.$

In particular, $\tau_{\infty\to 1}$ is a.s. finite. Let
$\mathcal{N}(t)=\sum_{x\in\mathbb{N}}\gamma_{t}(x)$ the process counting the
number of particles at time $t$. By (74) infinity is a continuous entrance
point. By Lemma 1.2. together with Remark 2.3. of [6], the finiteness
$\sup_{m}\mathbb{E}[\tau_{m\to 1}]$ implies that $(\mathcal{N}(t))_{t\geq 0}$
can be extended to a Feller process over $\mathbb{N}\cup\\{\infty\\}$ such
that under the extended law, $\mathcal{N}$ starts from $\infty$-many particles
and instantaneously enters a configuration with only finitely many particles.
The analogous statement for the whole process $\gamma$ now follows from the
Feller property of the particle count process together with Proposition 4.2
and Lemma 3.15 as well as the fact that we may encode the particles that
reside at $\infty$ at any time as an element of $2^{\mathbb{N}}$. ∎

##  6\. Proofs of the main results

In this Section, we finally prove Theorem 1.1, Theorem 1.2 and Theorem 1.3.

###### Proof of Theorem 1.1.

In Proposition 4.1, it is shown that the processes $Y^{(N)}$ under temporal
rescaling converge in distribution to a limit random walk $X$ on $\mathbb{N}$
which is governed by rates $\xi$ as in (1). Proposition 4.3 extends this to
systems $(\gamma_{t})_{t\geq 0}$ of finitely many coalescing random walks.
Hence, for any $t\geq 0$ and any $A\subset\mathbb{N}$ finite we may define the
corresponding probabilities for the limiting voter model through

(75) $\mathbb{P}\big{(}\eta_{t}(x)\equiv 1\text{ on
}A\big{|}\eta_{0}\big{)}:=\mathbb{P}\big{(}\eta_{0}(x)\equiv 1\text{ on
}\operatorname{supp}\gamma_{t}\big{|}\operatorname{supp}\gamma_{0}=A\big{)},$

where $\operatorname{supp}\gamma_{t}=\\{x:\gamma_{t}(x)=1\\},t\geq 0.$ The
extension of the definition to arbitrary $A\subset\mathbb{N}$ can now be
achieved by observing that the dual limiting system is well defined for an
arbitrary initial configuration of particles dispersed over $\mathbb{N}$ by
Proposition 4.2. ∎

###### Proofs of Theorems 1.2 and 1.3.

We only derive Theorem 1.2 explicitly, the statements for the finite system
can be obtained in the same fashion but without relying on properties of the
limiting random walk $X$. By duality, the expectation of the consensus time
$\tau$ is clearly bounded above by the expectation of the time
$\tau_{\infty\rightarrow 1}$ which is needed in the dual system to coalesce
the the initial configuration with a particle at every site into one single
particle. By Theorem 5.1, this time has a finite mean, which proves the first
part of Theorem 1.2. To prove the formula (2), note that at any fixed time
$t_{0}>0$, we may pick a site $x$ with probability proportional to $\xi_{x}$
and obtain the type $\eta_{t_{0}}(x)=\eta_{0}(y)$ by tracing the path of the
corresponding particle in the dual for time $t_{0}$ to some target site $y$.
The distribution of this path is independent of the fact whether consensus has
occurred or not and agrees with the distribution of a path of the limiting
walk $X$ initiated from its invariant distribution by Lemma 3.10. In
particular, $y$ has the invariant distribution
$(\xi_{x}/\sum_{z}\xi_{z})_{x\in\mathbb{N}}$ of $X$ if $t_{0}\geq\tau$, which
leads immediately to (2). ∎

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