# Long-range competition on the torus

Bas Lodewijks and Neeladri Maitra

###### Abstract

We study a competition between two growth models with long-range correlations
on the torus $\mathbb{T}_{n}^{d}$ of size $n$ in dimension $d$. We append the
edge set of the torus $\mathbb{T}_{n}^{d}$ by including all non-nearest-
neighbour edges, and from two source vertices, two first-passage percolation
(FPP) processes start flowing on $\mathbb{T}_{n}^{d}$ and compete to cover the
sites, where the cost of traversing an edge depends on the torus distance
between its endpoints. The FPP processes we consider are long-range first-
passage percolation processes, as studied by Chaterjee and Dey in [17]. Here,
we have two types, say Type-$1$ and Type-$2$, and the Type-$i$ transmission
time of an edge $e$ equals $\lambda_{i}^{-1}\|e\|^{\alpha_{i}}E_{e}$ for
$i\in\\{1,2\\}$, where $(E_{e})_{e}$ is a family of i.i.d. rate-one
exponential random variables, $\lambda_{1},\lambda_{2}>0$ are the global rate
parameters, and $\alpha_{1},\alpha_{2}\geq 0$ are the long-range parameters.
In particular, we consider the instantaneous percolation regime, where
$\alpha_{1},\alpha_{2}\in[0,d)$, and we allow all parameters to depend on $n$.
We study _coexistence_ , the event that both types reach a positive proportion
of the graph, and identify precisely when coexistence occurs. In the case of
absence of coexistence, we outline several phase transitions in the size of
the losing type, depending on the relation between the rates of both types.
One of the consequences of our results is that for constant intensity
competition, i.e. when the long-range parameters of the two processes are the
same, while their rates differ by a constant multiplicative factor,
coexistence of the two processes at the scale of the torus volume happen if
and only if their global rates are equal. On the other hand, when the long-
range parameters differ, it is possible for one of the types, e.g. Type-$2$,
to reach a significant number of vertices, even when its global rate parameter
$\lambda_{2}$ is much smaller than $\lambda_{1}$.

Keywords: First-passage percolation, long-range first-passage percolation,
competition, competing first-passage percolation, continuous-time branching
processes.

MSC Subject Classifications: Primary: 60K35, Secondary: 60C05.

##  1 Introduction

Suppose that a virus spreads through a population, when suddenly a mutated
variant appears. This mutated variant is less contagious, but has a longer
incubation period (the time until symptoms develop). As such, though the
variant spreads slower than the original virus locally, its increased
incubation period may result in infections to occur over greater distances.
People who carry the virus but show no symptoms may still go into work, visit
friends or family, or travel large distances. As such, the variant can more
easily spread along greater distances compared to the original virus.

An important question is which of the two viruses is the more dangerous one.
Indeed, in the outbreak of, for example, the COVID virus, it was observed that
mutated variants showed behaviour different to that of the original strain
[25, 47], thus requiring a shift in focus on how to combat the outbreak.
Depending on the difference in properties and behaviour of the two viruses
(e.g. their contagiousness and incubation periods), which of the two viruses
infects only a negligible part of the population, compared to the other, by
the time the viruses are endemic? Or, do both viruses infect a positive
proportion of the population and thus both form a threat?

The aim of this paper is to study a mathematical model of competing viral
infections in a population spread out through space, where the viral
infections can differ not only in the _local intensity_ at which they spread,
but also in the strength of their respective _long-range_ spreading rate. As
far as the authors are aware, this initiates the study of such competition
models where long-range effects are considered. We model the spread of
infections as two long-range first-passage percolation processes on the torus
$\mathbb{T}_{n}^{d}$ of size $n$ in dimension $d$. To describe the model, let
us briefly review the models of first-passage percolation and the competition
variants of them.

First-passage percolation (FPP) was first introduced by Hammersley and Welsh
in 1965 [26]. It has then been studied extensively as a prototypical model of
fluid-flow through porous media; we refer the interested reader to the survey
[6] and references therein. Informally, one considers a graph $G=(V,E)$, and
associates i.i.d. copies $\gamma_{e}$ of some random variable $\gamma$ to
every edge $e\in E$. We interpret $\gamma_{e}$ as the time it takes for a
fluid to traverse the edge $e$. For the special case when $\gamma$ is an
exponentially distributed random variable with mean $1/\lambda$, we say that
the FPP process ‘flows at rate $\lambda$’.

A model where two FPP processes compete to occupy sites in the two-dimensional
integer lattice $\mathbb{Z}^{2}$ was first introduced as the two-type
Richardson model in [36]. In this model, initially two different kinds of FPP
processes start flowing from two different sources, at potentially different
rates, and they compete to cover the entire lattice. The main interest is in
understanding when both processes cover infinitely many sites with positive
probability; an event known as _coexistence_. In this direction, the main
conjecture from [36] is that coexistence occurs if and only if the rates of
the two competing processes are equal. Note that by a straightforward scaling,
one can without loss of generality consider the case where the first fluid
flows at rate $1$, while the second at rate $\lambda\geq 1$. Many results
related to this conjecture have been obtained. The original paper [36]
introducing this model showed coexistence occurs for $\lambda=1$, [37] showed
non-coexistence for almost all parameter values of $\lambda$, and [4] showed
coexistence if and only if $\lambda=1$, on the half-plane
$\mathbb{Z}\times\mathbb{N}$. We refer the interested reader to the survey
[19] and the references therein for more in-depth information.

A model of _long-range first-passage percolation_ (LRFPP) that inspired the
model of the present paper, was introduced by Chatterjee and Dey in [17].
Here, the underlying graph
$\mathbb{G}_{\infty}=(V(\mathbb{G}_{\infty}),E(\mathbb{G}_{\infty}))$ has the
$d$-dimensional integer lattice $\mathbb{Z}^{d}$ as its vertex set, and each
vertex is connected to every other vertex; thus creating an ‘infinite complete
graph embedded in $\mathbb{Z}^{d}$’. Every edge $e$ is assigned a transmission
time $T_{e}:=\|e\|^{\alpha}E_{e}$, where $\|e\|$ denotes the spatial length of
the edge (one can take any $p$-norm to determine the length), $\alpha\geq 0$
is the _long-range_ parameter, and $(E_{e})_{e\in E(\mathbb{G}_{\infty})}$ is
a family of i.i.d. rate-one exponential random variables. One can view this
setting as a first-passage percolation model where transmission times are
penalised by the lengths of the edge, where long edges receive heavier
penalties and $\alpha$ controls the discrepancy between penalties of short and
long edges. Note that $\alpha=\infty$ can be interpreted as the case of
nearest-neighbour FPP on the integer lattice $\mathbb{Z}^{d}$.

In this paper, we study a competition variant of the LRFPP model on the torus
$\mathbb{T}_{n}^{d}$ of size $n$ in dimension $d$ obtained by including the
non-nearest-neighbor edges (instead of the entire lattice $\mathbb{Z}^{d}$ as
in the previous paragraph). Here, initially from two different source vertices
of $\mathbb{T}_{n}^{d}$, two different LRFPP processes start, say a Type-$1$
and Type-$2$ process, attempting to reach the vertices of the graph. Here,
Type-$i$ spreads through the graph according to transmission times
$T_{e}^{(i)}:=\lambda_{i}^{-1}\|e\|^{\alpha_{i}}E_{e}$, for $i\in\\{1,2\\}$,
where $\lambda_{1},\lambda_{2}>0$ are _global_ intensities and
$\alpha_{1},\alpha_{2}\geq 0$ are the long-range parameters. Referring back to
the real-world setting that served as motivation at the start, we can view the
different global intensities as the contagiousness of the viruses, and the
long-range parameters as the strength of the long-range spread of the viruses
(possibly determined by e.g. different incubation periods). Our main interest
is in understanding the size of the set of vertices infected by a particular
infection, when all the vertices of $\mathbb{T}_{n}^{d}$ have been infected.

We restrict our attention to long-range parameters strictly less than the
dimension $d$, which is the _instantaneous percolation regime_ for LRFPP
studied by Chatterjee and Dey [17] and can be thought of as a setting with
weak spatial dependence. We remark that, by [17, Theorem $1.2$], a LRFPP
process on $\mathbb{G}_{\infty}$ with long-range parameter $\alpha<d$ has the
following property: for any time $t>0$, the set of vertices that can be
reached from the origin equals $\mathbb{Z}^{d}$ almost surely. Hence, any
vertex is essentially reached instantaneously. This clarifies the choice to
restrict the competition process to the finite torus $\mathbb{T}_{n}^{d}$.
Indeed, such a process on $\mathbb{G}_{\infty}$ is not well-defined in the
instantaneous percolation regime, since one cannot distinguish which infection
reaches a vertex first. For long-range parameters exceeding $d$, this process
can be studied on $\mathbb{G}_{\infty}$ and is ongoing future work. In fact,
as remarked before, for the case when one has $\alpha_{1}=\alpha_{2}=\infty$,
and $\lambda_{1}=1,\lambda_{2}=\lambda\geq 0$, we recover the two-type
Richardson model. This link served as one of the motivations to consider the
model of the present paper. One particularly interesting question to
understand is to determine how large the long-range parameters have to be, so
that the model is essentially in the same _universality class_ as the two-type
Richardson’s model. That is, when do most qualitative aspects of the competing
LRFPP model behave in a similar manner to the two-type Richardson model?
Though this is beyond the scope of this paper, it serves as a first attempt to
understand the competing LRFPP model.

In this paper we are generally concerned with questions related to (absence
of) coexistence at a linear scale: after all vertices are infected, have both
LRFPP processes infected a positive proportion of the graph, or does one
process infect almost all vertices, leaving only $o(n)$ many for the other
process? In the case of non-coexistence, i.e. one type winning almost all
vertices of the torus, what can further be said about the size of the losing
type? We provide a complete picture in terms of the intensities
$\lambda_{1},\lambda_{2}$, and the long-range parameters
$\alpha_{1},\alpha_{2}$. In particular, we allow these parameters to depend on
$n$, which uncovers more subtle behaviour in certain cases that cannot be
observed from fixed parameters, independent of $n$, which is common in most of
the literature.

###  1.1 Model definition

Fix $d\in\mathbb{N}$ and let $\mathbb{T}_{n}^{d}$ be the $d$-dimensional torus
of size $n$, i.e. $\mathbb{T}_{n}^{d}=$
$[-n^{1/d}/2,n^{1/d}/2]\cap\mathbb{Z}^{d}$, where we identify vertices on
opposing boundaries to be the same. Let
$\lambda_{1},\lambda_{2}=\lambda_{1}(n),\lambda_{2}(n)>0$ and
$\alpha_{1},\alpha_{2}=\alpha_{1}(n),\alpha_{2}(n)\in[0,d)$. Throughout the
paper, when there is no confusion, we omit the argument $n$ and write
$\lambda_{i}$ (resp. $\alpha_{i}$) for $\lambda_{i}(n)$ (resp.
$\alpha_{i}(n)$), for $i=1,2$. Let $\|\cdot\|$ denote the torus norm, which we
can allow to be any $\ell^{p}$ distance on the discrete box
$Q_{n}:=[-n^{1/d}/2,n^{1/d}/2]\cap\mathbb{Z}^{d}$, with appropriate
periodicity conditions that come from identifying the boundaries of $Q_{n}$.
Let us define the quantities

$R_{n}^{(1)}:=\sum_{\begin{subarray}{c}v\in\mathbb{T}_{n}^{d}\\\ v\neq
0\end{subarray}}\|v\|^{-\alpha_{1}},\qquad
R_{n}^{(2)}:=\sum_{\begin{subarray}{c}v\in\mathbb{T}_{n}^{d}\\\ v\neq
0\end{subarray}}\lambda_{2}\|v\|^{-\alpha_{2}}.$ (1.1)

$R_{n}^{(1)}$ and $R_{n}^{(2)}$ are key quantities for the analysis of the
process, and thus we state them together. However, we do not need
$R_{n}^{(2)}$ in the model definition, while $R_{n}^{(1)}$ appears as a
scaling factor that does not influence the behaviour of the model, but is used
for convenience.

We append the edge set of the torus with all the non-nearest-neighbor edges,
and call this set of ‘complete’ edges $\mathrm{E}_{n}$, i.e.,

$\displaystyle\mathrm{E}_{n}:=\\{\\{x,y\\}:x,y\in\mathbb{T}_{n}^{d},{x\neq
y}\\}.$

To each edge $e\in\mathrm{E}_{n}$, we associate a pair
$(E^{(1)}_{e},E^{(2)}_{e})$ of independent exponential random variables with
rate parameter $1$, where the collection of the pairs over all the edges also
forms an independent collection, and we define

$T_{e}^{(i)}:=R_{n}^{(1)}\lambda_{i}^{-1}\|e\|^{\alpha_{i}}E^{(i)}_{e}$ (1.2)

to be the time it takes for the Type-$i$ infection to traverse the edge $e$,
for $i\in\\{1,2\\}$. For our results, it is not important that the random
variables $E^{(1)}_{e}$ and $E^{(2)}_{e}$ are independent for a fixed edge $e$
(in fact, we could use the _same_ random variable for both transmission
times). We still assume this independence, as this simplifies some techniques
that we employ. It is crucial, however, that for different edges $e_{1}$ and
$e_{2}$, $(E^{(1)}_{e_{1}},E^{(2)}_{e_{1}})$ and
$(E^{(1)}_{e_{2}},E^{(2)}_{e_{2}})$ are independent.

Let us now define the Long-Range Competition (LRC) process. Fix two distinct
vertices $v^{(1)},v^{(2)}\in\mathbb{T}_{n}^{d}$. Initially, the two vertices
$v^{(1)}$ and $v^{(2)}$ are infected with two distinct infection types, which
we call the Type-$1$ and Type-$2$ infection, respectively. Infections start
traversing edges in $\mathrm{E}_{n}$, incident to already infected vertices,
according to the transmission times in (1.2). When the Type-$i$ infection
traverses an edge $e$ and encounters an uninfected vertex $v$, it infects $v$
with Type-$i$ and starts to propagate to other vertices from $v$ according to
the times $T_{e^{\prime}}^{(i)}$ of edges $e^{\prime}\neq e$ incident to $v$.
If $v$ has already been infected by either Type-$1$ or Type-$2$, nothing
happens when the Type-$i$ infection reaches $v$.

There is a natural continuous-time Markovian description of this process,
which we adopt from [5]. Let
$(\mathcal{N}_{n}^{(1)}(t),\mathcal{N}_{n}^{(2)}(t))_{t\geq 0}$ be a Markov
process with state space the set of all pairs of disjoint subsets of
$\mathbb{T}_{n}^{d}$,

$\displaystyle\Omega=\\{(A,B):A,B\subset\mathbb{T}_{n}^{d},A\cap
B=\emptyset\\}.$

At $t=0$, we initialise the process by setting
$\mathcal{N}_{n}^{(1)}(0),\mathcal{N}_{n}^{(2)}(0)=(\\{v^{(1)}\\},\\{v^{(2)}\\})$
for the two distinct vertices $v^{(1)},v^{(2)}\in\mathbb{T}_{n}^{d}$. At any
time $t\geq 0$, the process evolves by including a vertex
$v\in(\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t))^{c}$ in the set
$\mathcal{N}_{n}^{(1)}(t)$ at rate
$\sum_{x\in\mathcal{N}_{n}^{(1)}(t)}(R_{n}^{(1)})^{-1}\lambda_{1}\|v-x\|^{-\alpha_{1}}$,
and to the set $\mathcal{N}_{n}^{(2)}(t)$ at rate
$\sum_{x\in\mathcal{N}_{n}^{(2)}(t)}(R_{n}^{(1)})^{-1}\lambda_{2}\|v-x\|^{-\alpha_{2}}$.
This Markov process describes the evolution of the sets of the two infection
types. We define the stopping times

$T_{k}:=\inf\\{t\geq
0:|\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t)|=k\\},\qquad
k\in\\{2,\ldots,n\\},$ (1.3)

where, by convention, $T_{2}=0$, and let $T_{\mathrm{cov}}=T_{n}$ be the
_cover time_ , when all vertices have been infected.

It is useful to also describe the corresponding discrete-time process
$(\mathcal{N}_{n}^{(1)}(T_{k}),\mathcal{N}_{n}^{(2)}(T_{k}))_{k=2}^{n}$, which
is the continuous-time process evaluated at the stopping times
$(T_{k})_{k=2}^{n}$. We set $\mathcal{N}_{n}^{(1)}(T_{2})=\\{v^{(1)}\\}$ and
$\mathcal{N}_{n}^{(2)}(T_{2})=\\{v^{(2)}\\}$. For any
$k\in\\{2,\ldots,n-1\\}$, conditionally on
$(\mathcal{N}_{n}^{(1)}(T_{k}),\mathcal{N}_{n}^{(2)}(T_{k}))$, the pair
$(\mathcal{N}_{n}^{(1)}(T_{k+1}),\mathcal{N}_{n}^{(2)}(T_{k+1}))$ is obtained
by first sampling a vertex
$v\in(\mathcal{N}_{n}^{(1)}(T_{k})\cup\mathcal{N}_{n}^{(2)}(T_{k}))^{c}$ with
probability

$\displaystyle\frac{\sum_{x\in\mathcal{N}_{n}^{(1)}(T_{k})}\lambda_{1}\|x-v\|^{-\alpha_{1}}+\sum_{y\in\mathcal{N}_{n}^{(2)}(T_{k})}\lambda_{2}\|y-v\|^{-\alpha_{2}}}{\sum_{z\in(\mathcal{N}_{n}^{(1)}(T_{k})\cup\mathcal{N}_{n}^{(2)}(T_{k}))^{c}}\left(\sum_{x\in\mathcal{N}_{n}^{(1)}(T_{k})}\lambda_{1}\|x-z\|^{-\alpha_{1}}+\sum_{y\in\mathcal{N}_{n}^{(2)}(T_{k})}\lambda_{2}\|y-z\|^{-\alpha_{2}}\right)}.$

Then, set
$(\mathcal{N}_{n}^{(1)}(T_{k+1}),\mathcal{N}_{n}^{(2)}(T_{k+1}))=(\mathcal{N}_{n}^{(1)}(T_{k})\cup\\{v\\},\mathcal{N}_{n}^{(2)}(T_{k}))$,
conditionally on both $v$ and on
$(\mathcal{N}_{n}^{(1)}(T_{k}),\mathcal{N}_{n}^{(2)}(T_{k}))$, with
probability

$\displaystyle\frac{\sum_{x\in\mathcal{N}_{n}^{(1)}(T_{k})}\lambda_{1}\|x-v\|^{-\alpha_{1}}}{\sum_{x\in\mathcal{N}_{n}^{(1)}(T_{k})}\lambda_{1}\|x-v\|^{-\alpha_{1}}+\sum_{y\in\mathcal{N}_{n}^{(2)}(T_{k})}\lambda_{2}\|y-v\|^{-\alpha_{2}}},$

and
$(\mathcal{N}_{n}^{(1)}(T_{k+1}),\mathcal{N}_{n}^{(2)}(T_{k+1}))=(\mathcal{N}_{n}^{(1)}(T_{k}),\mathcal{N}_{n}^{(2)}(T_{k})\cup\\{v\\})$
otherwise.

##### Standard notations.

For sequences $(a_{n})_{n\in\mathbb{N}}$ and $(b_{n})_{n\in\mathbb{N}}$ such
that $b_{n}>0$ for all $n$, we let $a_{n}=o(b_{n})$ and
$a_{n}=\mathcal{O}(b_{n})$ denote $\lim_{n\to\infty}\frac{|a_{n}|}{b_{n}}=0$
and $\limsup_{n\to\infty}\big{|}\frac{a_{n}}{b_{n}}\big{|}<\infty$,
respectively. For a sequence $(X_{n})_{n\in\mathbb{N}}$ of random variables
and a random variable $X$, we write
$X_{n}\overset{d}{\longrightarrow}X,X_{n}\overset{\mathbb{P}}{\longrightarrow}X,X_{n}\overset{\mathrm{a.s.}}{\longrightarrow}X$
to denote convergence of $X_{n}$ to $X$ in distribution, in probability, and
in the almost sure sense, respectively. For random variables $X$ and $Y$, we
let $X\stackrel{{\scriptstyle d}}{{=}}Y$ denote equality in distribution. We
say that a sequence of events $\mathcal{E}_{n}$ holds _with high probability_
, if $\mathbb{P}\left(\mathcal{E}_{n}\right)\to 1$ as $n\to\infty$.

###  1.2 Statements of main results

The main quantities of interest are the final sizes
$|\mathcal{N}_{n}^{(1)}(T_{\mathrm{cov}})|$ and
$|\mathcal{N}_{n}^{(2)}(T_{\mathrm{cov}})|$ of the two infections when the
entire torus is infected. In particular, we are interested whether both
infections reach a linear portion of the vertices in the graph, or if just one
of the infections reaches all but a sub-linear portion of the vertices. To
this end, we define notions _coexistence_ and _absence of coexistence_ as
follows:

######  Definition 1.1 ((Absence of) coexistence).

We say that _absence of coexistence_ occurs when, for some $i\in\\{1,2\\}$, it
holds that
$|\mathcal{N}_{n}^{(i)}(T_{\mathrm{cov}})|/n\overset{\mathbb{P}}{\longrightarrow}0$.
In this case, we call the Type-$i$ infection the _losing_ type, whereas the
Type-$(3-i)$ infection is called the _winning_ type.

We say that _coexistence_ occurs when

$\lim_{\epsilon\downarrow
0}\liminf_{n\to\infty}\mathbb{P}\left(|\mathcal{N}_{n}^{(i)}(T_{\mathrm{cov}})|/n>\epsilon\right)=1,$
(1.4)

for both $i\in\\{1,2\\}$.

Our main theorems describe precisely when (absence of) coexistence occurs. We
recall $R_{n}^{(1)}$ and $R_{n}^{(2)}$ from (1.1) and define

$Z_{n}:=\frac{R_{n}^{(2)}}{R_{n}^{(1)}}.$ (1.5)

Looking at the transmission times defined in (1.2), we see that a Type-$1$
vertex spreads at rate $1$ (assuming no other vertices are infected), whereas
a vertex of Type-$2$ spreads at rate $Z_{n}$. The ratio $Z_{n}$ is crucial in
understanding and formulating the behaviour of the LRC process. For the
presentation of the main results, it is convenient to further define

$c_{n}:=(Z_{n}-1)\log n,\quad\text{or, equivalently,}\quad
Z_{n}=1+\frac{c_{n}}{\log n}.$ (1.6)

The behaviour of the process can then by characterised fully by the behaviour
of $c_{n}$. We work under the following assumptions:

######  Assumption 1.2.

  * •

Without loss of generality, we set $\lambda_{1}=1$, and write
$\lambda_{2}(n)=\lambda(n)$.

  * •

Without loss of generality, $Z_{n}\geq 1$.

  * •

We assume that both
$\sup_{n\in\mathbb{N}}\alpha_{1}(n),\sup_{n\in\mathbb{N}}\alpha_{2}(n)\in[0,d)$.

The final assumption allows us to avoid certain technical difficulties that
can arise when $\alpha_{i}\uparrow d$ as $n\to\infty$ for $i\in\\{1,2\\}$.
Finally, we say that, for $p\in[0,1]$, a random variable $X$ is _geometric
distributed with success parameter $p$_, and write $X\sim\mathrm{Geo}(p)$,
when it has distribution

$\mathbb{P}\left(X=k\right)=p(1-p)^{k-1},\qquad k\in\mathbb{N}.$ (1.7)

We then have the following result, pertaining to absence of coexistence:

######  Theorem 1.3 (Absence of coexistence).

Let $(\mathcal{N}_{n}^{(1)}(t),\mathcal{N}_{n}^{(2)}(t))_{t\geq 0}$ denote a
LRC process on $\mathbb{T}_{n}^{d}$, where $\mathcal{N}_{n}^{(i)}(0)=v^{(i)}$,
$i\in\\{1,2\\}$, as defined in Section 1.1. We write
$N_{n}^{(i)}:=|\mathcal{N}_{n}^{(i)}(T_{\mathrm{cov}})|$. Recall $Z_{n}$ from
(1.5) and $c_{n}$ from (1.6). Under Assumption 1.2, when $c_{n}$ diverges,
$N_{n}^{(1)}/n\overset{\mathbb{P}}{\longrightarrow}0$. Furthermore:

  1. (i)

If $c_{n}/(\log n)^{2}\to\infty$, then $N^{(1)}_{n}=1$ with high probability.

  2. (ii)

If $c_{n}=(c+o(1))(\log n)^{2}$ for some $c>0$, then
$N_{n}^{(1)}\overset{d}{\longrightarrow}X$, where
$X\sim\mathrm{Geo}(\mathrm{e}^{-1/c})$.

  3. (iii)

If $c_{n}=o((\log n)^{2})$, then $\log(N_{n}^{(1)})\frac{Z_{n}}{\log
n}\overset{\mathbb{P}}{\longrightarrow}1$.

  4. (iv)

If $c_{n}=o(\log n)$ but $c_{n}$ diverges, then $c_{n}^{-1}(\log n-\log
N_{n}^{(1)})\overset{\mathbb{P}}{\longrightarrow}1$.

The following result identifies when coexistence occurs, showing that we have
a complete characterisation:

######  Theorem 1.4 (Coexistence).

Consider the same definitions and notation as in Theorem 1.3. When
$\sup_{n\in\mathbb{N}}c_{n}<\infty$, then $\lim_{\epsilon\downarrow
0}\liminf_{n\to\infty}\mathbb{P}(N_{n}^{(i)}/n>\epsilon)=1$ for
$i\in\\{1,2\\}$.

######  Remark 1.5 (Non-spatial results).

In the case that $\alpha_{1}=\alpha_{2}=0$ for all $n\in\mathbb{N}$, i.e. when
all transmission times $T_{e}^{(1)}$ and $T_{e}^{(2)}$ are i.i.d. exponential
random variables with rate $1$ and $\lambda(n)$, respectively, more precise
results can be proved. Indeed, as the process can then be exactly described by
a Pólya urn, it follows from [39, Theorem $1.4$] that the absence of
coexistence result and the results in Theorem 1.3 $(iii)$ and $(iv)$, as well
as Theorem 1.4, hold. Moreover, let $E_{1},E_{2}$ be i.i.d. rate-one
exponential random variables. When
$\lim_{n\to\infty}\lambda(n)=\lambda^{*}\geq 1$ such that $(\lambda(n)-1)\log
n$ diverges, then

$N_{n}^{(1)}n^{-1/\lambda(n)}\overset{d}{\longrightarrow}E_{1}E_{2}^{-1/\lambda^{*}},$
(1.8)

and when $\lim_{n\to\infty}(\lambda(n)-1)\log n=c\in[0,\infty)$, then

$\frac{N_{n}^{(1)}}{n}\overset{d}{\longrightarrow}\frac{E_{1}}{E_{1}+E_{2}\mathrm{e}^{c}}.$
(1.9)

In particular, when $c=0$, the limit has the uniform distribution on $[0,1]$.
We refer the interested reader to Section 1.4.3 for more details.
$\blacktriangleleft$

######  Remark 1.6 (Choice of the distance and negative long-range
parameters).

The torus norm $\|\cdot\|$ is some (unspecified) $\ell^{p}$ distance on the
torus $\mathbb{T}_{n}^{d}$. By the equivalence of all $p$-norms on
$\mathbb{R}^{d}$, changing the distance to other $p$-norm metrics introduces
only a constant factor in the intensity of the infections, so that the
characterisation of $c_{n}$ does not change. That is, changing $\lambda$ to
$\lambda^{\prime}=a\lambda$ for some constant $a>0$ does not change whether
$\sup_{n\in\mathbb{N}}c_{n}$ is finite or whether $c_{n}$ diverges.

Further, one could consider _negative_ long-range parameters. As long as one
assumes that
$\inf_{n\in\mathbb{N}}\alpha_{1}(n),\inf_{n\in\mathbb{N}}\alpha_{2}(n)>-\infty$,
the analysis for non-negative long-range parameters carries over with only
minor adjustments to this more general setting. When the long-range parameters
diverge to $-\infty$, penalising short edges significantly more than long
edges, similar technical issues arise as when $\alpha_{1},\alpha_{2}\uparrow
d$.  $\blacktriangleleft$

####  1.2.1 Examples

Let us discuss couple of examples in this section by taking some particular
choices of $\alpha_{1}(n)$, $\alpha_{2}(n)$, and $\lambda(n)$, for which we
can apply Theorems 1.3 and 1.4.

##### Equal and fixed long-range parameters.

Take $\alpha_{1}(n)=\alpha_{2}(n)=\alpha\in[0,d)$. In this case,
$Z_{n}=\lambda(n)$ and so $c_{n}=(\lambda(n)-1)\log{n}$. In this case, we
observe absence of coexistence when $\log(n)(\lambda(n)-1)$ diverges.
Moreover,

  1. $(i)$

$N_{n}^{(1)}=1$ with high probability when $\log{n}=o(\lambda(n))$;

  2. $(ii)$

$N_{n}^{(1)}$ converges to a geometric random variable when
$\lambda(n)/\log{n}$ converges;

  3. $(iii)$

$N_{n}^{(1)}=n^{(1+o(1))/\lambda(n)}$ when $\lambda(n)=o(\log{n})$;

  4. $(iv)$

$N_{n}^{(1)}=n\mathrm{e}^{-(1+o(1))(\lambda(n)-1)\log n}$ when
$\lambda(n)\downarrow 1$ but $(\lambda(n)-1)\log n$ diverges;

$N_{n}^{(1)}$ is linear in $n$ with high probability, when
$(\lambda(n)-1)\log{n}$ is bounded in $n$. In particular, when
$\lambda(n)=\lambda>1$ fixed, then $N_{n}^{(1)}=n^{1/\lambda+o(1)}$, while
when $\lambda(n)=1$, $N_{n}^{(1)}$ is linear. We remark that the particular
case where $\lambda(n)=\lambda\geq 1$ is fixed, is, in general, the setting of
competing first-passage percolation models most often studied in the
literature.

##### Unequal and fixed long-range parameters.

Take $\alpha_{1}(n)=\alpha_{1}\in[0,d)$ and $\alpha_{2}(n)=\alpha_{2}\in[0,d)$
constant, with $\alpha_{1}\neq\alpha_{2}$. The sum
$\sum_{x\in\mathbb{T}_{n}^{d},x\neq 0}\|x\|^{-\alpha}$ for $\alpha\in[0,d)$
equals $(R+o(1))n^{1-\alpha/d}$ for some constant $R=R(\alpha,d,p)$ depending
on $\alpha$, the dimension $d$, and $p$ signifying the $\ell^{p}$-norm on the
torus that we work with, see [31, Appendix]. When $\alpha_{1}\neq\alpha_{2}$,
we thus obtain that

$Z_{n}=\lambda(n)\Big{(}\frac{R(\alpha_{2},d,p)}{R(\alpha_{1},d,p)}+o(1)\Big{)}n^{(\alpha_{1}-\alpha_{2})/d}.$
(1.10)

Let us first consider the case $\alpha_{1}>\alpha_{2}$. The same conclusions
as in the three cases $(i)$ through $(iii)$ in the previous example can be
made, when substituting $\lambda(n)$ by
$\lambda(n)n^{(\alpha_{1}-\alpha_{2})/d}$. Case $(iv)$, as well as the case of
coexistence, are slightly more subtle, as one would need to understand the
precise order of the $o(1)$ term in (1.10). Rather, we have the more general
statement that $N_{n}^{(1)}=n\mathrm{e}^{-(1+o(1))(Z_{n}-1)\log n}$ when
$Z_{n}\downarrow 1$ but $(Z_{n}-1)\log n$ diverges, and that coexistence
occurs when $(Z_{n}-1)\log n$ is bounded in $n$. Here, in contrast to the
previous example, we see that the global intensity $\lambda(n)$ of the
Type-$2$ infection can tend to zero at a polynomial rate, and still the
Type-$2$ infection would reach almost all vertices before the Type-$1$
infection. We thus observe that the long-range advantage that the Type-$2$
infection has via $\alpha_{2}<\alpha_{1}$, which yields a less severe
penalisation for the transmission times of non-nearest neighbour edges,
results in a very strong advantage.

When $\alpha_{1}<\alpha_{2}$, similar conclusions can be drawn for
$N_{n}^{(2)}$, but now when $\lambda(n)$ diverges at a polynomial rate
$n^{(\alpha_{2}-\alpha_{1})/d}$.

##### Converging long-range parameters.

Take $\alpha_{2}(n)=\alpha_{2}\in[0,d)$ fixed and
$\alpha_{1}(n)\downarrow\alpha_{2}$ as $n\to\infty$. That is,
$\alpha_{1}=\alpha_{2}+d\mathrm{e}_{n}$ for some sequence
$\mathrm{e}_{n}\downarrow 0$. In this case, one can again obtain the same
expression for $Z_{n}$ as in (1.10), which we can now write as
$Z_{n}=(1+o(1))\lambda(n)n^{\mathrm{e}_{n}}$. In particular, absence of
coexistence or coexistence now depends on both $\lambda(n)$ and the sequence
$\mathrm{e}_{n}$, in particular at which rate $\mathrm{e}_{n}$ tends to zero.
The two processes may or may not coexist depending on how fast
$\mathrm{e}_{n}\downarrow 0$. We now have that

  1. $(i)$

$N_{n}^{(1)}=1$ with high probability, when
$n^{-\mathrm{e}_{n}}\log{n}=o(\lambda(n))$;

  2. $(ii)$

$N_{n}^{(1)}$ converges to a geometric random variable when
$\lambda(n)n^{\mathrm{e}_{n}}/\log{n}$ converges;

  3. $(iii)$

$N_{n}^{(1)}=n^{(1+o(1))n^{-\mathrm{e}_{n}}/\lambda(n)}$ when
$\lambda(n)=o(n^{-\mathrm{e}_{n}}\log{n})$;

  4. $(iv)$

$N_{n}^{(1)}=n\mathrm{e}^{-(1+o(1))(Z_{n}-1)\log n}$ when
$\lambda(n)n^{\mathrm{e}_{n}}\downarrow 1$ but $(Z_{n}-1)\log n$ diverges;

We observe coexistence when $(Z_{n}-1)\log n$ is bounded in $n$. In this
example, we see a weaker domination compared to the second example, where the
two long-range parameters are strictly bounded away from each other.

###  1.3 Overview of proof

Let us discuss, at a heuristic level, the main ideas that we use to prove
Theorems 1.3 and 1.4.

####  1.3.1 Coexistence

Let us start with the main proof idea of Theorem 1.4, as this is more
straightforward to describe. The key tool we deploy is a coupling of the
competing infection process with two independent branching processes. That is,
we couple the FPP flows from the vertices $v^{(1)}$ and $v^{(2)}$ with two
independent Yule processes (see Section 2 for definitions)
$\mathrm{BP}^{(1)}(t)$ and $\mathrm{BP}^{(2)}(t)$ with rates $1$ and $Z_{n}$
respectively, such that the sets of the infections can be seen as subsets of
the respective branching processes.

This roughly works in the following way. When vertices are infected, other
infected vertices can no longer spread to these newly infected vertices, which
decreases the rate at which the infection spreads. To counter this depletion
of vertices, we augment the graph with _artificial_ vertices to which the
infections can spread, so that the rate at which either the Type-$1$ or
Type-$2$ infection spreads is linear its current size. In a Yule process, the
rate at which individuals are born is also linear in the size of the process,
we can couple the augmented graph process to two Yule processes.

This yields an upper bound on the sizes of the infection sets at any time
$t\geq 0$ by the respective sizes of the branching processes. We express the
total size of the Type-$i$ infection at time $t$ as
$|\mathcal{N}_{n}^{(i)}(t)|=|\mathrm{BP}^{(i)}(t)|-C^{(i)}((0,t])$, where the
quantity $C^{(i)}((0,t])$ is to be interpreted as the ‘defect’ in coupling the
infection sets with branching processes. This defect arises by allowing
infected vertices to ‘re-infect’ other, already infected, vertices. These re-
infections correspond to births in the branching process whilst such re-
infections do not actually occur in the LRC process, thus causing an error or
defect in the coupling. Using the condition $\sup_{n}c_{n}<\infty$ of Theorem
1.4 and using a point-process representation of the defect, we show that with
high probability,

  * a.

by the time $t_{n}\approx\log(n)/Z_{n}$, the size of each Yule process is
$\delta n$ for some $\delta>0$, and

  * b.

the defect $C^{(i)}((0,t_{n}])$ is smaller than $\delta^{\prime}n$ for some
$\delta^{\prime}<\delta$ and for both $i\in\\{1,2\\}$.

As a result, the size of the FPP clusters $|\mathcal{N}_{n}^{(1)}(t_{n})|$ and
$|\mathcal{N}_{n}^{(2)}(t_{n})|$ themselves are both at least $\epsilon n$ for
some $\epsilon=\epsilon(\delta,\delta^{\prime})\in(0,\delta)$. In particular,
the final sizes $N_{n}^{(1)}$ and $N_{n}^{(2)}$ are both at least $\epsilon
n$, which implies coexistence.

####  1.3.2 Absence of coexistence

Let us now discuss how to prove
$N_{n}^{(1)}/n\overset{\mathbb{P}}{\longrightarrow}0$ when $c_{n}\to\infty$,
which is the main statement of Theorem 1.3. To prove this, we divide the time
interval $(0,T_{\mathrm{cov}}]$, which starts from the initial infections and
ends when all the vertices are infected, into two phases, namely $(0,t_{n}]$
and $(t_{n},T_{\mathrm{cov}}]$, where $t_{n}\approx\log(n)/Z_{n}$. We use the
coupling of the competition process to two Yule processes, as described above,
to show that at the end of the first phase,
$|\mathcal{N}_{n}^{(1)}(t_{n})|\approx\exp{\log(n)/Z_{n}}$, while
$|\mathcal{N}_{n}^{(2)}(t_{n})|\geq\xi n$ for some constant $\xi\in(0,1)$.
Again, this follows by using the growth rate of the respective Yule processes,
combined with showing that the defect $C^{(i)}((0,t_{n}])$ is not ‘too large’
for both $i\in\\{1,2\\}$. It is readily checked that
$\exp{\log(n)/Z_{n}}=o(n)$ when $c_{n}\to\infty$, so that, at the end of the
first phase, the Type-$1$ infection has covered a negligible portion of the
vertices, while the Type-$2$ has covered a positive fraction of the vertices.

For the second phase $(t_{n},T_{\mathrm{cov}})$, we use a martingale argument
to show that the significant advantage achieved by the Type-$2$ infection at
the end of the first phase is strong enough to ensure that the size of the
Type-$1$ infection does not increase beyond a multiplicative factor of its
size at the end of the first phase. Overall, at time $T_{\mathrm{cov}}$, the
size of the Type-$1$ infection is thus still $o(n)$, which shows there is no
coexistence of the two types. The martingale argument is based on the discrete
description of the competition process, as outlined in Section 1.1, and is
comparable to a Pólya urn argument, where the initial configuration of the urn
(which are the infection sizes at the end of phase 1), grows with $n$.

####  1.3.3 Size of the losing type

Finally we describe the more detailed results pertaining the exact asymptotic
size of $N_{n}^{(1)}$ in the different cases of Theorem 1.3.

For Theorem 1.3 $(i)$, we use a result from [31, Theorem 1.1], which tells us
that a single Type-$2$ infection takes roughly $2\log(n)/Z_{n}$ amount of time
to reach all vertices. On the other hand, the time for vertex $v^{(1)}$, which
is infected by the Type-$1$ infection at the start, to reach _any_ other
vertex, is an exponential random variable with rate roughly $1$. Hence, when
$c_{n}/(\log n)^{2}$ diverges or, equivalently, $Z_{n}/\log n$ diverges, it
follows that the Type-$2$ infection reaches all vertices by time
$2\log(n)/Z_{n}=o(1)$, whilst the Type-$1$ infection does not spread to any
vertex in that time, with high probability. As a result
$N_{n}^{(1)}\overset{\mathbb{P}}{\longrightarrow}1$.

Theorem 1.3 $(ii)$ uses almost the same approach, now using another result
from [31, Theorem 1.1], namely that by time $\log(n)/Z_{n}$, a Type-$2$
infection reaches _almost all_ vertices, i.e. only a negligible proportion of
vertices are not reached. At the same time, since $c_{n}=(c+o(1))(\log n)^{2}$
or, equivalently, $Z_{n}=(c+o(1))\log n$, it follows that in time
$\log(n)/Z_{n}=c^{-1}+o(1)$, the Type-$1$ infection can reach a (random but
finite) number of vertices, since it initially infects other vertices at rate
approximately $1$. To obtain the exact limiting distribution of the size of
the Type-$1$ infection, we use the coupling with the two independent Yule
processes, and show that up to time $t_{n}:=(1-\delta_{n})\log(n)/Z_{n}$, for
some $\delta_{n}=o(1)$, the coupling of the Type-$1$ infection with a rate-one
Yule process is in fact _exact_ with high probability. That is, the defect
$C^{(1)}((0,t_{n}])$ equals zero with high probability. We then use a similar
Pólya urn argument as discussed in Section 1.3.2 to show that the uninfected
vertices remaining at time $t_{n}$ (of which there are $o(n)$ many) are then
_all_ infected by the Type-$2$ infection, since it has such an overwhelming
advantage.

For Theorems 1.3 $(iii)$ and $(iv)$, the proof of absence of coexistence
discussed above, which uses that the Type-$1$ infection is of size at most
$An^{1/Z_{n}}$ for a large constant $A=A(\epsilon)$ with probability at least
$1-\epsilon$, yields a sufficient upper bound for $N_{n}^{(1)}$. A matching
lower bound follows by once more using the coupling to the Yule processes.
Here we again show that up to time $t_{n}\approx\log(n)/Z_{n}$, the defect
$C^{(1)}((0,t_{n}])$ of the coupling of the Type-$1$ infection with a Yule
process is not ‘too large’. Hence, we can bound the size of the Type-$1$
infection from below by the size of the Yule process at time $t_{n}$ (which is
roughly $n^{1/Z_{n}}$), minus a negligible term coming from the defect. This
implies a lower bound on $N_{n}^{(1)}$ of the right order.

###  1.4 Discussion

####  1.4.1 Related literature

As far as the authors are aware, the LRC model studied in this paper is the
first of its kind, in the sense that it is the first competing first-passage
percolation model where the edge-weights depend on the spatial embedding of
the vertices in the graph. It is inspired by the LRFPP model of Chatterjee and
Dey [17], where the behaviour of a single type infection on $\mathbb{Z}^{d}$
is studied. This behaviour is not limited to $\alpha<d$, and the study of the
competition version of this model for $\alpha\geq d$ is future work. The
qualitative results in [17] for the regime $\alpha<d$, however, play no role
in the analysis here. Instead, the quantitative results of the first author
and Van der Hofstad [31] regarding the single type LRFPP on
$\mathbb{T}_{n}^{d}$ are used to a greater extent. Moreover, a variation on
the coupling of the first-passage percolation process to a branching process
(a Crump-Mode-Jagers (CMJ) branching process, to be precise), as in [11],
underpins the analysis here, too.

Competing first-passage percolation on finite random graphs, has received a
significant interest, where many well-known models have been studied. These
include, regular random graphs [5], the configuration model with finite-
variance degrees [3], infinite-variance and finite-mean degrees [20, 28] (see
also the work of Baroni, Van der Hofstad, and Komjáthy on competing FPP with
deterministic edge-weights [7, 29]), and infinite-mean degrees [21]. The
preprint [28] shows a universal non-coexistence result for CMs with finite-
mean but infinite-variance degrees under general assumptions. The literature
on first-passage percolation processes with a single type on random graphs
extends to many other models as well. This includes, but does not limit
itself, to the complete graph [38, 1, 12, 34, 35, 11, 15, 22, 23, 24], the
Erdős-Rényi random graph [14, 32, 33], the configuration model [2, 8, 9, 13,
16], preferential attachment models [40, 41], and spatial models such as
scale-free percolation and geometric inhomogeneous random graphs [30, 42, 18].
Noteworthy is the recent work of Komjáthy et al. [43, 44, 45], who extensively
study a degree-penalised variation of first-passage percolation on geometric
inhomogeneous random graphs. In this model, edges incident to high-degree
vertices typically have higher transmission times. This is, at least in
spirit, similar to the spatial penalisation in LRFPP, though the underlying
random graph model is very different.

####  1.4.2 Main contributions

In many of the works mentioned above, both for single-type FPP processes and
competition processes, approximating first-passage percolation with branching
processes is a very useful and commonly used technique. We highlight here, in
particular, the work of Bhamidi and Van der Hofstad on FPP on the complete
graph [11], where a precise coupling between an FPP process on the complete
graph and a CMJ branching process is provided. The main contribution of the
current work is an adaptation of this coupling to branching processes in both
a competition setting and a spatial environment, as well as a more in-depth
analysis of the defect of the coupling. This analysis is crucial to many of
the proofs in this paper.

In [11], it is proved that the coupling between the FPP process and the CMJ
branching process approximation is _exact_ up to a certain time $T$, with high
probability. This result allows the authors to obtain precise asymptotic
results for the typical weighted distance in the complete graph, as well as
the hopcount (the number of edges on the shortest edge-weighted path). The
analysis of a similar coupling carried out here, additionally identifies the
_growth rate_ of the coupling defect. That is, when the coupling is no longer
exact, we control the error between the FPP processes and the CMJ branching
process approximation over time. In particular, we show that, up to some time
$T^{\prime}>T$, this error is negligible compared to the size of the branching
process. This allows us to use the coupling to a higher degree and obtain
useful quantitative bounds, much further into the competition process than
previously possible.

The work of Ahlberg, Deijfen, and Janson on competing FPP on the configuration
model with finite-variance degrees [3] is another example where branching-
process approximations up to the time that the approximations are exact, are
used. From that point onwards, their analysis shows that the competition
dynamics are essentially deterministic, in the sense that the ratio of the
sizes of the two competing infections stays roughly the same. These results
are harder to prove for the model studied here, due to the spatial embedding
of the vertices, even when the spatial dependence is weak (long-range
parameters close to zero), see also Problem 1.9 below. Instead, our improved
control of the coupling to branching processes, still allows us to obtain the
desired results, despite not being able to solve Problem 1.9.

####  1.4.3 Related models and open problems

##### Competition without spatial dependence.

When $\alpha_{1}=\alpha_{2}=0$, the spatial embedding of the vertices plays no
role, and the competition process takes place on the complete graph where the
Type-$1$ (resp. Type-$2$) infects through edges at rate $1$ (resp. $Z_{n}$,
which now simply equals $\lambda(n)$). As claimed in Remark 1.5, this setting
can be exactly represented as a Pólya urn. In particular, adopting the setting
of generalized Pólya Urns of Janson [39], we consider a Pólya Urn with two
types of balls - Type-$1$ and Type-$2$ balls, where upon drawing a ball of
some type, it is added back to the urn along with some additional balls
according to the replacement matrix

$\begin{pmatrix}1&0\\\ 0&Z_{n}\end{pmatrix}.$

This means, upon drawing a Type-$1$ ball (resp. Type-$2$ ball), one extra
Type-$1$ ball is (resp. $Z_{n}$ extra Type-$2$ balls are) added back to the
urn, along with the drawn ball. In this description, we implicitly think of
$Z_{n}$ as being a positive integer, but that need not be, and in particular,
the process has a description as a Markov process on the state space
$\\{(x,y)\in\mathbb{R}^{2}:x,y\geq 0,x+y>0\\}$, see [39, Remark $1.11$].
However, following Janson [39], we use phrases like ‘$Z_{n}$ balls are drawn’
for ease of understanding.

Let $M^{(i)}_{n}$ denote the number of Type-$i$ balls after $n$ draws, for
$i\in\\{1,2\\}$, where we initialise the urn with
$M^{(1)}_{0}=1,M^{(2)}_{0}=Z_{n}$. It readily follows that
$M_{n-2}^{(1)}\overset{d}{=}N_{n}^{(1)}$ and
$M^{(2)}_{n-2}/Z_{n}\overset{d}{=}N_{n}^{(2)}$ when $\alpha_{1}=\alpha_{2}=0$,
following the discrete description of the competition process in Section 1.1.
It thus follows that, when $Z_{n}\equiv Z$ for some fixed $Z\geq 1$, the
results of [39, Theorem $1.4$] can be applied to yield Theorems 1.3 and 1.4,
as well as the more in-depth results stated in Remark 1.5. The additional
results of Theorems 1.3 and 1.4 and Remark 1.5, for $Z_{n}$ depending on $n$,
can be obtained with minor adjustments only, as well. Though we are unable to
prove the results of Remark 1.5 when $\alpha_{1},\alpha_{2}$ are not equal
zero, we conjecture these to be true:

######  Conjecture 1.7 (Distributional convergence for (absence of)
coexistence).

Consider the same definitions and notation as in Theorem 1.3 and let
$E_{1},E_{2}$ be two i.i.d. rate-one exponential random variables. If $c_{n}$
diverges, but $\lim_{n\to\infty}Z_{n}=Z\geq 1$, then
$N_{n}^{(1)}n^{-1/Z_{n}}\overset{d}{\longrightarrow}E_{1}E_{2}^{-1/Z}$. If,
instead, $\lim_{n\to\infty}c_{n}=c\geq 0$ exists, then
$N_{n}^{(1)}/n\overset{d}{\longrightarrow}\frac{E_{1}}{E_{1}+E_{2}\mathrm{e}^{c}}$.

In fact we are even unable to show the following weaker result, which we leave
as a problem:

######  Problem 1.8 (Distributional asymmetry of the final sizes).

Consider the same definitions and notations as in Theorem 1.3. If
$\lim_{n\to\infty}c_{n}=c\geq 0$ exists, then
$p(c):=\lim_{n\to\infty}\mathbb{P}(N_{n}^{(1)}<N_{n}^{(2)})$ exists and
satisfies

$\begin{cases}p(c)>\frac{1}{2}&\text{if }c>0,\\\ p(c)=\frac{1}{2}&\text{if
}c=0.\end{cases}$ (1.11)

It is clear that Conjecture 1.7 implies Problem 1.8. Towards proving
Conjecture 1.7, using a branching process approximation, for sufficiently
small $\epsilon>0$, we can check that for $t_{\epsilon}:=\log(\epsilon n)$,
$|\mathcal{N}_{n}^{(1)}(t_{\epsilon})|$ and
$|\mathcal{N}_{n}^{(2)}(t_{\epsilon})|$ are roughly $E_{1}\epsilon n$ and
$E_{2}\epsilon\mathrm{e}^{c}n$ respectively, where $E_{1},E_{2}$ are two
i.i.d. rate-one Exponential random variables. Note that this implies that not
all vertices of the torus are infected by time $t_{\epsilon}$, for very small
$\epsilon$, although both the infections have reached linearly many vertices.
We were unable to apply similar techniques as used in e.g. [3], to show that,
at this stage, the ratios of the two infections remain roughly unchanged until
the end of the process. We leave this as an open problem:

######  Problem 1.9.

Consider the same definitions and notations as in Theorem 1.3. Assume
$c:=\lim_{n\to\infty}c_{n}$ exists. Suppose for two arbitrary subsets
$A,B\subset\mathbb{T}_{n}^{d}$ with $|A|=an$ and $|B|=bn$ for $a,b>0$ with
$A\cup B\neq\mathbb{T}_{n}^{d}$ and $A\cap B=\emptyset$, we set
$\mathcal{N}_{n}^{(1)}(0)=A$ and $\mathcal{N}_{n}^{(2)}(0)=B$. Then
$\frac{N_{n}^{(1)}}{n}\overset{\mathbb{P}}{\longrightarrow}\frac{a}{a+b}$, and
$\frac{N_{n}^{(2)}}{n}\overset{\mathbb{P}}{\longrightarrow}\frac{b}{a+b}$.

This problem, coupled with our earlier observation on the infection sizes up
to time $t_{\epsilon}$, resolves Conjecture 1.7 in the following way: we
divide time into intervals $[0,t_{\epsilon})$ and
$[t_{\epsilon},T_{\mathrm{cov}}]$. At the end of the first time interval both
the processes reach a linear size, with some arbitrary infection sets.
Applying the Markov property at time $t_{\epsilon}$ and using problem 1.9
shows that $N_{n}^{(1)}/n$ and $N_{n}^{(2)}/n$ converge in distribution to
$\frac{E_{1}}{E_{1}+\mathrm{e}^{c}E_{2}}$ and
$\frac{e^{c}E_{2}}{E_{1}+\mathrm{e}^{c}E_{2}}$, respectively. Using Pólya
urns, as discussed in Section 1.4.3, one can check Problem 1.9 is true for
$\alpha_{1}=\alpha_{2}=0$. When the long-range parameters are bounded away
from $0$, we were unable to resolve this problem.

##### Long-range parameters bounded away from $d$.

The main assumption that we make is that
$\sup_{n\in\mathbb{N}}\alpha_{1}(n),\sup_{n\in\mathbb{N}}\alpha_{2}(n)<d$.
This is necessary for parts of the analysis carried out in the paper. When
either $\alpha_{1}{(n)}$ or $\alpha_{2}{(n)}$ converges to $d$, estimates of
the quantities $R_{n}^{(1)}$ and $R_{n}^{(2)}$, as in (1.1), fail to hold,
which play a crucial role in the analysis. The case when either or both of
$\sup_{n\in\mathbb{N}}\alpha_{1}(n)$ and $\sup_{n\in\mathbb{N}}\alpha_{2}(n)$
equals $d$, remains open. In particular, we leave the following conjecture,
which is a first step beyond $\alpha_{1},\alpha_{2}\in[0,d)$ into more
unconstrained choices for the values of $\alpha_{1},\alpha_{2}\geq 0$:

######  Conjecture 1.10 ((Absence of) coexistence for long-range parameters
equal to $d$).

Consider the case when $\alpha_{1}(n)=\alpha_{2}(n)=d$ for all
$n\in\mathbb{N}$, and $\lambda(n)=\lambda\geq 1$ a fixed constant. Then,
$\lim_{n\to\infty}\mathbb{P}\left(N_{n}^{(i)}/n>0\right)=1$ for both
$i\in\\{1,2\\}$ when $\lambda=1$ and
$N_{n}^{(1)}/n\overset{\mathbb{P}}{\longrightarrow}0$ when $\lambda>1$.

The analysis of several crucial tools that we exploit in this paper does not
carry through when the long-range parameters are exactly equal to $d$. In
particular, we are unable to prove results equivalent to those in Proposition
3.2 and Lemma 4.3 (though we expect them to be true), both when we expect
coexistence and the absence of it to hold. Here, our approximations of the
rate at which infected vertices infect the remaining uninfected vertices seems
to be too crude.

##### On powers of exponentials.

Recall the pair of random variables $(E_{e}^{(1)},E_{e}^{(2)})$ for every edge
$e$, that we use to define the transmission times
$T_{e}^{(i)}=(R_{n}^{(1)})^{-1}\lambda_{i}^{-1}\|e\|^{\alpha_{i}}E_{e}^{(i)}$,
for $i\in\\{1,2\\}$, and where the collection
$(E_{e}^{(i)})_{i\in\\{1,2\\},e\in E(\mathbb{G}_{n})}$ is an i.i.d. collection
of exponential random variables with rate parameter $1$. We expect similar
results like Theorems 1.3 and 1.4 to hold when the $E_{e}^{(i)}$ are equal to
rate-one exponential random variables, raised to some power $\gamma>0$. Proofs
would use similar techniques as used in this paper, as is the case in [11],
though we have not pursued this. In this situation, following [17, Remark
1.9], one would have to assume
$\sup_{n\in\mathbb{N}}\alpha_{1}(n),\sup_{n\in\mathbb{N}}\alpha_{2}(n)<d/\gamma$
(instead being smaller than $d$, as in the statement of Theorems 1.3 and 1.4).
We leave this as an open problem:

######  Problem 1.11.

Prove results similar to Theorems 1.3 and 1.4 when
$E_{e}^{(i)}=\big{(}\widetilde{E}_{e}^{(i)}\big{)}^{\gamma}$, for some
$\gamma>0$, where the collection
$(\widetilde{E}_{e}^{(i)})_{i\in\\{1,2\\},e\in E(\mathbb{G}_{n})}$ is an
i.i.d. collection of rate-one exponential random variables. In particular, do
the phase transitions of the scale $c_{n}$ stay in the same order, i.e. orders
of $(\log{n})^{2},\log{n}$ and constant. Do analogues of Conjecture 1.7 or
Problem 1.8 hold?

##### Competition on the long-range percolation cluster.

Consider the _long-range percolation_ graph constructed on the torus
$\mathbb{T}_{n}^{d}$ as follows. We include all the nearest-neighbour edges.
Further, for pairs $x,y\in\mathbb{T}_{n}^{d}$ that are non-nearest neighbours,
we include the edge $\\{x,y\\}$ independently over all other edges with
probability $\|x-y\|^{-\alpha}$, for some fixed $\alpha\in[0,d)$ and where
$\|\cdot\|$ is the torus norm. Note that this graph is always connected as it
contains $\mathbb{T}_{n}^{d}$ as a subgraph. In fact, it has constant diameter
$\lceil\frac{d}{d-\alpha}\rceil$, see [10]. Consider a standard competing FPP
process on this graph started from two uniformly chosen vertices $v^{(1)}$ and
$v^{(2)}$, and let us call these the Type-$1$ and Type-$2$ infections,
respectively, as before. Here, the Type-$1$ infection spreads at rate $1$ and
the Type-$2$ at rate $\lambda=\lambda(n)$ along edges. That is, an uninfected
vertex $v$ is infected with Type-$1$ (resp. Type-$2$) at rate $|\\{u\sim
v:u\text{ infected by Type-$1$}\\}|$ (resp. $\lambda(n)|\\{u\sim v:u\text{
infected by Type-$2$}\\}|$). When $\lambda(n)$ is a constant $\lambda\geq 1$,
we expect asymptotics similar to Theorems 1.3 and 1.4 to hold for the sizes
$N_{n}^{(1)}$ and $N_{n}^{(2)}$ of the two types of infections when all the
vertices have been infected.

######  Conjecture 1.12.

Let $\lambda\geq 1$, and consider the competing FPP on the long-range
percolation cluster with $\alpha<d$ as described above. Then,

  * •

If $\lambda>1$, $N_{n}^{(2)}/n\overset{\mathbb{P}}{\longrightarrow}1$ and
$\log(N_{n}^{(1)})/\log n\overset{\mathbb{P}}{\longrightarrow}1/\lambda$.

  * •

If $\lambda=1$, then
$\lim_{n\to\infty}\mathbb{P}\left(N^{(i)}_{n}/n>0\right)=1$ for
$i\in\\{1,2\\}$.

More interesting is the case $\lambda(n)=1+\frac{c_{n}}{\log{n}}$ for some
sequence $c_{n}$. In particular, it would be interesting to see whether for
this case one obtains asymptotic sizes similar to Theorems 1.3 and 1.4.

######  Problem 1.13.

Consider competing FPP on the long-range percolation cluster as described
above, with intensities $1$ and $\lambda(n)=1+c_{n}/\log{n}$ for the Type-$1$
and Type-$2$ infections. Is the asymptotic behaviour of $N_{n}^{(1)}$ similar
to that in Theorems 1.3 and 1.4, when $c_{n}$ behaves in the different ways as
described in those theorem statements? Further, do analogues of Conjecture 1.7
(with possibly different limits) or Problem 1.8 hold?

##### Organisation of the paper.

The remainder of the paper is organized as follows. In Section 2 we state and
prove several preliminary results. In Section 3, we describe an important tool
that is used throughout the paper, namely a coupling of the infection growths
to branching processes. In particular, Section 3.2 contains the main technical
contribution, formalizing that the branching-process coupling can be
controlled up to a certain time. In Section 4 we prove the upper bounds on the
size of the Type-$1$ infection that are claimed in Theorem 1.3. In particular,
these upper bounds imply when absence of coexistence occurs. We prove matching
lower bounds on the size of the Type-$1$ infection and complete the proofs of
the main theorems in Section 5.

##  2 Preliminary results

In this section we collect a number of minor technical preliminary results
that we need throughout the paper. They consist of estimating the quantities
$R_{n}^{(1)}$ and $R_{n}^{(2)}$, as in (1.1), as well as some basic well-known
results on the growth rate of branching processes.

###  2.1 Rate estimation

Recall the quantities $R_{n}^{(1)}$ and $R_{n}^{(2)}$ from (1.1) (with
$\lambda_{2}=\lambda(n)$). In this section we provide quantitative bounds and
asymptotic results for these quantities. To this end, we define, with
$\alpha=\alpha(n)\in[0,d)$,

$R_{n}(\alpha):=\sum_{\begin{subarray}{c}v\in\mathbb{T}_{n}^{d}\\\
v\neq\mathbf{0}\end{subarray}}\|v\|^{-\alpha}.$ (2.1)

It follows from [31, Appendix] that, by switching from summation to
integration and applying appropriate variable substitutions, if $\alpha$ is
fixed and independent of $n$, then $R_{n}=(R+o(1))n^{1-\alpha/d}$ as
$n\to\infty$, where

$R=R(\alpha,d,p)=2^{\alpha}\int_{[0,1]^{d}}\|x\|^{-\alpha}\,\mathrm{d}x=2^{d}\int_{[0,\frac{1}{2}]^{d}}\|x\|^{-\alpha}\,\mathrm{d}x.$
(2.2)

Here, $R$ depends on $\alpha,d$, and the $\ell_{p}$-norm that is used, for
$p\in[1,\infty)\cup\\{\infty\\}$. As we are only interested in the dependence
on $\alpha$ in what follows, we omit the arguments $d$ and $p$. The final
representation on the right-hand side follows from a variable substitution.
This representation, however, allows us to directly conclude that $R(\alpha)$
is increasing in $\alpha$, regardless of which $\ell^{p}$-norm is used. It is
also readily verified that the asymptotic result
$R_{n}=(R+o(1))n^{1-\alpha/d}$ also holds when $\alpha=\alpha(n)\in[0,d)$
depends on $n$, as long as $\sup_{n\in\mathbb{N}}\alpha(n)<d$ is satisfied.
Moreover, it holds along any diverging sub-sequence $(j(n))_{n\in\mathbb{N}}$,
in the sense that

$R_{j(n)}(\alpha(n))=(R(\alpha(n))+o(1))j(n)^{1-\alpha(n)/d},\qquad\text{as
}n\to\infty,$ (2.3)

where $R(\alpha(n))$ as in (2.2) now also depends on $n$. We have the
following useful bounds for $R_{j}(\alpha(n))$ for all $j\in\mathbb{N}$, as
well as bounds for the sum of the rates between disjoint sets
$A,B\subset\mathbb{T}_{n}^{d}$, based on $R_{n}$:

######  Lemma 2.1.

Let $\alpha=\alpha(n)\in[0,d)$ be such that
$\sup_{n\in\mathbb{N}}\alpha(n)<d$. There exist constants
$0<\underline{c}<\overline{c}<\infty$, such that, uniformly over
$j\in\mathbb{N}$ and $n\in\mathbb{N}$,

$\underline{c}\leq\frac{R_{j}(\alpha(n))}{j^{1-\alpha(n)/d}}\leq\overline{c}.$
(2.4)

###### Proof.

Set $\overline{\alpha}:=\sup_{n\in\mathbb{N}}\alpha(n)$. It follows from (2.3)
that $R_{j}(\alpha(n))=(R(\alpha(n))+o(1))j^{1-\alpha(n)/d}$ as $j\to\infty$,
where $j$ can either be independent of $n$ and tend to infinity, or $j$ can
depend on $n$ such that $n\to\infty$ and $j(n)\to\infty$. Then, since $R$ is
increasing in $\alpha$, we bound $1=R(0)\leq R(\alpha(n))\leq
R(\overline{\alpha})<\infty$, uniformly over the sequence
$(\alpha(n))_{n\in\mathbb{N}}$. As a result, taking $\underline{c}$
sufficiently small and $\overline{c}$ sufficiently large, we obtain the
desired result. ∎

######  Lemma 2.2.

Let $\alpha\in[0,d)$, fix $A,B\subset\mathbb{T}_{n}^{d}$ such that $A\cap
B=\emptyset$, and write $A^{c}:=\mathbb{T}_{n}^{d}\setminus A$ for the
complement of $A$. Then,

$\displaystyle\sum_{v\in A}\sum_{u\in B}\|u-v\|^{-\alpha}$
$\displaystyle\leq\min\\{|A|R_{|B|}(\alpha),|B|R_{|A|}(\alpha)\\},$ (2.5) and
$\displaystyle\sum_{v\in A}\sum_{u\in B}\|u-v\|^{-\alpha}$
$\displaystyle\geq\max\\{|A|(R_{n}(\alpha)-R_{|B^{c}|}(\alpha)),|B|(R_{n}(\alpha)-R_{|A^{c}|}(\alpha))\\}.$
(2.6)

In particular, for $B=A^{c}$,

$\displaystyle\sum_{v\in A}\sum_{u\in A^{c}}\|u-v\|^{-\alpha}$
$\displaystyle\leq\min\\{|A|R_{|A^{c}|}(\alpha),|A^{c}|R_{|A|}(\alpha)\\}\leq\min\\{|A|,|A|^{c}\\}R_{n}(\alpha),$
(2.7) and $\displaystyle\sum_{v\in A}\sum_{u\in A^{c}}\|u-v\|^{-\alpha}$
$\displaystyle\geq\max\\{|A|(R_{n}(\alpha)-R_{|A|}(\alpha)),|A^{c}|(R_{n}(\alpha)-R_{|A^{c}|}(\alpha))\\}.$
(2.8)

###### Proof.

For the upper bound we observe that for each $v\in A$, the sum of the inverse
distances from $v$ to all vertices $u\in B$ is always smaller than the sum of
the inverted distances of the $|B|$ many closest vertices to $v$. Hence,

$\sum_{v\in A}\sum_{u\in A^{c}}\|u-v\|^{-\alpha}\leq\sum_{v\in
A}R_{|B|}(\alpha)=|A|R_{|B|}(\alpha).$ (2.9)

By symmetry the upper bound $|B|R_{|A|}(\alpha)$ also holds. The final upper
bound (with $B=A^{c}$) directly follows since $R_{k}(\alpha)$ is increasing in
$k$.

For the lower bound we observe that for each $v\in A$,

$\sum_{u\in B}\|u-v\|^{-\alpha}=R_{n}(\alpha)-\sum_{\begin{subarray}{c}u\in
B^{c}\\\ u\neq v\end{subarray}}\|u-v\|^{-\alpha}\geq
R_{n}(\alpha)-R_{|B^{c}|}(\alpha),$ (2.10)

where we again use that the sum over $u\in B^{c},u\neq v$, is at most the sum
of the inverted distances of the $|B^{c}|$ many closest vertices to $v$. Since
the lower bound does not depend on $v$, we obtain the first argument of the
maximum. Symmetry again concludes the proof. ∎

###  2.2 Elementary results on branching processes

Let us start by defining a Yule process:

######  Definition 2.3 (Yule process).

Let $\lambda>0$. We define a rate-$\lambda$ _Yule process_
$(\mathcal{Y}(t))_{t\geq 0}$ as a branching process in continuous time,
started from a root $\varnothing$ at time $t=0$. Each individual born into the
system $($including $\varnothing)$ has an associated i.i.d. exponential clock
of rate $\lambda$. Every time the clock of an individual rings, it gives birth
to a child $($receiving its own associated exponential clock$)$ and resets its
clock. Children are connected to their parent by an edge.

The following is a useful lemma on the size of a Yule process. To this end,
recall the distribution of a geometric random variable $X$ from (1.7):
$X\sim\mathrm{Geo}(p)$, when it has distribution
$\mathbb{P}\left(X=k\right)=p(1-p)^{k-1},k\in\mathbb{N}$.

######  Lemma 2.4.

Fix $\lambda>0$ and consider a rate-$\lambda$ Yule process
$(\mathrm{BP}(t))_{t\geq 0}$.

  1. 1.

For any $t\geq 0$, it holds that
$|\mathrm{BP}(t)|\sim\mathrm{Geo}(\mathrm{e}^{-\lambda t})$.

  2. 2.

It holds that

$\displaystyle\lim_{A\to\infty}\mathbb{P}\left(\\{\sup_{t>0}|\mathrm{BP}(t)|\mathrm{e}^{-\lambda
t}\leq A\\}\cap\\{\inf_{t>0}|\mathrm{BP}(t)|\mathrm{e}^{-\lambda t}\geq
A^{-1}\\}\right)=1.$

We do not prove Lemma 2.4 (1); it is quite standard, and we refer the reader
to [27, Section 3.4.1] for a proof.

###### Proof of Lemma 2.4 (2).

Since a Yule process of rate $\lambda$, evaluated at time $t$, is equal in
distribution to a Yule process of rate $1$ evaluated at time $\lambda t$, we
can without of generality set $\lambda=1$ in the proof. It is well known that,
with $E\sim\mathrm{Exp}(1)$,

$|\mathrm{BP}(t)|\mathrm{e}^{-t}\overset{\mathrm{a.s.}}{\longrightarrow}E,$
(2.11)

see e.g. [27, Exercise 3.16]. Moreover, we have by $(1)$ that
$\mathbb{E}\left[|\mathrm{BP}(t)|\right]=\mathrm{e}^{t}$ for any $t\geq 0$. We
use both results in the following. First, we can bound

$\mathbb{P}\left(\sup_{t\geq
0}|\mathrm{BP}(t)|\mathrm{e}^{-t}>A\right)\leq\mathbb{P}\left(\sup_{t\leq\log(A)/2}|\mathrm{BP}(t)|\mathrm{e}^{-t}>A\right)+\mathbb{P}\left(\sup_{t\geq\log(A)/2}|\mathrm{BP}(t)|\mathrm{e}^{-t}>A\right).$
(2.12)

For the first probability, we bound the exponential term from above by $1$ and
observe that $|\mathrm{BP}(t)|$ is monotone increasing in $t$. As a result,
using Markov’s inequality yields

$\mathbb{P}\left(\sup_{t\leq\log(A)/2}|\mathrm{BP}(t)|\mathrm{e}^{-t}>A\right)\leq
A^{-1}\mathbb{E}\left[|\mathrm{BP}(\log(A)/2)|\right]=A^{-1/2}.$ (2.13)

For the second probability, we use the almost sure convergence in (2.11) to
obtain the upper bound

$\mathbb{P}\left(\Big{|}\sup_{t\geq\log(A)/2}\\!\\!\\!\\!|\mathrm{BP}(t)|\mathrm{e}^{-t}-E\Big{|}>1\right)+\mathbb{P}\left(E>A-1\right)=\mathbb{P}\left(\Big{|}\sup_{t\geq\log(A)/2}\\!\\!\\!\\!|\mathrm{BP}(t)|\mathrm{e}^{-t}-E\Big{|}>1\right)+\mathrm{e}^{1-A}.$
(2.14)

The first term vanishes with $A$ by (2.11). Thus, for any $\epsilon>0$ we can
choose $A$ sufficiently large such that the probability on the right-hand side
is at most $\epsilon/6$, as well as $\mathrm{e}^{1-A}<\epsilon/6$ and
$A^{-1/2}<\epsilon/6$, to arrive at the desired bound

$\mathbb{P}\left(\sup_{t\geq
0}|\mathrm{BP}(t)|\mathrm{e}^{-t}>A\right)<\epsilon/2.$ (2.15)

For the infimum, we first observe that
$|\mathrm{BP}(t)|\mathrm{e}^{-t}\geq\mathrm{e}^{-t}>A^{-1}$ for any $t<\log
A$. Hence,

$\mathbb{P}\left(\inf_{t\geq
0}|\mathrm{BP}(t)|\mathrm{e}^{-t}<A^{-1}\right)=\mathbb{P}\left(\inf_{t\geq\log
A}|\mathrm{BP}(t)|\mathrm{e}^{-t}<A^{-1}\right).$ (2.16)

Similar calculations as for the supremum yield that for any $\epsilon>0$, we
can choose $A=A(\epsilon)$ sufficiently large such that the right-hand side is
at most $\epsilon/2$. Combined with the bound for the supremum and since
$\epsilon$ is arbitrary, we obtain the desired result. ∎

##  3 Coupling the infection growths to Yule processes

In this section, we introduce and analyse a coupling between the LRC process
and two Yule processes, that underpins the analysis carried out in the
upcoming sections. This coupling is inspired by the work of Bhamidi and Van
der Hofstad [11], where a similar coupling for first-passage percolation (FPP)
on the complete graph is used to determine precise results regarding the
typical distance between vertices and the hopcount (the number of edges on the
edge-weighted shortest path between two vertices). Here, we extend the
definition coupling to a _spatial setting_ with two _different_ FPP processes.
Moreover, we extend the analysis of the coupling beyond the point when the
coupling is no longer exact, and instead derive quantitative bounds on the
defect made in the coupling.

We rigorously define the coupling in Section 3.1 and carry out the analysis of
the coupling defect in Section 3.2.

###  3.1 Coupling definition

Let us use the notation $\mathcal{K}_{n}$ for the embedding of the complete
graph obtained by taking the torus $\mathbb{T}_{n}^{d}$ and adding all the
non-nearest-neighbor edges, where we associate to each edge $e$ a pair of
exponential random variables $E_{e}^{(1)}$ and $E_{e}^{(2)}$, with means
$R_{n}^{(1)}\|e\|^{\alpha_{1}}$ and
$R_{n}^{(1)}\lambda^{-1}\|e\|^{\alpha_{2}}$, respectively. We think of the
infections starting from the initial vertices $v^{(1)}$ and $v^{(2)}$ as the
flow of two types of fluid from these vertices, say Type-$1$ and Type-$2$.
Here, we think of the variable $E_{e}^{(i)}$ as the time it takes for the
Type-$i$ flow to traverse the edge $e$ once it reaches either of the vertices
incident to $e$. We continue the process until the time that all vertices have
been reached by the two fluids. At this time, letting $\mathcal{N}_{n}^{(i)}$
denote the set of vertices that has been reached by the Type-$i$ fluid, it is
immediate to see that $|\mathcal{N}_{n}^{(i)}|$ has the same distribution as
$N_{n}^{(i)}$ for $i\in\\{1,2\\}$.

To study the sets $\mathcal{N}_{n}^{(1)}$ and $\mathcal{N}_{n}^{(2)}$, let us
first write $\mathcal{N}_{n}^{(i)}(t)$ for the set of vertices in
$\mathcal{K}_{n}$, reached by the Type-$i$ fluid at time $t\geq 0$. We couple
the LRC processes $(\mathcal{N}_{n}^{(1)}(t))_{t\geq 0}$ and
$(\mathcal{N}_{n}^{(2)}(t))_{t\geq 0}$ to two independent Yule processes
$(\mathrm{BP}^{(1)}(t))_{t\geq 0}$ and $(\mathrm{BP}^{(2)}(t))_{t\geq 0}$ with
intensities $1$ and $Z_{n}$, respectively. The coupling is such that the total
population sizes of the branching processes dominate the individual growths of
the infections at all time, i.e.
$\mathcal{N}_{n}^{(i)}(t)\subseteq\mathrm{BP}^{(i)}(t)$ for $i\in\\{1,2\\}$
and all $t\geq 0$.

To rigorously set up this coupling, we need the following randomization
ingredients:

  * a.

The complete graph $\mathcal{K}_{n}$ with random edge weights
$(E_{e}^{(1)},E_{e}^{(2)})$ for each edge $e$.

  * b.

For each $x\in\mathcal{K}_{n}$, an infinite i.i.d. sequence
$\mathbf{A}_{x}=(A_{x}^{j})_{j\in\mathbb{N}_{0}}$, where, for each
$j\in\mathbb{N}_{0}$ and $x\in\mathcal{K}_{n}$, $A_{x}^{j}$ is an i.i.d.
random vector of length $n-1$ with independent coordinates, which correspond
to the edges incident to $x$. We think of the coordinates of these vectors as
being indexed by the vertices in $\mathcal{K}_{n}\setminus\\{x\\}$, where each
coordinate represents the tuple of random variables associated to the
corresponding edge emanating from $x$. That is, $A_{x}^{j}(y)$ denotes a tuple
of random variables associated with the edge $\\{x,y\\}$, with
$y\in\mathcal{K}_{n}\setminus\\{x\\}$. This tuple consists of independent
copies $E_{(x,y)}^{(1),j}$ and $E_{(x,y)}^{(2),j}$ of the exponential random
variables $E_{(x,y)}^{(1)}$ and $E_{(x,y)}^{(2)}$, with means
$R_{n}^{(1)}\|x-y\|^{\alpha_{1}}$ and
$R_{n}^{(1)}\lambda^{-1}\|x-y\|^{\alpha_{2}}$, respectively, as described
above. Further, for fixed $x,y\in\mathcal{K}_{n}$, the collection
$(A_{x}^{j}(y))_{j\in\mathbb{N}_{0}}$ is an i.i.d. collection, and the
collection $\mathbf{A}=(\mathbf{A}_{x})_{x\in\mathcal{K}_{n}}$ is an
independent collection among the vertices $x$.

  * c.

For each $x\in\mathcal{K}_{n}$, an infinite i.i.d. sequence of rate-one Yule
processes $\mathrm{BP}_{x}^{(1)}=(\mathrm{BP}_{x,j}^{(1)})_{j\in\mathbb{N}}$
and a sequence of rate-$Z_{n}$ Yule processes
$\mathrm{BP}_{x}^{(2)}=(\mathrm{BP}_{x,j}^{(2)})_{j\in\mathbb{N}}$. Further,
the collection
$\mathrm{BP}=(\mathrm{BP}_{x}^{(1)},\mathrm{BP}_{x}^{(2)})_{x\in\mathcal{K}_{n}}$
is an independent collection.

Roughly, one should have the following idea in mind. In the FPP process, an
infection cannot travel from an infected vertex to another infected vertex
simply because the latter is already infected. This is called _depletion_ and
causes the rate at which the infection reaches uninfected vertices from some
fixed infected vertex to decrease over time. To couple the infection growth to
a branching process, we counter the depletion by creating artificial ‘proxy’
vertices and edges for already infected vertices, to which the infection can
travel through these artificial edges from infected vertices. This involves
augmenting the network with these extra vertices and edges throughout the
processes, which yields a domination of the total size of the infection growth
in the FPP process, which is exactly described by the total population size of
a Yule process. Keeping this heuristic in mind, we recall the tuple
$A_{x}^{j}(y)=(E^{(1),j}_{(x,y)},E^{(2),j}_{(x,y)})$. One should think of
$E^{(i),j}_{(x,y)}$ as the time it takes for the Type-$i$ flow to traverse the
edge $\\{x,y\\}$, when it enters this edge for the $j^{\mathrm{t}h}$ time
(using a ‘real’ edge if $j=0$, or using an ‘artificial’ edge if $j>0$).

We now describe this intuitive idea in a rigorous manner. To this end, recall
that $\mathcal{N}_{n}^{(i)}(t)$ is the set of vertices first reached by the
Type-$i$ fluid within time $t$.

##### Initialization step.

We describe the initialisation of the process starting from two vertices
$v^{(1)}$ and $v^{(2)}$. First, we remove the edge $\\{v^{(1)},v^{(2)}\\}$,
since the vertices $v^{(1)}$ and $v^{(2)}$ are already infected, so that the
fluids cannot flow along this edge to reach the other vertex. We also create
two _inactive artificial vertices_ $v^{(1)}_{1}$ and $v^{(2)}_{1}$, and create
_artificial edges_ between $v^{(i)}_{1}$ and all vertices in
$\mathcal{K}_{n}\setminus\\{v^{(i)}\\}$, for $i\in\\{1,2\\}$. The removal of
the edge $\\{v^{(1)},v^{(2)}\\}$ and construction of the two artificial
vertices and edges is to ensure that the flows from vertices $v^{(1)}$ and
$v^{(2)}$ occur at rates exactly equal to $1$ and $Z_{n}$, respectively, which
we achieve by associating appropriate edge weights to the edges. Further,
through the creation of these artificial edges, it is possible for the
Type-$i$ flow from $v^{(i)}$ to traverse to a ‘proxy’ $v^{(3-i)}_{1}$ of the
‘original’ vertex $v^{(3-i)}$, for $i\in\\{1,2\\}$.

Let us now describe the edge weights. To all edges $\\{x,y\\}$ with $x\neq
y\in\mathcal{K}_{n}$ (but not $x=v^{(1)},y=v^{(2)}$ or vice versa) we assign
tuples $A_{x}^{0}(y)=(E^{(1),0}_{(x,y)},E^{(2),0}_{(x,y)})$. The first (resp.
second) argument of the tuple encodes the time it takes the Type-$1$ (resp.
Type-$2$) flow to traverse the edge $\\{x,y\\}$ once it reaches either $x$ or
$y$. For $y\in\mathcal{K}_{n}\setminus\\{v^{(1)},v^{(2)}\\}$, we assign to
each _artificial_ edge $\\{v^{(i)}_{1},y\\}$ a tuple
$A_{y}^{1}(v^{(i)})=(E^{(1),1}_{(y,v^{(i)})},E^{(2),1}_{(y,v^{(i)})})$, for
$i\in\\{1,2\\}$. Here, the first (resp. second) argument of the tuple encodes
the time it takes the Type-$1$ (resp. Type-$2$) flow to traverse the
artificial edge $\\{y,v^{(i)}_{1}\\}$ once it reaches $y$. Finally, we assign
the artificial edges $\\{v^{(1)},v^{(2)}_{1}\\}$ and
$\\{v^{(2)},v^{(1)}_{1}\\}$ the weights $E^{(1),1}_{(v^{(1)},v^{(2)})}$ and
$E^{(2),1}_{(v^{(2)},v^{(1)})}$, respectively. We assign the other artificial
edges a tuple of edge weights, as we do not yet know whether the Type-$1$ or
Type-$2$ flow will infect the real vertex incident to this edge. On the other
hand, we already know that $v^{(1)}$ and $v^{(2)}$ are infected by the
Type-$1$ and Type-$2$ infection, respectively. As such, we need only assign
the artificial edges $\\{v^{(1)},v^{(2)}_{1}\\}$ and
$\\{v^{(2)},v^{(1)}_{1}\\}$ the weights $E^{(1),1}_{(v^{(1)},v^{(2)})}$ and
$E^{(2),1}_{(v^{(2)},v^{(1)})}$, respectively.

Now, the Type-$i$ fluid starts to flow from vertex $v^{(i)}$ along the edges
and artificial edges incident to $v^{(i)}$, for both $i\in\\{1,2\\}$. The
initialisation of the process is displayed in the top left part of Figure 1.

Figure 1: From the top left, moving clockwise: $(1)$ Initialisation of the
process. The blue vertex $v^{(1)}$ is Type-$1$, the red vertex $v^{(2)}$ is
Type-$2$. The edge $\\{v^{(1)},v^{(2)}\\}$ is removed, inactive artificial
vertices $v^{(1)}_{1},v^{(2)}_{1}$ are created (dotted line denotes the link
between a vertex and its artificial copy), and (dashed) artificial edges
$\\{v^{(1)},v^{(2)}_{1}\\}$ and $\\{v^{(2)},v^{(1)}_{1}\\}$ are created (other
artificial edges, which are not used by the infection from the start, are
omitted for clarity). $(2)$ The infections have spread and the Type-$2$
infection has reached vertex $x$ at time $T_{x}$. $(3)$ We create an inactive
artificial vertex $x_{1}$, and connect it to the other infected vertices
$v^{(1)}$ and $v^{(2)}$ by artificial edges (again, unused artificial edges
are omitted for clarity). $(4)$ The edge $\\{v^{(1)},x\\}$ is removed and the
flow from $v^{(1)}$ to $x$ is redirected along the artificial edge
$\\{v^{(1)},x_{1}\\}$.

##### Creation of inactive artificial vertices and edges.

We now proceed to describe how to continue this process when the fluid of one
of the two types reach a vertex $x\in\mathcal{K}_{n}$. In particular, we do
not yet discuss what happens when the fluid reaches an _inactive artificial
vertex_.

Consider a vertex $x\in\mathcal{K}_{n}$ which is reached by the Type-$i$ flow
at time $T_{x}$, say. Let $a_{0}(x)$ be the vertex in
$\mathcal{N}_{n}^{(i)}(T_{x})$ such that the Type-$i$ flow used the edge
$\\{a_{0}(x),x\\}$ to reach $x$. At this time instance, we do the following:

  * 1.

Remove all edges $\\{x,y\\}$ with
$y\in\mathcal{N}_{n}^{(1)}(T_{x})\cup\mathcal{N}_{n}^{(2)}(T_{x})$, _except_
for the edge $\\{a_{0}(x),x\\}$.

  * 2.

Create an artificial vertex $x_{1}$.

  * 3.

Add artificial edges from $x_{1}$ to all the vertices in the set
$\mathcal{N}_{n}^{(i)}(T_{x})\setminus\\{x\\}$. For each
$y\in\mathcal{N}_{n}^{(i)}(T_{x})\setminus\\{x\\}$, assign an edge weight
$E^{(i),1}_{(x,y)}+(T_{x}-T_{y})$ to the artificial edge $\\{x_{1},y\\}$,
where $T_{y}$ denotes the time the Type-$i$ fluid reached $y$. We think of the
Type-$i$ flow to have already travelled a distance of $T_{x}-T_{y}$ on the
newly created artificial edge $\\{x_{1},y\\}$, and say that
$E^{(i),1}_{(x,y)}$ is the _remaining_ edge weight. By the _memoryless
property_ of the exponential distribution, conditionally on having travelled a
distance $T_{x}-T_{y}$ on the edge $\\{x_{1},y\\}$, the remaining edge weight
is an exponential random variable $E^{(i),1}_{(x,y)}$.

  * 4.

Add artificial edges from $x_{1}$ to all the vertices in the set
$\mathcal{N}_{n}^{(3-i)}(T_{x})$. For each
$y\in\mathcal{N}_{n}^{(3-i)}(T_{x})$, we assign the artificial edge
$\\{x_{1},y\\}$ an edge weight $E^{(3-i),1}_{(x,y)}+(T_{x}-T_{y})$, where
$T_{y}$ denotes the time the Type-$(3-i)$ fluid reached $y$. We think of the
Type-$(3-i)$ flow to have already travelled a distance of $T_{x}-T_{y}$ on the
newly created artificial edge $\\{x_{1},y\\}$, and say that
$E^{(3-i),1}_{(x,y)}$ is the _remaining_ edge weight.

  * 5.

Add artificial edges from $x_{1}$ to all the vertices in the set
$\mathcal{K}_{n}\setminus\big{(}\mathcal{N}_{n}^{(1)}(T_{x})\cup\mathcal{N}_{n}^{(2)}(T_{x})\big{)}$.
For each
$y\notin\mathcal{N}_{n}^{(1)}(T_{x})\cup\mathcal{N}_{n}^{(2)}(T_{x})$, we
assign the tuple $A_{x}^{1}(y)=(E^{(1),1}_{(x,y)},E^{(2),1}_{(x,y)})$ to the
artificial edge $\\{x_{1},y\\}$. We think of the Type-$1$ and Type-$2$ flows
as not yet having entered this edge. Further, if the Type-$i$ flow enters this
edge, it needs $E^{(i),1}_{(x,y)}$ time to traverse it.

All these additional artificial vertices and edges are now considered part of
the graph, and the Type-$i$ flow can now start to flow along the artificial
edges incident to the vertices in $\mathcal{N}_{n}^{(i)}(T_{x})$, for
$i\in\\{1,2\\}$. As before, only the Type-$i$ flow can access the artificial
edges between $x_{1}$ and vertices in
$\mathcal{N}_{n}^{(i)}(T_{x})\setminus\\{x\\}$, for $i\in\\{1,2\\}$. The
creation of inactive artificial vertices and edges when vertices are infected
is displayed in Figures 1 and 2.

Figure 2: From left to right: $(1)$ Starting from the final set-up in Figure
1, the flows continue. $(2)$ The Type-$1$ infection reaches vertex $z$ at time
$T_{z}$. An inactive artificial vertex $z_{1}$ is created and connected by
artificial edges to $v^{(1)},v^{(2)}$, and $x$ (unused artificial edges are
omitted for clarity). Moreover, the edges $\\{v^{(2)},z\\}$ and $\\{x,z\\}$
are removed and the flow from $v^{(2)}$ and $x$ to $z$ is redirected along the
artificial edges $\\{v^{(2)},z_{1}\\}$ and $\\{x,z_{1}\\}$, respectively.

##### Activation of artificial vertices.

We call a newly created artificial vertex _inactive_ at the time of its
creation. Eventually, when one of the two flows reaches the artificial vertex
using an artificial edge, it gets _activated_. Keeping track of artificial
vertices and activating them is important, as this allows us to precisely
describe the _defect_ of the coupling between the LRC process and two Yule
processes. Let us describe the activation steps in more detail now.

Suppose that the Type-$i$ flow reaches an inactive artificial vertex $z_{j}$,
for some $j\in\mathbb{N}$, and let $T_{z_{j}}$ denote the time at which this
occurs. We now _activate_ the artificial vertex $z_{j}$. We observe that, when
no inactive artificial vertex has been activated yet, all the inactive
artificial vertices are indexed by $j=1$ (as in the description of the
creation of inactive artificial vertices above). Furthermore, we let
$a_{j}(z)\in\mathcal{N}_{n}^{(i)}(T_{z_{j}})$ denote the vertex such that the
Type-$i$ flow used edge $\\{a_{j}(z),z_{j}\\}$ to reach $z_{j}$. At this time
instance, we do the following:

  * 1.

Remove all edges incident to $z_{j}$, _except_ for the edge
$\\{a_{j}(z),z_{j}\\}$.

  * 2.

Start a Yule process $(\mathrm{BP}^{(i)}_{z,j}(t-T_{z_{j}}))_{t\geq
T_{z_{j}}}$ with $z_{j}$ as its root, and say that the artificial vertex
$z_{j}$ has been _activated_. The vertices born in this Yule process have no
relation with the flows on $\mathcal{K}_{n}$ and any other inactive artificial
vertices. We call the vertices born in $\mathrm{BP}^{(i)}_{z,j}(t-T_{z_{j}})$
the _descendants_ of $z_{j}$ at time $t$. In particular, we call any such
descendant born in a Yule process due to the Type-$i$ fluid activating an
inactive artificial vertex, a _Type- $i$ descendant_, and their root, the
artificial vertex $z_{j}$ which just got activated, a _Type- $i$ root_.

  * 3.

Create a new inactive artificial vertex $z_{j+1}$.

  * 4.

Add artificial edges from $z_{j+1}$ to all the vertices in the set
$\mathcal{N}_{n}^{(i)}(T_{z_{j}})\setminus\\{z\\}$. For each
$y\in\mathcal{N}_{n}^{(i)}(T_{z_{j}})\setminus\\{z\\}$, we assign an edge
weight $E^{(i),j+1}_{(x,y)}+(T_{z_{j}}-T_{y})$ to the artificial edge
$\\{y,z_{j}\\}$, where $T_{y}$ denotes the time the Type-$i$ fluid reached
$y$. We think of the Type-$i$ flow to have already travelled a distance of
$T_{z_{j}}-T_{y}$ on this edge, and say that $E^{(i),j+1}_{(x,y)}$ is the
_remaining_ edge weight.

  * 5.

Add artificial edges from $z_{j+1}$ to all the vertices in the set
$\mathcal{N}_{n}^{(3-i)}(T_{z_{j}})$. For each
$y\in\mathcal{N}_{n}^{(3-i)}(T_{z_{j}})$, we assign an edge weight
$E^{(3-i),j+1}_{(x,y)}+(T_{z_{j}}-T_{y})$ to the artificial edge
$\\{y,z_{j}\\}$. We think of the Type-$(3-i)$ flow to have already travelled a
distance of $T_{z_{j}}-T_{y}$ on this edge.

  * 6.

Add artificial edges from $z_{j+1}$ to all the vertices in the set
$\mathcal{K}_{n}\setminus\big{(}\mathcal{N}_{n}^{(1)}(T_{z_{j}})\cup\mathcal{N}_{n}^{(2)}(T_{z_{j}})\big{)}$.
For each
$y\in\mathcal{K}_{n}\setminus\big{(}\mathcal{N}_{n}^{(1)}(T_{z_{j}})\cup\mathcal{N}_{n}^{(2)}(T_{z_{j}})\big{)}$,
we assign the tuple $A_{z}^{j+1}(y)=(E^{(1),j+1}_{(z,y)},E^{(2),j+1}_{(z,y)})$
to the artificial edge $\\{y,z_{j}\\}$. We think of the Type-$1$ and Type-$2$
flows as not yet having entered this edge. Further, if the Type-$i$ flow first
enters this edge, it needs $E^{(i),j+1}_{(z,y)}$ time to traverse it.

All these extra artificial vertices and edges are now considered part of the
graph, and the Type-$i$ flow can now start to flow along the artificial edges
incident to the vertices in $\mathcal{N}_{n}^{(i)}(T_{z_{j}})$, for
$i\in\\{1,2\\}$. The activation of artificial vertices by one of the flows
reaching an inactive artificial vertex is displayed in Figure 3.

Figure 3: From the top left, moving clockwise: $(1)$ Starting from the final
set-up in Figure 2, the flows continue. $(2)$ The Type-$1$ infection reaches
the inactive artificial vertex $z_{1}$ at time $T_{z_{1}}$; the inactive
artificial vertex $z_{1}$ is activated and becomes a Type-$1$ root. A rate-one
Yule process $\mathrm{BP}^{(1)}_{z,1}$, rooted at $z_{1}$, is started at time
$T_{z_{1}}$, and an inactive artificial vertex $z_{2}$ is created. $(3)$ The
inactive artificial vertex $z_{2}$ is connected by artificial edges to
$v^{(1)},v^{(2)}$, and $x$ (unused artificial edges are omitted for clarity).
Moreover, the artificial edges $\\{v^{(2)},z_{1}\\}$ and $\\{x,z_{1}\\}$ are
removed and the flow from $v^{(2)}$ and $x$ to $z_{1}$ is redirected along the
artificial edges $\\{v^{(2)},z_{2}\\}$ and $\\{x,z_{2}\\}$, respectively.

We let $(\mathrm{BP}^{(i)}(t))_{t\geq 0}$ denote the process that consists of
the Type-$i$ flow, the Type-$i$ roots, and Type-$i$ descendants, for
$i\in\\{1,2\\}$. Moreover, let $\mathcal{R}^{(i)}(t)$ and
$\mathcal{D}^{(i)}(t)$ denote the set of Type-$i$ roots and the set of
Type-$i$ descendants at time $t$, respectively. We thus have

$\mathrm{BP}^{(i)}(t)=\mathcal{N}_{n}^{(i)}(t)\cup\mathcal{R}^{(i)}(t)\cup\mathcal{D}^{(i)}(t),\qquad
t\geq 0,i\in\\{1,2\\}.$

Let us recall the standard _memoryless_ property of the exponential
distribution. For two i.i.d. exponential random variables $E,E^{\prime}$,
conditionally on the event $\\{E>t\\}$, $E$ has the same distribution as
$t+E^{\prime}$. Using this fact, and from the definition of the coupling, the
following result follows.

######  Proposition 3.1.

  * 1.

$(\mathrm{BP}^{(i)}(t))_{t\geq 0}$ is equal in distribution to a Yule process
with rate $1$ if $i=1$ or $Z_{n}$ if $i=2$.

  * 2.

$(\mathrm{BP}^{(1)}(t))_{t\geq 0}$ and $(\mathrm{BP}^{(2)}(t))_{t\geq 0}$ are
independent.

###### Proof.

Fix $t\geq 0$ and condition on
$\mathcal{N}_{n}^{(1)}(t),\mathcal{R}^{(1)}(t),\mathcal{D}^{(1)}(t)$, as well
as $\mathcal{N}_{n}^{(2)}(t)$. Each real vertex $v\in\mathcal{N}_{n}^{(1)}(t)$
is infected with the Type-$1$ infection, and spreads this to the uninfected
real vertices in $(\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t))^{c}$
at rate

$\frac{1}{R_{n}^{(1)}}\sum_{u\in(\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t))^{c}}\|u-v\|^{-\alpha_{1}}.$
(3.1)

Furthermore, for each infected vertex
$v^{\prime}\in\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t)$ with
$v^{\prime}\neq v$, there exists a unique artificial vertex $v^{\prime}_{j}$
for some $j\in\mathbb{N}$ such that $v$ is connected to $v^{\prime}_{j}$ and
spreads to $v^{\prime}_{j}$ at rate
$(R_{n}^{(1)})^{-1}\|v^{\prime}-v\|^{-\alpha_{1}}$. As a result, by the
definition $R_{n}^{(1)}$ in (1.1), the rate at which $v$ spreads the Type-$1$
infection equals

$\frac{1}{R_{n}^{(1)}}\sum_{u\in(\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t))^{c}}\|u-v\|^{-\alpha_{1}}+\frac{1}{R_{n}^{(1)}}\sum_{\begin{subarray}{c}v^{\prime}\in\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t)\\\
v^{\prime}\neq
v\end{subarray}}\|v^{\prime}-v\|^{-\alpha_{1}}=\frac{1}{R_{n}^{(1)}}\sum_{\begin{subarray}{c}u\in\mathbb{T}_{n}^{d}\\\
u\neq v\end{subarray}}\|u-v\|^{-\alpha_{1}}=1.$ (3.2)

Note that the sum of the two terms is equal to one, irrespective of which
vertices are in $\mathcal{N}_{n}^{(2)}(t)$, so the rate at which the Type-$1$
infection spreads from $v$ equals one, independently of
$\mathcal{N}_{n}^{(2)}(t)$.

Then, consider the activated artificial vertices, activated by the Type-$1$
flow, and the descendants of Type-$1$ activated artificial vertices in
$\mathcal{R}^{(1)}(t)$ and $\mathcal{D}^{(1)}(t)$, respectively. We know, by
the construction of the coupling, that such vertices give birth to children at
rate $1$, as they are either the root of or a child in a rate-one Yule
process. As all individuals in
$\mathrm{BP}^{(1)}(t)=\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{R}^{(1)}(t)\cup\mathcal{D}^{(1)}(t)$
give birth at rate one for any $t\geq 0$, it thus follows that
$(\mathrm{BP}^{(1)}(t))_{t\geq 0}$ is equal in distribution to a rate-one Yule
process.

For $\mathrm{BP}^{(2)}(t)$, we substitute $\alpha_{2}$ for $\alpha_{1}$ and
introduce a factor $\lambda$ to each rate. Recalling $R_{n}^{(2)}$ from (1.1)
and $Z_{n}$ from (1.5), applying the same argument yields that each real
vertex in $\mathrm{BP}^{(2)}(t)$ spreads the infection at rate $Z_{n}$.
Similarly, each Type-$2$ activated artificial vertex and each descendant of a
Type-$2$ activated artificial vertex in $\mathrm{BP}^{(2)}(t)$ gives birth to
children at rate $Z_{n}$, by definition. As a result,
$(\mathrm{BP}^{(2)}(t))_{t\geq 0}$ is equal in distribution to a rate $Z_{n}$
Yule process, proving $1$.

The independence of $(\mathrm{BP}^{(1)}(t))_{t\geq 0}$ and
$(\mathrm{BP}^{(2)}(t))_{t\geq 0}$ follows from the fact that the infections
spread along disjoint sets of edges, whose transmission times are independent,
and by the rerouting of flow along edges to artificial edges, once vertices
are infected. Finally, as the Yule processes $\mathrm{BP}^{(i)}_{v,j}$ for
$i\in\\{1,2\\},v\in\mathbb{T}_{n}^{d}$, and $j\in\mathbb{N}$, started when
inactive artificial vertices are activated, are also independent, we arrive at
$2$., which concludes the proof. ∎

###  3.2 Tail bounds on the coupling defect

To be able to accurately apply the coupling introduced in the previous
section, it is important to keep track of the _defect_ we incur while coupling
CLR process with the two Yule processes $\mathrm{BP}^{(1)}$ and
$\mathrm{BP}^{(2)}$. That is, we require control of the size of the sets
$\mathcal{D}^{(i)}(t)\cup\mathcal{R}^{(i)}(t)$ for $i\in\\{1,2\\}$ and $t\geq
0$. In this section we provide bounds for this defect, which form the main
technical tool used throughout the proofs of Theorems 1.3 and 1.4. Controlling
this defect is essential to be able to use quantitative results for Yule
processes, like some of the preliminary results stated in Section 2.

Analysis of a similar kind has been done in [11], up to the point when defects
first start to appear in the coupling - an event the authors of [11] call
_collisions_. At this point in time, the defect is of constant order. We move
significantly beyond this, by also controlling the size of the defect when it
is of linear order and show that it is still ‘sufficiently small’ to be of use
for our purposes.

We let $\tau^{(1)}_{0}=\tau^{(2)}_{0}=0$ and, for $k\in\mathbb{N}$, define

$\tau^{(i)}_{k}:=\inf\\{t>\tau^{(i)}_{k-1}:|\mathcal{R}^{(i)}(t)|=k\\}.$ (3.3)

So, the number of Type-$i$ artificial vertices increase by one at times
$\tau_{k}^{(i)}$. Furthermore, let
$(\mathrm{BP}_{j}^{(i)}(\cdot))_{j\in\mathbb{N}}$ be an i.i.d. collection of
Yule processes with rate $1$ if $i=1$ and with rate $Z_{n}$ if $i=2$, and let
us use the convention that $|\mathrm{BP}_{j}^{(i)}(a)|=0$ for any
$j\in\mathbb{N}$ when $a\leq 0$. Then, define the random measure $C^{(i)}$ for
$i\in\\{1,2\\}$ and $0\leq a\leq b<\infty$, such that

$C^{(i)}((a,b])=\sum_{j=1}^{\infty}\mathbbm{1}_{\\{\tau_{j}^{(i)}\in(a,b]\\}}|\mathrm{BP}_{j}^{(i)}(b-\tau_{j}^{(i)})|.$
(3.4)

Observe that the total size $|\mathcal{R}^{(i)}(t)\cup\mathcal{D}^{(i)}(t)|$
of Type-$i$ activated artificial vertices and their descendants at time $t$ is
exactly distributed as $C^{(i)}((0,t])$. That is,

$|\mathcal{R}^{(i)}(t)\cup\mathcal{D}^{(i)}(t)|\overset{d}{=}C^{(i)}((0,t]).$
(3.5)

As such, to show that the Yule process $\mathrm{BP}^{(i)}(t)$ is a good
approximation for the FPP cluster $\mathcal{N}_{n}^{(i)}(t)$ at some time $t$,
is equivalent to showing that $C^{(i)}((0,t])$ is not too big. To this end, we
have the following result:

######  Proposition 3.2.

Fix $\epsilon>0$ and
$\delta\in(0,\inf_{n\in\mathbb{N}}\min\\{1-\alpha_{1}(n)/d,1-\alpha_{2}{(n)}/d\\})$,
and recall $Z_{n}$ from (1.5). If $Z_{n}$ does not converge to one, then for
any $m\in\mathbb{N}$ there exists $N=N(m)\in\mathbb{N}$, such that for all
$n\geq N$,

$\mathbb{P}\left(C^{(1)}\Big{(}\Big{(}0,\frac{\log{n}-m}{Z_{n}}\Big{]}\Big{)}>n\mathrm{e}^{-m(1+\delta)}\right)<\epsilon.$
(3.6)

If $\lim_{n\to\infty}Z_{n}=1$ instead, then there exist
$M=M(\epsilon,\delta)\in\mathbb{N}$ and $N\in\mathbb{N}$, _independent_ of
$M$, such that for all $m\geq M$ and $n\geq N$, the inequality (3.6) holds.
Moreover,

$\mathbb{P}\left(C^{(2)}\Big{(}\Big{(}0,\frac{\log{n}-m}{Z_{n}}\Big{]}\Big{)}>n\mathrm{e}^{-m(1+\delta)}\right)<\epsilon.$
(3.7)

Furthermore, if $c_{n}$ diverges, let $\delta_{n}$ be such that
$\limsup_{n\to\infty}\delta_{n}<1$ and $\delta_{n}\log n$ diverges. Then,

$C^{(1)}\Big{(}\Big{(}0,\frac{(1-\delta_{n})\log
n}{Z_{n}}\Big{]}\Big{)}\mathrm{e}^{-(1-\delta_{n})\log(n)/Z_{n}}\overset{\mathbb{P}}{\longrightarrow}0.$
(3.8)

To prove Proposition 3.2, we first define the $\sigma$-algebra

$\mathcal{F}=\sigma\\{(\mathcal{N}_{n}^{(1)}(s),\mathcal{N}_{n}^{(2)}(s)):0<s<\infty\\}.$
(3.9)

Clearly, $C^{(i)}$ depends on $\mathcal{F}$, as artificial vertices are
activated via the flow from infected real vertices. Towards proving
Proposition 3.2, let us start with an upper bound on the expected value of
$C^{(i)}((a,b])$ for some interval $(a,b]$.

######  Lemma 3.3.

Let $\mathcal{F}$ be the $\sigma$-algebra in (3.9), and recall $C^{(i)}$ from
(3.4). Then, for any $0\leq a\leq b<\infty$ and $i\in\\{1,2\\}$,

$\mathbb{E}\left[C^{(i)}((a,b])\,\Big{|}\,\mathcal{F}\right]=\int_{a}^{b}\Bigg{(}\frac{1}{R_{n}^{(1)}}\sum_{u\in\mathcal{N}_{n}^{(i)}(t)}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t)\\\
v\neq
u\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\mathbbm{1}_{\\{i=1\\}}+\mathbbm{1}_{\\{i=2\\}}\lambda)\|u-v\|^{-\alpha_{i}}\Bigg{)}\mathrm{e}^{b-t}\,\mathrm{d}t.$
(3.10)

###### Proof.

We recall the sequence $(\tau_{j}^{(i)})_{k\in\mathbb{N}}$ defined in (3.3);
the stopping times at which artificial vertices are activated by the Type-$i$
flow. By conditioning on both $\mathcal{F}$ and
$(\tau_{j}^{(i)})_{k\in\mathbb{N}}$, we can use Lemma 2.4 to obtain

$\mathbb{E}\left[C^{(i)}((a,b])\,\Big{|}\,\mathcal{F},(\tau_{j}^{(i)})_{j\in\mathbb{N}}\right]=\sum_{j=1}^{\infty}\mathbbm{1}_{\\{\tau_{j}^{(i)}\in(a,b]\\}}\mathrm{e}^{b-\tau_{j}^{(i)}}.$
(3.11)

We now want to take the expected value with respect to the sequence
$(\tau_{j}^{(i)})_{j\in\mathbb{N}}$. We first define the function
$f_{b}(t):=\mathrm{e}^{b-t}$, and note that

$\sum_{j=1}^{\infty}\mathbbm{1}_{\\{\tau_{j}^{(i)}\in(a,b]\\}}\mathrm{e}^{b-\tau_{j}^{(i)}}=\int_{a}^{b}f_{b}(t)\,\eta^{(i)}(\mathrm{d}t),\quad\text{where}\quad\eta^{(i)}:=\sum_{j=1}^{\infty}\delta_{\tau_{j}^{(i)}},$
(3.12)

with $\delta$ a Dirac measure. Recall the sequence of stopping times
$(T_{k})_{k=2}^{n}$ from (1.3), and let $T_{n+1}=\infty$. We observe that the
sequence $(T_{k})_{k=2}^{n+1}$ is measurable with respect to $\mathcal{F}$.
For any $k\in\\{2,\ldots,n\\}$, the point process $\eta^{(i)}$, restricted to
the interval $[T_{k},T_{k+1})$, is distributed as a Poisson point process
$\mathcal{P}^{(i)}_{k}$, restricted to the interval $[T_{k},T_{k+1})$, with
constant intensity

$\frac{1}{R_{n}^{(1)}}\sum_{u\in\mathcal{N}_{n}^{(i)}(T_{k})}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{n}^{(1)}(T_{k})\cup\mathcal{N}_{n}^{(2)}(T_{k})\\\
v\neq
u\end{subarray}}(\mathbbm{1}_{\\{i=1\\}}+\mathbbm{1}_{\\{i=2\\}}\lambda)\|u-v\|^{-\alpha_{i}}.$
(3.13)

Moreover, as
$[0,T_{3})=[T_{2},T_{3}),[T_{3},T_{4}),\ldots,[T_{n},T_{n+1})=[T_{n},\infty)$
partitions $[0,\infty)$, we can write

$\displaystyle\mathbb{E}\left[C^{(i)}((a,b])\,\Big{|}\,\mathcal{F}\right]$
$\displaystyle=\mathbb{E}\left[\int_{a}^{b}f_{b}(t)\,\eta^{(i)}(\mathrm{d}t)\,\bigg{|}\,\mathcal{F}\right]$
(3.14)
$\displaystyle=\sum_{k=2}^{n+1}\mathbb{E}\left[\int_{[a,b)\cap[T_{k},T_{k+1})}f_{b}(t)\,\eta^{(i)}(\mathrm{d}t)\,\bigg{|}\,\mathcal{F}\right]$
$\displaystyle=\sum_{k=2}^{n+1}\mathbb{E}\left[\int_{[a,b)\cap[T_{k},T_{k+1})}f_{b}(t)\,\mathcal{P}^{(i)}_{k}(\mathrm{d}t)\,\bigg{|}\,\mathcal{F}\right]$

Campbell’s formula [46, Proposition 2.7] then yields that this equals

$\sum_{k=2}^{\infty}\int_{[a,b)\cap[T_{k},T_{k+1})}\Bigg{(}\frac{1}{R_{n}^{(1)}}\sum_{u\in\mathcal{N}_{n}^{(i)}(T_{k})}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{n}^{(1)}(T_{k})\cup\mathcal{N}_{n}^{(2)}(T_{k})\\\
v\neq
u\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\mathbbm{1}_{\\{i=1\\}}+\mathbbm{1}_{\\{i=2\\}}\lambda)\|u-v\|^{-\alpha_{i}}\Bigg{)}f_{b}(t)\,\mathrm{d}t.$
(3.15)

Further, if we consider the quantity

$\frac{1}{R_{n}^{(1)}}\sum_{u\in\mathcal{N}_{n}^{(i)}(t)}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t)\\\
v\neq
u\end{subarray}}(\mathbbm{1}_{\\{i=1\\}}+\mathbbm{1}_{\\{i=2\\}}\lambda)\|u-v\|^{-\alpha_{i}}$
(3.16)

as a function of $t$, then, by definition of the stopping times $T_{k}$, this
function is a constant function on the interval $[T_{k},T_{k+1})$. As a
result, also recalling that $f_{b}(t):=\mathrm{e}^{b-t}$, we can write (3.15)
as

$\displaystyle\sum_{k=2}^{\infty}{}$
$\displaystyle\int_{[a,b)\cap[T_{k},T_{k+1})}\Bigg{(}\frac{1}{R_{n}^{(1)}}\sum_{u\in\mathcal{N}_{n}^{(i)}(t)}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t)\\\
v\neq
u\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\mathbbm{1}_{\\{i=1\\}}+\mathbbm{1}_{\\{i=2\\}}\lambda)\|u-v\|^{-\alpha_{i}}\Bigg{)}\mathrm{e}^{b-t}\,\mathrm{d}t$
(3.17)
$\displaystyle=\int_{a}^{b}\Bigg{(}\frac{1}{R_{n}^{(1)}}\sum_{u\in\mathcal{N}_{n}^{(i)}(t)}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t)\\\
v\neq
u\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\mathbbm{1}_{\\{i=1\\}}+\mathbbm{1}_{\\{i=2\\}}\lambda)\|u-v\|^{-\alpha_{i}}\Bigg{)}\mathrm{e}^{b-t}\,\mathrm{d}t,$

as desired. ∎

We are now ready to prove Proposition 3.2.

###### Proof of Proposition 3.2.

We combine Lemma 3.3 with Lemma 2.2. First, we apply Lemma 2.2, with
$A=\mathcal{N}_{n}^{(i)}(t)$ and
$B=\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t)$, to bound

$\sum_{u\in\mathcal{N}_{n}^{(i)}(t)}\sum_{\begin{subarray}{c}v\in\mathcal{N}_{n}^{(1)}(t)\cup\mathcal{N}_{n}^{(2)}(t)\\\
v\neq
u\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\mathbbm{1}_{\\{i=1\\}}+\mathbbm{1}_{\\{i=2\\}}\lambda)\|u-v\|^{-\alpha_{i}}\leq|\mathcal{N}_{n}^{(i)}(t)|R^{(i)}_{|\mathcal{N}_{n}^{(1)}(t)|+|\mathcal{N}_{n}^{(2)}(t)|},\qquad
i\in\\{1,2\\}.$ (3.18)

We then combine this upper bound with Lemma 3.3 to obtain the upper bound

$\mathbb{E}\left[C^{(i)}((a,b])\,\Big{|}\,\mathcal{F}\right]\leq\int_{a}^{b}\frac{1}{R_{n}^{(1)}}|\mathcal{N}_{n}^{(i)}(t)|R^{(i)}_{|\mathcal{N}_{n}^{(1)}(t)|+|\mathcal{N}_{n}^{(2)}(t)|}\mathrm{e}^{b-t}\,\mathrm{d}t.$
(3.19)

Next, we define for $A>0$ the event

$\mathcal{E}(A):=\\{\sup_{t>0}|\mathcal{N}_{n}^{(1)}(t)|\mathrm{e}^{-t}<A\\}\cap\\{\sup_{t>0}|\mathcal{N}_{n}^{(2)}(t)|\mathrm{e}^{-Z_{n}t}<A\\}.$
(3.20)

First, it follows from Lemma 2.4 for any $\epsilon>0$ and all $A=A(\epsilon)$
sufficiently large, that $\mathbb{P}\left(\mathcal{E}(A)\right)\geq
1-\epsilon/2$, since we have from the coupling between the LRC process and the
Yule processes $\mathrm{BP}^{(1)}(t)$ and $\mathrm{BP}^{(2)}(t)$ that
$|\mathcal{N}_{n}^{(i)}(t)|\leq|\mathrm{BP}^{(i)}(t)|$ for $i\in\\{1,2\\}$,
almost surely. Moreover, since $\mathcal{E}(A)$ is measurable with respect to
the $\sigma$-algebra $\mathcal{F}$, we obtain using Markov’s inequality, with
$\theta>0$ and $A$ large,

$\displaystyle\mathbb{P}\left(C^{(i)}((a,b])\geq\theta\right)$
$\displaystyle\leq\frac{1}{\theta}\mathbb{E}\left[C^{(i)}((a,b])\mathbbm{1}_{\mathcal{E}(A)}\right]+\mathbb{P}\left(\mathcal{E}(A)^{c}\right)$
(3.21)
$\displaystyle\leq\frac{1}{\theta}\mathbb{E}\left[\mathbbm{1}_{\mathcal{E}(A)}\int_{a}^{b}\frac{1}{R_{n}^{(1)}}|\mathcal{N}_{n}^{(i)}(t)|R^{(i)}_{|\mathcal{N}_{n}^{(1)}(t)|+|\mathcal{N}_{n}^{(2)}(t)|}\mathrm{e}^{b-t}\,\mathrm{d}t\right]+\epsilon/2.$

Using the upper bounds in the event $\mathcal{E}(A)$ and then bounding the
indicator from above by one, yields the upper bounds

$\begin{cases}\frac{1}{\theta}\int_{a}^{b}\frac{A}{R_{n}^{(1)}}R^{(1)}_{\lfloor
A(\mathrm{e}^{t}+\mathrm{e}^{Z_{n}t})\rfloor}\mathrm{e}^{b}\,\mathrm{d}t+\epsilon/2&\mbox{if
}i=1,\\\ \frac{1}{\theta}\int_{a}^{b}\frac{A}{R_{n}^{(1)}}R^{(2)}_{\lfloor
A(\mathrm{e}^{t}+\mathrm{e}^{Z_{n}t})\rfloor}\mathrm{e}^{b+(Z_{n}-1)t}\,\mathrm{d}t+\epsilon/2&\mbox{if
}i=2.\end{cases}$ (3.22)

When can then bound

$\underline{c}^{(1)}\leq\frac{R_{j}^{(1)}}{j^{1-\alpha_{1}/d}}\leq\overline{c}^{(1)},\quad\text{and}\quad\underline{c}^{(2)}\leq\frac{R_{j}^{(2)}}{\lambda
j^{1-\alpha_{2}/d}}\leq\overline{c}^{(2)},$ (3.23)

for all $j\in\mathbb{N}$ and for constants
$0<\underline{c}^{(i)}<\overline{c}^{(i)}<\infty$ and for $i\in\\{1,2\\}$, as
follows from Lemma 2.1. We then use this in (3.22) to arrive at, when $i=1$,

$\frac{1}{\theta}\int_{a}^{b}\frac{A^{2-\alpha_{1}/d}\overline{c}^{(1)}}{\underline{c}^{(1)}}n^{-(1-\alpha_{1}/d)}(\mathrm{e}^{t}+\mathrm{e}^{Z_{n}t})^{1-\alpha_{1}/d}\mathrm{e}^{b}\,\mathrm{d}t+\epsilon/2.$
(3.24)

Since $(x+y)^{a}\leq x^{a}+y^{a}$ for all $a\in[0,1]$ and $x,y\geq 0$, we can
further bound this from above, for some constants
$C_{1},C_{2}=C_{1}(A),C_{2}(A)>0$, by

$\displaystyle\frac{1}{\theta}{}$
$\displaystyle\int_{a}^{b}C_{1}n^{-(1-\alpha_{1}/d)}\mathrm{e}^{(1-\alpha_{1}/d)t+b}\,\mathrm{d}t+\frac{1}{\theta}\int_{a}^{b}C_{1}n^{-(1-\alpha_{1}/d)}\mathrm{e}^{(1-\alpha_{1}/d)Z_{n}t+b}\,\mathrm{d}t+\epsilon/2$
(3.25)
$\displaystyle\leq\frac{C_{2}}{\theta}n^{-(1-\alpha_{1}/d)}\mathrm{e}^{(2-\alpha_{1}/d)b}+\frac{C_{2}}{\theta}Z_{n}^{-1}n^{-(1-\alpha_{1}/d)}\mathrm{e}^{(1+(1-\alpha_{1}/d)Z_{n})b}+\epsilon/2$
$\displaystyle=\frac{C_{2}}{\theta}n^{-(1-\alpha_{1}/d)}\mathrm{e}^{(1+(1-\alpha_{1}/d)Z_{n})b}\big{[}\mathrm{e}^{-(1-\alpha_{1}/d)(Z_{n}-1)b}+Z_{n}^{-1}\big{]}+\epsilon/2.$

As $Z_{n}\geq 1$, the term in the square brackets can be bounded from above by
a constant to then lead to

$\mathbb{P}\left(C^{(1)}((a,b])\geq\theta\right)\leq\frac{C_{3}}{\theta}n^{-(1-\alpha_{1}/d)}\mathrm{e}^{(1+(1-\alpha_{1}/d)Z_{n})b}+\epsilon/2,$
(3.26)

for some $C_{3}>C_{2}$. When $i=2$ instead, we write
$R_{n}^{(1)}=R_{n}^{(2)}/Z_{n}$ and use (3.23) in (3.22) to, with a similar
approach, arrive at the upper bound

$\displaystyle\frac{1}{\theta}{}$
$\displaystyle\int_{a}^{b}\frac{A^{2-\alpha_{2}/d}\overline{c}^{(2)}}{\underline{c}^{(2)}}Z_{n}n^{-(1-\alpha_{2}/d)}(\mathrm{e}^{t}+\mathrm{e}^{Z_{n}t})^{1-\alpha_{2}/d}\mathrm{e}^{b+(Z_{n}-1)t}\,\mathrm{d}t+\epsilon/2$
(3.27)
$\displaystyle\leq\frac{1}{\theta}\int_{a}^{b}\widetilde{C}_{1}Z_{n}n^{-(1-\alpha_{2}/d)}\mathrm{e}^{b+(Z_{n}-\alpha_{2}/d)t}\,\mathrm{d}t+\frac{1}{\theta}\int_{a}^{b}\widetilde{C}_{1}Z_{n}n^{-(1-\alpha_{2}/d)}\mathrm{e}^{b+(Z_{n}(2-\alpha_{2}/d)-1)t}\,\mathrm{d}t+\epsilon/2$
$\displaystyle\leq\frac{\widetilde{C}_{1}}{\theta}\frac{Z_{n}}{Z_{n}-\alpha_{2}/d}n^{-(1-\alpha_{2}/d)}\mathrm{e}^{(Z_{n}+1-\alpha_{2}/d)b}+\frac{\widetilde{C}_{1}}{\theta}\frac{Z_{n}}{Z_{n}(2-\alpha_{2}/d)-1}n^{-(1-\alpha_{2}/d)}\mathrm{e}^{Z_{n}(2-\alpha_{2}/d)b}+\epsilon/2,$

where $\widetilde{C}_{1}=\widetilde{C}_{1}(A)>0$ is a constant. We then
observe that both the fractions $\frac{Z_{n}}{Z_{n}-\alpha_{2}/d}$ and
$\frac{Z_{n}}{Z_{n}(2-\alpha_{2}/d)-1}$ are bounded in $n$, irrespective of
the choice of $\alpha_{2}$ and whether $Z_{n}=1+o(1)$ or $Z_{n}$ diverges,
since $\sup_{n\in\mathbb{N}}\alpha_{2}(n)<d$. We can thus omit the fractions
and replace $\widetilde{C}_{1}$ with some large constant
$\widetilde{C}_{2}=\widetilde{C}_{2}(A)>\widetilde{C}_{1}$. We then arrive at
the inequality

$\displaystyle\mathbb{P}\left(C^{(2)}((a,b])\geq\theta\right)$
$\displaystyle\leq\frac{\widetilde{C}_{2}}{\theta}n^{-(1-\alpha_{2}/d)}\mathrm{e}^{(Z_{n}+1-\alpha_{2}/d)b}+\frac{\widetilde{C}_{2}}{\theta}n^{-(1-\alpha_{2}/d)}\mathrm{e}^{Z_{n}(2-\alpha_{2}/d)b}+\epsilon/2$
(3.28)
$\displaystyle=\frac{\widetilde{C}_{2}}{\theta}n^{-(1-\alpha_{2}/d)}\mathrm{e}^{Z_{n}(2-\alpha_{2}/d)b}\big{[}\mathrm{e}^{-(1-\alpha_{2}/d)(Z_{n}-1)b}+1\big{]}+\epsilon/2$
$\displaystyle\leq\frac{\widetilde{C}_{3}}{\theta}n^{-(1-\alpha_{2}/d)}\mathrm{e}^{Z_{n}(2-\alpha_{2}/d)b}+\epsilon/2,$

where we again use that $Z_{n}\geq 1$ to bound the term in the square brackets
from above by a constant.

We now apply the general upper bounds in (3.26) and (3.28) to derive the
desired results. First, we prove (3.6), for which we take $a=0,b=(\log
n-m)/Z_{n}$, and $\theta=n\mathrm{e}^{-m(1+\delta)}$, with $m$ a large
constant to be determined and
$\delta\in(0,\inf_{n\in\mathbb{N}}\min\\{1-\alpha_{1}/d,1-\alpha_{2}/d\\})$.
Applying (3.26) yields

$\displaystyle\mathbb{P}\bigg{(}C^{(1)}\Big{(}\Big{(}0,\frac{\log
n-m}{Z_{n}}\Big{]}\Big{)}\geq n\mathrm{e}^{-m(1+\delta)}\bigg{)}$
$\displaystyle\leq
C_{3}\mathrm{e}^{m(1+\delta)}n^{-(2-\alpha_{1}/d)}\mathrm{e}^{(1+(1-\alpha_{1}/d)Z_{n})(\log
n-m)/Z_{n}}+\epsilon/2$ (3.29)
$\displaystyle=C_{3}\mathrm{e}^{m(1+\delta)-m(1-\alpha_{1}/d)-m/Z_{n}}n^{-1}\mathrm{e}^{\log(n)/Z_{n}}+\epsilon/2.$

We now distinguish between three cases, depending on the behaviour of $Z_{n}$.

#####  $Z_{n}$ converges to one.

As $Z_{n}=1+o(1)$ (that is, $c_{n}/\log n=o(1)$), we have that
$m/Z_{n}=m-o(1)$ and $\log(n)/Z_{n}\leq\log n-c_{n}/2$ for $n\geq N$ and
$N\in\mathbb{N}$ large and fixed, so that for some constant $C_{4}>0$ we
obtain the upper bound

$C_{4}\mathrm{e}^{m(\delta-(1-\alpha_{1}/d))-c_{n}/2}+\epsilon/2\leq
C_{4}\mathrm{e}^{m(\delta-(1-\alpha_{1}/d))}+\epsilon/2.$ (3.30)

By the choice of $\delta$ we can thus let $M=M(\delta,\epsilon)\in\mathbb{N}$
be so large that for all $m\geq M$ the first term is at most $\epsilon/2$ to
arrive at the desired upper bound.

#####  $Z_{n}$ bounded away from one.

We let $\underline{z}:=\liminf_{n\to\infty}Z_{n}>1$ and bound
$\log(n)/Z_{n}\leq 2(1+\underline{z})^{-1}\log n$ for all $n$ sufficiently
large to obtain the upper bound

$\displaystyle
C_{3}\mathrm{e}^{m(1+\delta)-m(1-\alpha_{1}/d)-m/Z_{n}}n^{-1}\mathrm{e}^{\log(n)/Z_{n}}+\epsilon/2$
$\displaystyle\leq
C_{3}\mathrm{e}^{m(1+\delta)}n^{-1+2(1+\underline{z})^{-1}}+\epsilon/2$ (3.31)
$\displaystyle=C_{3}\mathrm{e}^{m(1+\delta)}n^{-(\underline{z}-1)/(\underline{z}+1)}+\epsilon/2.$

For any $m$, this upper bound is smaller than $\epsilon$ for $n\geq N$, where
now $N=N(m)\in\mathbb{N}$ depends on $m$.

#####  $Z_{n}$ converges to one along a sub-sequence.

The positive integers can be partitioned into two sub-sequences
$(n^{1}_{k})_{k\in\mathbb{N}}$ and $(n^{2}_{k})_{k\in\mathbb{N}}$ such that
$Z_{n^{1}_{k}}=1+o(1)$ as $k\to\infty$ and
$\liminf_{k\to\infty}Z_{n^{2}_{k}}>1$. The desired result follows by combining
the approaches of the cases one and two in the sub-sequences
$(n^{1}_{k})_{k\in\mathbb{N}}$ and $(n^{2}_{k})_{k\in\mathbb{N}}$,
respectively.

We then prove (3.7) and let $i=2$. We use (3.28) with $a=0,b=(\log
n-m)/Z_{n}$, and $\theta=n\mathrm{e}^{m(1+\delta)}$, and obtain with similar
calculations, for all $n\geq N$ and some large, fixed $N\in\mathbb{N}$,

$\displaystyle\mathbb{P}\bigg{(}C^{(2)}\Big{(}\Big{(}0,\frac{\log
n-m}{Z_{n}}\Big{]}\Big{)}\geq n\mathrm{e}^{-m(1+\delta)}\bigg{)}$
$\displaystyle\leq\widetilde{C}_{3}\mathrm{e}^{m(1+\delta)}n^{-(2-\alpha_{2}/d)}\mathrm{e}^{(2-\alpha_{2}/d)(\log
n-m)}+\epsilon/2$ (3.32)
$\displaystyle=\widetilde{C}_{3}\mathrm{e}^{m(\delta-(1-\alpha_{2}/d))}+\epsilon/2.$

By the choice of $\delta$ we can thus let $M$ be so large that the first term
is at most $\epsilon/2$ for all $m\geq M$ to arrive at the desired upper
bound, which concludes the proof of (3.7). To then prove (3.8), we use (3.26)
with $a=0,b=(1-\delta_{n})\log(n)/Z_{n}$ and $\theta=\zeta\mathrm{e}^{b}$,
where $\zeta>0$ is fixed, and $(\delta_{n})_{n\in\mathbb{N}}$ is a sequence
such that $\limsup_{n\to\infty}\delta_{n}<1$ and $\delta_{n}\log n$ diverges.
We then obtain

$\displaystyle\mathbb{P}\bigg{(}C^{(1)}\Big{(}\Big{(}0,\frac{(1-\delta_{n})\log
n}{Z_{n}}\Big{]}\Big{)}\geq\zeta\mathrm{e}^{(1-\delta_{n})\log(n)/Z_{n}}\bigg{)}$
$\displaystyle\leq
C_{3}\zeta^{-1}n^{-(1-\alpha_{1}/d)}\mathrm{e}^{(1-\alpha_{1}/d)(1-\delta_{n})\log
n}+\epsilon/2$ (3.33)
$\displaystyle=C_{3}\zeta^{-1}\mathrm{e}^{-(1-\alpha_{1}/d)\delta_{n}\log
n}+\epsilon/2.$

As $1-\alpha_{1}/d$ is bounded away from zero since
$\sup_{n\in\mathbb{N}}\alpha_{1}(n)<d$, it follows that the first term on the
right-hand side converges to zero as $n$ tends to infinity, irrespective of
the choice of $\epsilon$ and $\zeta$. Thus, we obtain

$\limsup_{n\to\infty}\mathbb{P}\bigg{(}C^{(1)}\Big{(}\Big{(}0,\frac{(1-\delta_{n})\log
n}{Z_{n}}\Big{]}\Big{)}\geq\zeta\mathrm{e}^{(1-\delta_{n})\log(n)/Z_{n}}\bigg{)}\leq\epsilon/2.$
(3.34)

However, as the left-hand side does not depend on $\epsilon$, which was
arbitrary, we in fact obtain that the limit of the probability equals zero,
for any $\zeta>0$. This proves (3.8) and concludes the proof.

∎

##  4 Absence of coexistence

In this section, we prove an upper bound on $N_{n}^{(1)}$ when
$c_{n}\to\infty$, implied by the following key proposition:

######  Proposition 4.1.

Fix $d\in\mathbb{N}$, let $\alpha_{1},\alpha_{2}\in[0,d)$ such that
$\sup_{n\in\mathbb{N}}\alpha_{i}(n)<d$ for $i\in\\{1,2\\}$, and recall $Z_{n}$
from (1.5). Let $N_{n}^{(1)}$ denote the size of the Type-$1$ infection of a
LRC process, started from two distinct initial vertices
$v^{(1)},v^{(2)}\in\mathbb{T}_{n}^{d}$. When $c_{n}$ diverges,
$(N_{n}^{(1)}n^{-1/Z_{n}})_{n\in\mathbb{N}}$ is a tight sequence of random
variables.

To prove Proposition 4.1, we divide the competition process in two phases. In
phase $\mathrm{I}$, the Type-$1$ infection grows to a sub-linear size of the
‘correct order’, and the Type-$2$ infection grows linearly large. In phase
$\mathrm{II}$, the Type-$1$ infection only increases its size up to a
multiplicative constant, that is, it stays of the same order, whereas the
Type-$2$ infection is able to infect all the remaining vertices. This
intuitive two-phase approach is outlined more precisely in the following two
lemmas. To this end, we define, for $A,m>0$ and $\xi\in(0,1)$, the following
event:

$\mathcal{E}_{n}(A,m,\xi):=\bigg{\\{}\Big{|}\mathcal{N}_{n}^{(1)}\Big{(}\frac{\log
n-m}{Z_{n}}\Big{)}\Big{|}\leq A\exp\Big{(}\frac{\log
n-m}{Z_{n}}\Big{)}\bigg{\\}}\cap\bigg{\\{}\Big{|}\mathcal{N}_{n}^{(2)}\Big{(}\frac{\log
n-m}{Z_{n}}\Big{)}\Big{|}\geq\xi n\bigg{\\}}.$ (4.1)

We can then formulate the following result, regarding phase $\mathrm{I}$:

######  Lemma 4.2 (Phase $\mathrm{I}$).

Fix $d\in\mathbb{N}$, let $\alpha_{1},\alpha_{2}\in[0,d)$ such that
$\sup_{n\in\mathbb{N}}\alpha_{i}(n)<d$ for $i\in\\{1,2\\}$, and recall $Z_{n}$
from (1.5). When $c_{n}$ diverges, for any $\epsilon>0$ and all fixed $A,m$
sufficiently large and $\xi$ sufficiently small,

$\mathbb{P}\left(\mathcal{E}_{n}(A,m,\xi)\right)\geq 1-\epsilon.$ (4.2)

When $c_{n}$ diverges, it follows that $\log(n)/Z_{n}=\log
n-\mathcal{O}(c_{n}\wedge\log n)$, so that the upper bound on the size of the
Type-$1$ infection in Proposition 4.2 is indeed sub-linear, whereas the
Type-$2$ infection has a linear size by time $(\log n-m)/Z_{n}$. Additionally
we have the following result for Phase $\mathrm{II}$:

######  Lemma 4.3 (Phase $\mathrm{II}$).

Fix $A,m>0$ and $\xi\in(0,1)$. Conditionally on the event
$\mathcal{E}_{n}(A,m,\xi)$, $(N_{n}^{(1)}n^{-1/Z_{n}})_{n\in\mathbb{N}}$ is a
tight sequence of random variables.

Proposition 4.1 then readily follows:

###### Proof of Proposition 4.1 subject to Lemmas 4.2 and 4.3..

Fix $\epsilon>0$. First, choose $A$ and $m$ sufficiently large and $\xi$
sufficiently small so that
$\mathbb{P}\left(\mathcal{E}_{n}(A,m,\xi)^{c}\right)\leq\epsilon/2$, which is
possible by Lemma 4.2. Then, choose $K>0$ sufficiently large so that
$\mathbb{P}\big{(}N_{n}^{(1)}n^{-1/Z_{n}}\geq
K\,\big{|}\,\mathcal{E}_{n}(A,m,\xi)\big{)}\leq\epsilon/2$, which is possible
by Lemma 4.3. It then follows that

$\mathbb{P}\left(N_{n}^{(1)}n^{-1/Z_{n}}\geq
K\right)\leq\mathbb{P}\left(N_{n}^{(1)}n^{-1/Z_{n}}\geq
K\,\Big{|}\,\mathcal{E}_{n}(A,m,\xi)\right)+\mathbb{P}\left(\mathcal{E}_{n}(A,m,\xi)^{c}\right)\leq\epsilon,$
(4.3)

as desired. ∎

It thus remains to prove the two lemmas. The bulk of the work for Lemma 4.2 is
already carried out in Section 3.2, in Proposition 3.2 in particular, so that
we can provide the proof straight away with relative ease. Lemma 4.3 requires
a bit more work, and is based on an approximation of the competition process
by a supermartingale, which we take care of in Section 4.1.

###### Proof of Lemma 4.2.

We set $t_{n}:=(\log n-m)/Z_{n}$ for ease of writing. We prove that the event
$\mathcal{E}_{n}$, as in (4.1), holds with probability at least $1-\epsilon$
in two steps. First, as we can stochastically dominate the Type-$1$ infection
set $\mathcal{N}_{n}^{(1)}(t)$ by the rate-one Yule process
$\mathrm{BP}^{(1)}(t)$ for all $t\geq 0$, we directly obtain for any
$\epsilon>0$ and $A$ sufficiently large, that

$\mathbb{P}\left(|\mathcal{N}_{n}^{(1)}(t_{n})|>A\mathrm{e}^{t_{n}}\right)\leq\mathbb{P}\left(|\mathrm{BP}^{(1)}(t_{n})|>A\mathrm{e}^{t_{n}}\right)\leq\mathbb{P}\left(\sup_{t\geq
0}|\mathrm{BP}^{(1)}(t)|\mathrm{e}^{-t}>A\right)<\epsilon/2,$ (4.4)

where the final step follows from Lemma 2.4. Then, recalling $C^{(2)}$ from
(3.4), we observe that

$|\mathcal{N}_{n}^{(2)}(t_{n})|=|\mathrm{BP}^{(2)}(t_{n})|-C^{(2)}((0,t_{n}]).$
(4.5)

As a result, for
$\delta\in(0,\inf_{n\in\mathbb{N}}\min\\{1-\alpha_{1}/d,1-\alpha_{2}/d\\})$,
the event

$\mathcal{G}_{n}:=\\{|\mathrm{BP}^{(2)}(t_{n})|\geq
m^{-1}\mathrm{e}^{-m}n,C^{(2)}((0,t_{n}])<n\mathrm{e}^{-m(1+\delta)}\\}$ (4.6)

implies that, with $\xi<(m^{-1}-\mathrm{e}^{-\delta m})\mathrm{e}^{-m}$ (where
the right-hand side is positive for $m$ large),

$|\mathcal{N}_{n}^{(2)}(t_{n})|\geq(m^{-1}-\mathrm{e}^{-\delta
m})\mathrm{e}^{-m}n>\xi n.$ (4.7)

The event $\mathcal{G}_{n}$ holds with probability at least $1-\epsilon/2$ for
all $m$ and $n$ large, by Lemma 2.4 (using a similar proof as in (4.4), now
using the infimum rather than the supremum) and Propposition 3.2 (by the
choice of $\delta$). Combining this with the bound in (4.4), we thus arrive at
the desired conclusion. ∎

###  4.1 Phase II: Proof of Lemma 4.3

By Lemma 4.2 we know that the event $\mathcal{E}_{n}(A,m,\xi)$ (as in (4.1))
holds with probability at least $1-\epsilon$, when choosing $A$ and $m$ large
and $\xi$ small. This event tells us that, by time $t_{n}:=(\log n-m)/Z_{n}$,
the Type-$2$ infection has infected at least $k_{n}^{(2)}:=\xi n$ many
vertices and the Type-$1$ infection has infected at most
$k_{n}^{(1)}:=A\exp((\log n-m)/Z_{n})$ many vertices.

In Phase II of the process, we aim to show that the Type-$1$ infection only
reaches a small portion of the remaining number of uninfected vertices.
Heuristically this should be the case, independently of _which_ vertices are
exactly infected so far, due to the overwhelming advantage that the Type-$2$
infection already has.

From here on out, we work conditionally on the event
$\mathcal{E}_{n}(A,m,\xi)$, and omit its arguments for ease of writing.
Furthermore, let
$\ell_{n}:=n-(|\mathcal{N}_{n}^{(1)}(t_{n})|+|\mathcal{N}_{n}^{(2)}(t_{n})|)$
and let $(A_{j})_{j=0}^{\ell_{n}},(B_{j})_{j=0}^{\ell_{n}}$ denote the sets of
Type-$1$ and Type-$2$ vertices when exactly
$|\mathcal{N}_{n}^{(1)}(t_{n})|+|\mathcal{N}_{n}^{(2)}(t_{n})|+j$ many
vertices have been infected. In particular,
$A_{0}=\mathcal{N}_{n}^{(1)}(t_{n})$ and $B_{0}=\mathcal{N}_{n}^{(2)}(t_{n})$.
The evolution of these sets can be described using the discrete analogue of
the competition process, as introduced in Section 1.1.

For ease of writing, we let $S_{j}:=|A_{j}|$, for
$j\in\\{0,\ldots,\ell_{n}\\}$, with $S_{0}=|\mathcal{N}_{n}^{(1)}(t_{n})|\leq
k_{n}^{(1)}$, conditionally on $\mathcal{E}_{n}$. We then have the following
result:

######  Lemma 4.4.

Let $\mathcal{F}_{j}$ denote the $\sigma$-algebra generated by the sets
$(A_{i})_{i=0}^{j}$ and $(B_{i})_{i=0}^{j}$. Then $(M_{j})_{j\geq 0}$, given
by

$M_{j}:=S_{j}\prod_{i=0}^{j-1}\Big{(}1+\frac{R^{(1)}_{|(A_{i}\cup
B_{i})^{c}|}}{|(A_{i}\cup
B_{i})^{c}|(R^{(2)}_{n}-R^{(2)}_{|B_{i}^{c}|})}\Big{)}^{-1},\qquad
j\in\\{0,\ldots,\ell_{n}\\},$ (4.8)

is a supermartingale with respect to $(\mathcal{F}_{j})_{j\geq 0}$.

###### Proof.

We observe that

$\mathbb{E}\left[S_{j+1}\,|\,\mathcal{F}_{j}\right]=S_{j}+\frac{\sum_{v\in
A_{j}}\sum_{u\in(A_{j}\cup B_{j})^{c}}\|u-v\|^{-\alpha_{1}}}{\sum_{v\in
A_{j}}\sum_{u\in(A_{j}\cup B_{j})^{c}}\|u-v\|^{-\alpha_{1}}+\sum_{v\in
B_{j}}\sum_{u\in(A_{j}\cup B_{j})^{c}}\lambda\|u-v\|^{-\alpha_{2}}}.$ (4.9)

We first bound the denominator from below by omitting the first sum. Then,
using Lemma 2.2, we bound the numerator from above by
$|A_{j}|R^{(1)}_{|(A_{j}\cup B_{j})^{c}|}=S_{j}R_{|(A_{j}\cup
B_{j})^{c}|}^{(1)}$ and the remaining term in the denominator from below by
$|(A_{j}\cup B_{j})^{c}|(R_{n}^{(2)}-R_{|B_{j}^{c}|}^{(2)})$. Overall, we
obtain the upper bound

$\mathbb{E}\left[S_{j+1}\,|\,\mathcal{F}_{j}\right]\leq
S_{j}\Big{(}1+\frac{R_{|(A_{j}\cup B_{j})^{c}|}^{(1)}}{|(A_{j}\cup
B_{j})^{c}|(R^{(2)}_{n}-R^{(2)}_{|B_{j}^{c}|})}\Big{)}.$ (4.10)

It then readily follows that $M_{i}$ is a supermartingale, as desired. ∎

With Lemma 4.4 at hand, we can prove Lemma 4.3:

###### Proof of Lemma 4.3.

First, we use the following estimate. Since, conditionally on the event
$\mathcal{E}_{n}$, $|B_{0}^{c}|\leq(1-\xi)n$ and as $|B_{j}^{c}|$ is non-
increasing in $j$, it follows that, uniformly in $j$,

$R_{n}^{(2)}-R_{|B_{j}^{c}|}^{(2)}\geq
R_{n}^{(2)}-R_{|B_{0}^{c}|}^{(2)}\geq\gamma R_{n}^{(2)},$ (4.11)

almost surely, where $\gamma\in(0,1)$ is a constant. We now aim to bound the
expected value of $N_{n}^{(1)}$, conditionally on $\mathcal{E}_{n}$. We write

$\mathbb{E}\left[N_{n}^{(1)}\,\Big{|}\,\mathcal{E}_{n}\right]=\mathbb{E}\left[S_{\ell_{n}}\,|\,\mathcal{E}_{n}\right]=\mathbb{E}\left[M_{\ell_{n}}\prod_{j=0}^{\ell_{n}-1}\Big{(}1+\frac{R^{(1)}_{|(A_{j}\cup
B_{j})^{c}|}}{|(A_{j}\cup
B_{j})^{c}|(R^{(2)}_{n}-R^{(2)}_{|B_{j}^{c}|})}\Big{)}\,\Bigg{|}\,\mathcal{E}_{n}\right].$
(4.12)

By combining the the deterministic lower bound in (4.11), the fact that
$|(A_{j}\cup
B_{j})^{c}|=n-(|\mathcal{N}_{n}^{(1)}(t_{n})|+|\mathcal{N}_{n}^{(2)}(t_{n})|)-j=\ell_{n}-j$,
and that $S_{0}\leq k_{n}^{(1)}$, on $\mathcal{E}_{n}$, together with Lemma
4.4, we obtain

$\mathbb{E}\left[N_{n}^{(1)}\,\Big{|}\,\mathcal{E}_{n}\right]\leq
k_{n}^{(1)}\mathbb{E}\left[\prod_{j=0}^{\ell_{n}-1}\Big{(}1+\frac{1}{\gamma
R^{(2)}_{n}}\frac{R^{(1)}_{\ell_{n}-j}}{\ell_{n}-j}\Big{)}\,\Bigg{|}\,\mathcal{E}_{n}\right]=k_{n}^{(1)}\mathbb{E}\left[\prod_{j=1}^{\ell_{n}}\Big{(}1+\frac{1}{\gamma
R_{n}^{(2)}}\frac{R_{j}^{(1)}}{j}\Big{)}\,\Bigg{|}\,\mathcal{E}_{n}\right].$
(4.13)

Since, conditionally on $\mathcal{E}_{n}$, we have that $\ell_{n}\leq
n-k_{n}^{(2)}\leq(1-\xi)n$, we arrive at the deterministic upper bound

$k_{n}^{(1)}\exp\bigg{(}\frac{1}{\gamma
R^{(2)}_{n}}\sum_{j=1}^{(1-\xi)n}\frac{R^{(1)}_{j}}{j}\bigg{)}\leq
k_{n}^{(1)}\exp\bigg{(}\frac{K_{1}}{Z_{n}n^{1-\alpha_{1}/d}}\sum_{j=1}^{(1-\xi)n}j^{-\alpha_{1}/d}\bigg{)},$
(4.14)

for some constant $K_{1}>0$. In the last step we use the inequalities in Lemma
2.1. Now, we fix $\eta>0$. By bounding the sum from above by an integral, we
obtain, for some constants $K_{2}>0$ and $C>\mathrm{e}^{K_{2}}/\eta$, and
using that $Z_{n}\geq 1$, the upper bound

$\displaystyle\mathbb{P}\left(N_{n}^{(1)}\geq
Ck_{n}^{(1)}\,\Big{|}\,\mathcal{E}_{n}\right)\leq\frac{\mathbb{E}\left[N_{n}^{(1)}\,\Big{|}\,\mathcal{E}_{n}\right]}{Ck_{n}^{(1)}}\leq
C^{-1}\mathrm{e}^{K_{2}/Z_{n}}\leq C^{-1}\mathrm{e}^{K_{2}}<\eta.$

As a result, conditionally on $\mathcal{E}_{n}$, $N_{n}^{(1)}/k_{n}^{(1)}$ is
tight. Since $k_{n}^{(1)}=A\exp((\log n-m)/Z_{n})\leq An^{1/Z_{n}}$, it
follows that $N_{n}^{(1)}n^{-1/Z_{n}}$ is tight, conditionally on
$\mathcal{E}_{n}$, as well, so that the desired result follows. ∎

##  5 Coexistence and asymptotic behaviour of $N_{n}^{(1)}$

In this section, we prove Theorems 1.3 and 1.4. We remark that Proposition 4.1
in Section 4 already implies that the size $N_{n}^{(1)}$ of the Type-$1$
infection is $o(n)$ when $c_{n}$ diverges, which partially proves Theorem 1.3.
We verify this in the proof of Theorem 1.3, in Section 5.3. To obtain the
asymptotic results of Theorems 1.3 (i) and (ii), we improve on Proposition 4.2
in Sections 5.1 and 5.2, respectively, by using the approximation of the
competition process to two Yule processes, as described in Section 3. Though
we already used this to prove Proposition 4.1, we use it in a more refined way
here to prove that $N_{n}^{(1)}$ converges in distribution in these two cases.
Next, we prove a lower bound for $N_{n}^{(1)}$ for Theorems 1.3 (iii) and (iv)
in Section 5.3, that match the upper bound in Proposition 4.1. This section
also contains the proof of Theorem 1.3. Finally, in Section 5.4, we prove
Theorem 1.4.

###  5.1 The winner takes it all but one

To be able to show that the Type-$2$ infection reaches all but a finite number
of vertices in this section and in Section 5.2, we require some notation.
Recall from (1.2) that $T^{(i)}_{e}$ denotes the transmission time of the
Type-$i$ infection to traverse the edge $e$, which is an exponential random
variable with rate $(R_{n}^{(1)})^{-1}\|e\|^{-\alpha_{1}}$ when $i=1$ and rate
$(R_{n}^{(1)})^{-1}\lambda\|e\|^{-\alpha_{2}}$ when $i=2$. We then let
$X_{u,v}^{(i)}$ denote the edge-weighted distance between the vertices $u$ and
$v$, formally defined as

$X_{u,v}^{(i)}:=\inf_{\pi\in\Gamma_{u,v}}\sum_{e\in\pi}T_{e}^{(i)},$ (5.1)

where $\Gamma_{u,v}$ is the set that consists of all self-avoiding paths from
$u$ to $v$ in $\mathbb{T}_{n}^{d}$. This was the object of study in a recent
article of the first author and Van der Hofstad [31]. We generalise this to
the following extent. Let $V\subset\mathbb{T}_{n}^{d}$ such that
$u,v\in\mathbb{T}_{n}^{d}\setminus V$, and let $\Gamma_{u,v}^{(V)}$ be the set
of self-avoiding paths from $u$ to $v$ in $\mathbb{T}_{n}^{d}\setminus V$.
Then, define

$X^{(i),(V)}_{u,v}:=\inf_{\pi\in\Gamma^{(V)}_{u,v}}\sum_{e\in\pi}T^{(i)}_{e},$
(5.2)

as the length of the shortest edge-weighted path from $u$ to $v$ in
$\mathbb{T}_{n}^{d}\setminus V$. With this at hand, we prove Theorem 1.3 (i).

###### Proof of Theorem 1.3 (i).

Fix two distinct vertices $v^{(1)},v^{(2)}\in\mathbb{T}_{n}^{d}$, set
$V=\\{v^{(1)}\\}$, and start the Type-$i$ infection from $v^{(i)}$ for
$i\in\\{1,2\\}$. For $\beta>2$ fixed, we introduce the following two events:

$\mathcal{E}_{n}^{(1)}:=\Big{\\{}\min_{\begin{subarray}{c}u\in\mathbb{T}_{n}^{d}\\\
u\neq v^{(1)},v^{(2)}\end{subarray}}T_{v^{(1)},u}^{(1)}>\frac{\beta\log
n}{Z_{n}}\Big{\\}},\quad\text{and}\quad\mathcal{E}_{n}^{(2)}:=\Big{\\{}\max_{\begin{subarray}{c}u\in\mathbb{T}_{n}^{d}\\\
u\neq v^{(1)},v^{(2)}\end{subarray}}X_{v^{(2)},u}^{(2),(V)}<\frac{\beta\log
n}{Z_{n}}\Big{\\}}.$ (5.3)

In words, $\mathcal{E}_{n}^{(1)}$ states that it takes the Type-$1$ infection
more than $\beta\log n/Z_{n}$ amount of time to traverse any edge incident to
$v^{(1)}$, and thus the Type-$i$ infection potentially reaches its first
vertex after time $\beta\log n/Z_{n}$. On the other hand,
$\mathcal{E}_{n}^{(2)}$ states that the maximal weighted distance from
$v^{(2)}$ in $\mathbb{T}_{n}^{d}\setminus
V=\mathbb{T}_{n}^{d}\setminus\\{v^{(1)}\\}$ among all other vertices (not
including $v^{(1)}$) for the Type-$2$ infection is at most $\beta\log
n/Z_{n}$. As a result, the event
$\mathcal{E}_{n}^{(1)}\cap\mathcal{E}_{n}^{(2)}$ implies that the Type-$2$
infection reaches _all_ vertices in $\mathbb{T}_{n}^{d}\setminus\\{v^{(1)}\\}$
before the Type-$1$ infection spreads to any vertex. Hence, it suffices to
prove that the probability of $\mathcal{E}_{n}^{(1)}\cap\mathcal{E}_{n}^{(2)}$
converges to one to prove Case (i).

We observe that the two events $\mathcal{E}_{n}^{(1)}$ and
$\mathcal{E}_{n}^{(2)}$ are independent. Indeed, the two events consider two
disjoint sets of edges (and thus disjoint sets of independent edge-weights).
It hence suffices to show that either event has a probability converging to
one. We can (almost) directly use [31, Theorem $1.1$] to obtain that
$\mathbb{P}(\mathcal{E}_{n}^{(2)})=1-o(1)$, as $\beta>2$. Indeed, this result
tells us that the _flooding time_ , $\max_{u\in\mathbb{T}_{n}^{d},u\neq
v^{(2)}}X^{(2)}_{v^{(2)},u}$, the time to reach all vertices in
$\mathbb{T}_{n}^{d}$ of a Type-$2$ infection, starting from any initial vertex
$v^{(2)}\in\mathbb{T}_{n}^{d}$, is of the order $(2+o(1))\log n/Z_{n}$. As the
required result considers this same time, but on
$\mathbb{T}_{n}^{d}\setminus\\{v^{(1)}\\}$ rather than $\mathbb{T}_{n}^{d}$,
we refer the reader to [31] to see that all details indeed carry through
rather than provide an almost identical proof here. Moreover, we note that
[31, Theorem $1.1$] also holds with $\alpha_{2}$ that can depend on $n$ such
that $\sup_{n\in\mathbb{N}}\alpha_{2}(n)<d$, rather than being fixed (as in
the assumption in [31]). Finally, note that a single-type LRFPP model on the
complete graph, as studied in [31], does not include the time scaling
$R_{n}^{(1)}$ in the rates of the exponential random variables, which here
leads to the term $Z_{n}$ rather than $R_{n}^{(2)}$ in the denominator.

For the event $\mathcal{E}_{n}^{(1)}$ we use that the minimal edge-weight
among
$(T_{v^{(1)},u}^{(1)})_{u\in\mathbb{T}_{n}^{d}\setminus\\{v^{(1)},v^{(2)}\\}}$
is an exponential random variable with rate
$1-\|v^{(1)}-v^{(2)}\|^{-\alpha_{1}}/R_{n}^{(1)}\leq 1$, to arrive at

$\mathbb{P}\left(\mathcal{E}_{n}^{(1)}\right)=\exp\Big{(}-\big{(}1-\|v^{(1)}-v^{(2)}\|^{-\alpha_{1}}/R_{n}^{(1)}\big{)}\frac{\beta\log
n}{Z_{n}}\Big{)}\geq\exp\Big{(}-\frac{\beta\log n}{Z_{n}}\Big{)}.$ (5.4)

If $c_{n}/(\log n)^{2}$ diverges or, equivalently, $Z_{n}/\log n$ diverges, it
follows that $\mathbb{P}(\mathcal{E}_{n}^{(1)})=1-o(1)$, which yields the
desired result. ∎

###  5.2 The winner takes all but finitely many

In Section 5.1, we saw that the proof of Theorem 1.3 (i) is based on the fact
that the Type-$2$ infection can spread to all vertices within time
$\beta\log(n)/Z_{n}$ with high probability for any $\beta>2$, since
$c_{n}\gg(\log n)^{2}$ and thus $Z_{n}\gg\log n$. In this section, we discuss
Theorem 1.3 (ii), where $c_{n}=(c+o(1))(\log n)^{2}$ and thus
$Z_{n}=(c+o(1))\log n$. In this case, it is not sufficient to run the process
up to time $2\log(n)/Z_{n}$. Instead, the aim is to run it up to time
$(1+\delta_{n})\log(n)/Z_{n}$, for some small $\delta_{n}$, and show that by
this time the Type-$2$ infection has reached all but a negligible proportion
of the vertices, whilst the approximation of the slower Type-$1$ infection by
the corresponding Yule process, as described in Section 3, is in fact _exact_
with high probability.

####  5.2.1 Typical distance on the torus with a small set removed

To be able to show that the Type-$2$ infection reaches all but a small set of
vertices in time $(1+\delta_{n})\log(n)/Z_{n}$, we need to strengthen a part
of the first result in [31, Theorem $1.1$]. This result is concerned with
$X^{(2)}_{v,U}$, as in (5.1), the edge-weighted distance between a fixed
vertex $v$ and a uniform vertex $U$ in $\mathbb{T}_{n}^{d}$, in a single-type
LRFPP process on the torus $\mathbb{T}_{n}^{d}$, with long-range parameter
$\alpha\in[0,d)$. Theorem $1.1$ in [31] states that
$X^{(2)}_{v,U}\frac{Z_{n}}{\log n}\overset{\mathbb{P}}{\longrightarrow}1$. In
particular, for any $\epsilon>0$, the left-hand side is at most $1+\epsilon$
with high probability. Here, we generalise this result by, first, allowing
$\alpha$ to depend on $n$, and by considering $X^{(2),(V)}_{v,U}$, as in
(5.2), for sets $V\subset\mathbb{T}_{n}^{d}$ such that $|V|\leq\log n$. We
also strengthen it by allowing $\epsilon=\epsilon(n)$ to tend to zero with
$n$. This is summarised in the following result.

######  Proposition 5.1.

Let $V\subset\mathbb{T}_{n}^{d}$ such that $|V|\leq\log n$, and take
$v\in\mathbb{T}_{n}^{d}\backslash V$, and let $U$ be a vertex selected
uniformly at random from $\mathbb{T}_{n}^{d}\backslash(V\cup\\{v\\})$. Run a
single Type-2 LRFPP on $\mathbb{T}_{n}^{d}\backslash V$, started from $v$,
with a distance metric as in (5.2). Finally let
$t_{n}:=(1+\delta_{n})\log(n)/Z_{n}$, where $\delta_{n}$ is such that
$\delta_{n}=o(1)$ and $\delta_{n}\log n\to\infty$. Then, there exists a
constant $K>0$ such that, uniformly over all sets $V$ of size at most $\log n$
and vertices $v\in\mathbb{T}_{n}^{d}\setminus V$,

$\mathbb{P}\left(X_{v,U}^{(2),(V)}\geq
t_{n}\right)\leq\mathrm{e}^{-K\delta_{n}\log n}.$ (5.5)

###### Proof.

We can bound the probability in the statement of the lemma from above by
replacing $U$ with a supremum over all vertices in
$\mathbb{T}_{n}^{d}\setminus V$ that are not $v$. That is,

$\mathbb{P}\left(X_{v,U}^{(2),(V)}\geq
t_{n}\right)\leq\sup_{\begin{subarray}{c}w\in\mathbb{T}_{n}^{d}\backslash V\\\
w\neq v\end{subarray}}\mathbb{P}\left(X_{v,w}^{(2),(V)}\geq t_{n}\right).$
(5.6)

We bound $X_{v,w}^{(2),(V)}$ from above by the time it takes two independent
LRFPP processes on $\mathbb{T}_{n}^{d}\backslash V$, started at $v$ and $w$,
to reach $a_{n}:=\lfloor\sqrt{n}\rfloor$ many individuals, plus the shortest
transmission time $T^{(2)}_{(x,y)}$, among vertices $x$ and $y$ in the process
started at $v$ and $w$, respectively. We let $\tau_{j}^{(v)}$ denote the time
it takes for the infection started from $v$ to reach $j\in[n-2]$ many
vertices, and let $B_{v}(\tau_{j}^{(v)})$ denote the set of vertices, reach by
$v$ at time $\tau_{j}^{(v)}$. We then have the upper bound

$\sup_{\begin{subarray}{c}w\in\mathbb{T}_{n}^{d}\backslash V\\\ w\neq
v\end{subarray}}\mathbb{P}\left(\tau_{a_{n}}^{(v)}+\tau_{a_{n}}^{(w)}+\min_{\begin{subarray}{c}x\in
B_{w}(\tau^{(w)}_{a_{n}}),y\in
B_{v}(\tau^{(v)}_{a_{n}})\end{subarray}}T^{(2)}_{(x,y)}\geq t_{n}\right).$
(5.7)

Indeed, if the two sets $B_{v}(\tau^{(v)}_{a_{n}})$ and
$B_{w}(\tau^{(w)}_{a_{n}})$ are disjoint, then the sum of these three terms
provides the edge-weighted distance of one possible path from $w$ to $v$, and
hence is an upper bound for $X_{v,w}^{(2),(V)}$. If the two sets are not
disjoint, then the minimum equals zero and the first two terms already are an
upper bound for $X_{v,w}^{(2),(V)}$, since $v$ and $w$ can be connected within
time $\tau_{a_{n}}^{(v)}+\tau_{a_{n}}^{(w)}$. We further split this
probability into the two terms

$2\mathbb{P}\left(\tau_{a_{n}}^{(v)}\geq\Big{(}\frac{1}{2}+\frac{\delta_{n}}{4}\Big{)}\frac{\log
n}{Z_{n}}\right)+\mathbb{P}\left(\min_{\begin{subarray}{c}x\in
B_{w}(\tau^{(w)}_{a_{n}}),y\in
B_{v}(\tau^{(v)}_{a_{n}})\end{subarray}}T^{(2)}_{(x,y)}\geq\frac{\delta_{n}}{2}\frac{\log
n}{Z_{n}}\right),$ (5.8)

where the multiplicative factor in front of the first term is due to symmetry
for $v$ and $w$. Let us start by bounding the second term from above. As a
minimum of exponential random variables is an exponential random variable,
whose rate equals the sum of all the individual rates, we can condition on
$B_{w}(\tau^{(w)}_{a_{n}})$ and $B_{v}(\tau^{(v)}_{a_{n}})$ to obtain

$\mathbb{P}\bigg{(}\min_{\begin{subarray}{c}x\in
B_{w}(\tau^{(w)}_{a_{n}}),y\in
B_{v}(\tau^{(v)}_{a_{n}})\end{subarray}}\\!\\!\\!\\!T^{(2)}_{(x,y)}\geq\frac{\delta_{n}\log
n}{2Z_{n}}\bigg{)}=\mathbb{E}\Bigg{[}\exp\bigg{(}\\!-\frac{\delta_{n}}{2}\frac{\log
n}{R_{n}^{(2)}}\\!\\!\\!\\!\sum_{x\in B_{w}(\tau^{(w)}_{a_{n}})}\sum_{y\in
B_{v}(\tau^{(v)}_{a_{n}})}\\!\\!\\!\\!\\!\\!\\!\lambda\|x-y\|^{-\alpha_{2}}\bigg{)}\Bigg{]}.$
(5.9)

We now bound $\|x-y\|^{-\alpha_{2}}\geq Cn^{-\alpha_{2}/d}$ for some large
constant $C>0$, since we can bound the distance between $x$ and $y$ from above
by a constant times $n^{1/d}$ for any $x,y\in\mathbb{T}_{n}^{d}$. Using this,
and the fact that $R_{n}^{(2)}\lambda^{-1}=\Theta(n^{1-\alpha_{2}/d})$ by
Lemma 2.1, we obtain for some constants $C,C^{\prime},\kappa_{1}>0$,