# Coexistence in competing first passage percolation with conversion

Thomas<EMAIL_ADDRESS>University of Bath, Deptartment of
Mathematical Sciences, supported by a scholarship from the EPSRC Centre for
Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under
the project EP/L015684/1. Alexandre<EMAIL_ADDRESS>Università Roma Tre, Dipartimento di Matematica e Fisica; University of Bath,
Department of Mathematical Sciences, supported by EPSRC Fellowship
EP/N004566/1.

###### Abstract

We introduce a two-type first passage percolation competition model on
infinite connected graphs as follows. Type 1 spreads through the edges of the
graph at rate $1$ from a single distinguished site, while all other sites are
initially vacant. Once a site is occupied by type 1, it converts to type 2 at
rate $\rho>0$. Sites occupied by type 2 then spread at rate $\lambda>0$
through vacant sites _and_ sites occupied by type 1, whereas type 1 can only
spread through vacant sites. If the set of sites occupied by type 1 is non-
empty at all times, we say type 1 _survives_. In the case of a regular $d$-ary
tree for $d\geq 3$, we show type 1 can survive when it is slower than type 2,
provided $\rho$ is small enough. This is in contrast to when the underlying
graph is $\mathbb{Z}^{d}$, where for any $\rho>0$, type 1 dies out almost
surely if $\lambda>1$.

##  1 Introduction

Consider the following two-type first passage percolation model on an infinite
connected graph $G$. Each site can either be occupied by type 1, type 2 or be
vacant according to the following dynamics. At time $0$, a distinguished site
is occupied by type 1 while every other site is vacant. Sites occupied by type
1 attempt to occupy neighbouring vacant sites at rate 1. Once a site is
occupied by type 1, it is converted to type 2 at rate $\rho>0$. That is, we
define the collection of random variables
$\left\\{\mathcal{I}_{x}\right\\}_{x\in G}$ of conversion times, that are
i.i.d. exponential random variables of rate $\rho>0$, assigned to each site of
$G$. Once a site is occupied by type 1, it waits for its respective conversion
time to expire before converting to type 2. Sites occupied by type 2 then
spread type 2 to vacant sites _and_ sites occupied by type 1 at rate
$\lambda>0$.

This model can be seen as a variant of the chase-escape dynamics in predator-
prey models (see Section 1.2 for more details). In these models, there is
initially a single predator that evolves to block the spread of a species of
prey. A natural interpretation for our model is as a spreading infection where
individuals are either aware or unaware of their infected status. Unaware
individuals spread the infection to nearby individuals but become aware of
their infected status after a certain time elapses. Aware individuals try to
warn neighbouring individuals which become aware of the spread of infection
even if it it has not been infected. Uninfected individuals that are aware
take the necessary measures (for example, self-isolating) to avoid infection
and do not get infected. Can the unaware individuals coexist with aware
individuals for all time? Equivalently, can the infection reach an unbounded
number of individuals?

Another motivation for us to introduce this model comes from a recent way of
analysing strongly interacting particle systems through growth models; for
example, the analysis of multiparticle diffusion limited aggregation by
Sidoravicius and Stauffer [18] and the analysis of a heterogeneous spread of
infection model by Dauvergne and Sly [8]. We believe the competition process
we introduce is a natural model for such applications; we discuss this further
in Section 1.2.

The main interest of this paper is understanding coexistence regimes in this
model on different graphs. We prove on the regular tree, type 1 can survive
even if it is slower than type $2$ (i.e. $\lambda$ is larger than one), so
long as $\rho$ is sufficiently small (c.f. Theorem 1.1). Then, we prove such
behaviour on the lattice is impossible and type 1 dies out even if it is just
faster than type 2, for all $\rho>0$ (c.f. Theorem 1.3).

A major difficulty in analysing this model is the counter-intuitive lack of
monotonicity. If we increase $\rho$ or $\lambda$, it seems we can only
decrease the probability that type 1 survives. Surprisingly, proving this
remains an open problem. The issue is the model is non-monotone in the sense
that the standard coupling argument fails to hold (unlike other competition
models like the two-type Richardson model as discussed in Section 1.2). Models
that lack monotonicity require a careful analysis as they include the
possibility of many phase transitions occurring. The subtle behaviour of
processes that lack monotonicity has been studied in related models in
Candellero and Stauffer [6] and Deijfen and Häggström [9]. Another difficulty
the model poses is non-equilibrium dynamics and long-range correlations
between the occupancy of sites. For example, to determine whether a site is
ever occupied by type 1, one may need non-local information about the first
times other sites are occupied by type 1 and how type 2 spreads from them.

###  1.1 Our results

The main result of this paper is the following. On the $d$-ary tree for $d\geq
3$ (i.e. the infinite tree where all vertices have _degree_ equal to $d$), we
show type 1 can survive even in a regime where it is slower than type 2.

######  Theorem 1.1.

Fix $d\geq 3$ and consider the $d$-ary tree. There exists
$\lambda_{0}=\lambda_{0}(d)>1$ such that if $\lambda\in(0,\lambda_{0})$ and
$\rho$ is small enough, then type 1 survives with positive probability.

Recalling the interpretation of the model as the spread of infection and
awareness, we deduce the following from Theorem 1.1. In the case of the
$d$-ary tree, the infection can survive even if awareness spreads faster than
the infection provided infected individuals do not become aware until a
typically large time has passed.

######  Remark 1.2.

Theorem 1.1 may naturally be extended to all supercritical Galton-Watson trees
of uniformly bounded degree. The requirement of uniformly bounded degrees is
needed to appeal to a result in branching random walks by Addario-Berry and
Reed [1] (see Theorem 3.4). The behaviour of the model on more general trees
remains an open problem. For example, if the degree distribution follows a
power law, can type 1 survive for all $\lambda\in(0,\infty)$?

Our next result is establishing the behaviour on the regular tree is
fundamentally different than on the lattice. More precisely, on
$\mathbb{Z}^{d}$ for $d\geq 2$, for all $\rho>0$, there exists a constant
_smaller_ than one such that if $\lambda$ is larger than this constant, type 1
dies out almost surely. In other words, there exists a regime where type 1 is
faster than type 2 but still dies out almost surely. We note it is immediate
for $d=1$ that type 1 dies out almost surely for all $\lambda,\rho>0$.

######  Theorem 1.3.

Consider the model on $\mathbb{Z}^{d}$ for $d\geq 2$. For all $\rho>0$, there
exists $\lambda^{\prime}=\lambda^{\prime}(\rho,d)<1$ such that if
$\lambda>\lambda^{\prime}$, type 1 dies out almost surely.

The fact that for any $\rho>0$, type 1 dies out almost surely for all
$\lambda>1$ is a simple consequence of the classical _shape theorem_ for first
passage percolation on $\mathbb{Z}^{d}$ (see Theorem 4.1). Hence, on
$\mathbb{Z}^{d}$, the infection cannot survive forever if awareness spreads
faster than the infection. Theorem 1.3 sharpens this result so that for any
conversion rate, unaware individuals cannot survive forever if the infection
spreads only just faster than awareness.

A complete picture of the survival regimes on $\mathbb{Z}^{d}$ is an open
problem but the expected behaviour is given in the following conjecture.

######  Conjecture 1.4.

Consider the model on $\mathbb{Z}^{d}$ for $d\geq 2$.

  * •

For all $\lambda<1$, if $\rho$ is sufficiently small, type 1 survives with
positive probability.

  * •

There exists $\rho_{c}=\rho_{c}(d)\in(0,\infty)$ such that:

    1. 1.

if $\rho<\rho_{c}$ and $\lambda$ is sufficiently small, type 1 survives with
positive probability.

    2. 2.

if $\rho>\rho_{c}$ and $\lambda>0$, then type 1 dies out almost surely.

The first half of Conjecture 1.4 is closely related to the strong survival
phase for first passage percolation in a hostile environment, studied in
Sidoravicius and Stauffer [18] (see Section 1.2). We believe the proof in our
case will be similar to the encapsulation arguments in [18], though there are
some caveats in applying that argument directly.

A priori there is no reason $\rho_{c}$ is even well-defined in Conjecture 1.4
due to the non-monotone nature of the model. Indeed, several phase transitions
may occur. We provide a partial answer to Conjecture 1.4. The following result
gives the expected subcritical and supercritical behaviour holds so long as
$\rho$ is sufficiently small or sufficiently large, respectively.

######  Theorem 1.5.

Consider the model on $\mathbb{Z}^{d}$ for $d\geq 2$.

  * •

There exists $\rho_{\ell}=\rho_{\ell}(d)>0$ such that if $\rho<\rho_{\ell}$
and $\lambda$ is sufficiently small, type 1 survives with positive
probability.

  * •

There exists $\rho_{u}=\rho_{u}(d)<\infty$ such that if $\rho>\rho_{u}$, type
1 dies out almost surely for all $\lambda>0$.

Proving there exists $\rho_{\ell}>0$ in Theorem 1.5 relies on a careful
coupling with a percolation process analysed by Dauvergne and Sly [8]. The
proof for the existence of $\rho_{u}<\infty$ follows through a direct
comparison with Bernoulli percolation. Details for both these proofs are given
in Section 4.

###  1.2 Related work

The model introduced in this paper is closely related to _chase-escape
dynamics in the predator-prey model_. In the language of our model, the
predator-prey dynamics can be viewed in the following manner. At time 0, the
origin is occupied by type 2 and a neighbour is occupied by type 1. Type 1
spreads at rate $\lambda>0$ to unoccupied sites while type 2 spreads at rate
$1$ to both unoccupied sites and sites occupied by type 1. Type 1 can be seen
as _prey_ while type 2 is the _predator_. Note we have swapped the role of
$\lambda$ to the rate of type 1 to remain consistent with the notation in the
predator-prey literature. The model described above was introduced by
Kordzakhia [14], who proved a phase transition occurs and computed the exact
critical value in the case of the $d$-ary tree. More precisely, there exists
$\lambda_{c}>0$ such that if $\lambda>\lambda_{c}$, type 1 survives with
positive probability while if $\lambda<\lambda_{c}$, type dies out almost
surely. Bordenave [4] extended the results by Kordzakhia to Galton-Watson
trees and proved type 1 dies out almost surely at criticality. We also direct
the reader to Kortchemski [15] and references therein for more information
about predator-prey dynamics.

The question of coexistence in competing random processes on $\mathbb{Z}^{d}$
has attracted significant attention in recent years. The _two-type Richardson
model_ was introduced by Häggström and Pemantle [12] as a model for competing
first passage percolation. In this model, each type starts from a finite set
of sites and spreads at rate $1$ and $\lambda$, respectively. so that once a
site is occupied by one process, it remains occupied by that process
henceforth. They conjecture coexistence occurs with positive probability if
and only if $\lambda=1$, assuming non-degenerate initial conditions. See [10]
for a background in coexistence regimes in the two-type Richardson model and
progress towards this conjecture.

A more recent competition model that has been studied is _first passage
percolation in a hostile environment_ (FPPHE), introduced by Sidoravicius and
Stauffer [18]. The first type in FPPHE initially only occupied the origin
while the second is dormant in _seeds_ that are distributed as a product of
Bernoulli measures of parameter $p$. Type 1 spreads at rate 1 through edges
and when it encounters a seed, the occupancy is suppressed and the seed is
activated. Activated seeds then spread at rate $\lambda>0$ to vacant sites.
Once a site is occupied by either type, it remains that type henceforth.

Sidoravicius and Stauffer [18] considered FPPHE on $\mathbb{Z}^{d}$ for $d>1$.
They proved for all $\lambda<1$, if $p$ is sufficiently small, then type 1
survives and all components of type 2 are bounded with positive probability -
a regime called strong survival. Finn and Stauffer [11] proved a regime of
coexistence exists on $\mathbb{Z}^{d}$ for $d>2$, in which both types occupy
unbounded connected regions. Coexistence on transitive, hyperbolic, non-
amenable graphs has also been established in Candellero and Stauffer [5].

FPPHE has been used as an analytical tool in Sidoravicius and Stauffer [18] to
analyse a challenging aggregation model called _multiparticle diffusion
limited aggregation_. A streamlined version of FPPHE, called
Sidoravicius–Stauffer percolation (SSP), was utilised in Dauvergne and Sly [8]
to study a non-homogeneous spread of infection model (see Section 4.2). In the
above works, coupling the evolution of the desired process with FPPHE or SSP
allowed an efficient way to analyze a process with non-equilibrium dynamics
without the need to carry out an involved multi-scale analysis from scratch.

We believe our competition process can also be used in this regard. The main
idea is that type 1 represents the _propagation front_ through “typically
good” regions of the process being analysed, being it the front of the growth
of an aggregate or the front of the propagation of the infection, in the above
cases. A type-1 site being converted to type 2 represents that enough time has
passed so that that site is not anymore contributing to the propagation front.
Being it a random time, a site could have a very short conversion time, not
allowing type 1 to spread from it to its neighbors. This models situations
where the propagation front may pass through _atypically bad_ regions of
space-time, which locally blocks the propagation of the front. The spread of
type 2 then represents the spread of the influence that bad regions may have
on neighboring areas. If one can show that type 1 grows indefinitely despite
the expansion of type 2, then such a reasoning would imply that the typical
regions are dense enough to compensate the presence of bad regions in space-
time, allowing the propagation front to survive indefinitely.

###  1.3 Overview of paper

In Section 2, we provide a rigorous construction of the model on the $d$-ary
tree that will facilitate later proofs and recall some notation from first
passage percolation. In Section 3, we prove Theorem 1.1 through a careful
renormalisation scheme that controls how type 1 and type 2 spreads through the
tree. In Section 4, we switch focus to the lattice and prove Theorem 1.3 and
Theorem 1.5.

##  2 Preliminaries

###  2.1 Construction of the model on the $d$-ary tree

In this section we provide a particular construction of the model on the
$d$-ary tree that will later be helpful in proofs.

For $d\geq 3$, let $\mathbb{T}_{d}$ be the $d$-ary tree with a distinguished
site called the root, written as $\sigma$. Fix constants $\rho>0$ and
$\lambda>0$. Let $V(\mathbb{T}_{d})$ and $E(\mathbb{T}_{d})$ be the vertex set
and edge set of $\mathbb{T}_{d}$, respectively. Let
$\\{\mathcal{I}_{x}\\}_{x\in V(\mathbb{T}_{d})}$ be an i.i.d. collection of
exponential random variables of rate $\rho$ that correspond to the conversion
time of a site. That is, given a site $x\in V(\mathbb{T}_{d})$, once $x$ is
occupied by type 1, after a further time of $\mathcal{I}_{x}$ has expired, it
converts to type 2. Let $\\{t_{1,e}\\}_{e\in E(\mathbb{T}_{d})}$ be a
collection of i.i.d. exponential random variables of rate 1 that correspond to
the passage times for type 1. Similarly, let $\\{t_{u,e}\\}_{e\in
E(\mathbb{T}_{d})}$ and $\\{t_{d,e}\\}_{e\in E(\mathbb{T}_{d})}$ be two
collections of i.i.d. exponential random variables of rate $\lambda$ that
correspond to the passage times for type 2 upwards and downwards,
respectively. That is, if $e=xy$ is an edge with $d(\sigma,x)<d(\sigma,y)$,
where $d(\cdot,\cdot)$ is the graph distance metric on $\mathbb{T}_{d}$, the
passage time for type 2 spreading from $x$ to $y$ is given by $t_{d,e}$ and
the passage time for type 2 spreading from $y$ to $x$ is given by $t_{u,e}$.
In this case, we say $y$ is a _descendent_ of $x$.

At time $t=0$, the root $\sigma$ is occupied by type 1 and all other sites are
vacant. If a site $x$ is occupied by type 1 and $y$ neighbours $x$, then $y$
is attempted to be occupied by type 1 after waiting $t_{1,xy}$ time. The
occupation is successful if $y$ is vacant and suppressed otherwise.
Additionally, once a site is occupied by type 1, after waiting
$\mathcal{I}_{x}$ time, it converts to type 2. If a site $x$ is occupied by
type 2 and $y$ neighbours $x$, then $y$ is attempted to be occupied by type 2
after waiting $t_{u,xy}$ or $t_{d,xy}$ time, depending on whether $y$ is
closer or further than $x$ to the root with respect to the graph-distance
metric, respectively. The occupation is successful if $y$ is unoccupied or is
occupied by type 1. The requirement to distinguish between downward and upward
passage times for type 2 is to decouple the different ways type 2 may spread
and is an important ingredient in our proofs.

Given a site $x\in V(G)$, let $\tau_{1}(x)$ denote the first time that type 1
occupies $x$ and $\tau_{2}(x)$ denote the first time that type 2 occupies $x$,
so that

$\tau_{i}(x)=\inf\left\\{t\geq 0:\mbox{type }i\mbox{ occupies
}x\right\\}\quad\mbox{for }i=1,2.$

If type 1 never occupies the site $x$, we write $\tau_{1}(x)=\infty$.

###  2.2 First passage percolation

A _path_ is a sequence of distinct sites
$\left(x_{1},x_{2},\ldots,x_{n}\right)$ such $x_{i}x_{i+1}\in
E(\mathbb{T}_{d})$ for each $i\in\left\\{1,2,\ldots,n-1\right\\}$. Given a
path $\gamma=\left(x_{1},x_{2},\ldots,x_{n}\right)$, we define the _type #
passage time for_ $\gamma$ as the random variable

$T_{\\#}(\gamma)=\sum_{i=1}^{n-1}t_{\\#,x_{i}x_{i+1}},$ (1)

where $\\#\in\\{1,u,d\\}$.

Given $x,y\in V(\mathbb{T}_{d})$, define the _type # passage time from_ $x$
_to_ $y$ as the random variable

$T_{\\#}(x\rightarrow y)=T_{\\#}(\gamma_{x,y})$

where $\gamma_{x,y}$ is the unique shortest path from $x$ to $y$ and
$\\#\in\left\\{1,u,d\right\\}$. Note the direction of the arrow in the
definition of the passage time will always agree with the descendent structure
of the tree. That is, if $y$ is a descendent of $x$, the type 2 passage time
from $x$ to $y$ is downward, while the type 2 passage time from $y$ to $x$ is
upward. This is purely a notational subtlety and will not alter any of our
arguments.

##  3 Survival on the $d$-ary tree

In this section we prove Theorem 1.1. We begin with a roadmap of the proof to
help guide the reader in how we establish survival of type 1 with positive
probability.

###  3.1 Roadmap of proof of Theorem 1.1

Given $d\geq 3$, recall $\mathbb{T}_{d}$ is the $d$-ary tree with $\sigma$ as
a distinguished site called the root. For any $n\in\mathbb{Z}_{+}$, let
$V_{n}$ denote the set of sites up to graph distance $n$ from the root and
$\partial V_{n}$ denote the set of sites with graph distance exactly $n$ from
the origin, so that

$V_{n}=\left\\{x\in V(\mathbb{T}_{d}):d(\sigma,x)\leq
n\right\\}\quad\mbox{and}\quad\partial V_{n}=\left\\{x\in
V(\mathbb{T}_{d}):d(\sigma,x)=n\right\\},$

where $d(\cdot,\cdot)$ is the metric on $\mathbb{T}_{d}$ induced by shortest-
paths between sites. We refer to the set $V_{n}$ as a _box_ of depth $n$ from
$\sigma$, or just a box for brevity.

The aim is to partition $\mathbb{T}_{d}$ into boxes that are labelled as
_good_ or _bad_ according to events measurable with respect to the box, so
that a good box implies that type 1 is able to propagate well through the box
while type 2 is hindered. Let $k$ and $r$ be two large integers we set later.
Let $B_{1}=V_{kr}$ and $B_{2}=V_{kr+k^{2}}\backslash V_{kr}$ be the two
segments of our box $B=B_{1}\cup B_{2}=V_{kr+k^{2}}$. The reader should have
in mind that we will eventually set $r$ much larger than $k$ so that $kr\gg
k^{2}$.

In $B_{1}$, the first section of the box, we want to prove that there are
sufficiently many _highways_ down to $\partial V_{kr}$ with high probability.
Roughly speaking, highways are paths where type 1 is typically faster than
type 2, and we will prove that type 1 survives along these highways with
positive probability. We aim to prove that many such highways exist in $B_{1}$
even if $\lambda$ is slightly larger than 1. This is made rigorous in Section
3.2. An illustration of highways in a good box can be seen in Figure 1.

Assuming we can show the existence of many highways through $B_{1}$, we wish
to prove the following holds in $B_{2}$. From the set of sites in $\partial
V_{kr}$ that are connected to $\sigma$ through highways, there are at least
two such sites that can be extended down to depth $kr+k^{2}$ so that the type
1 passage time on the path is fast and the type 2 passage time on each edge
with a site incident to the path is very large. The probability a given path
satisfies these properties is extremely small, which is precisely why we
require many highways existing through $B_{1}$ to ensure that two such paths
exist with high probability. These paths that extend highways we refer to as
_spines_. The reason why we need such a strong property from the spines is
that they are very close to the boundary of the box, and we need to ensure
that conversions of type 2 from outside the a good box cannot propagate into
the box and block type 1. These notions are made rigorous in Section 3.3. An
illustration of spines can be seen in Figure 1.

We have not yet considered the fact that type 1 converts to type 2 at rate
$\rho$. The idea is that we set $\rho$ small enough so that no conversions can
take place in a good box until type 1 can spread far down the tree. However,
conversions could occur outside of a good box, allowing type 2 to spread
upwards the tree into a highway or a spine. In Section 3.4, we control how
type 2 can spread in a good box through conversions occurring outside that
good box, so that type 1 is not impeded on the previously constructed highways
or spines.

In Section 3.5, we put together the previous sections to define a good box and
then use this construction of good boxes to prove Theorem 1.1 via a branching
argument in Section 3.6. The idea is that with positive probability, a
branching structure of good boxes gives rise to at least one infinite path of
sites such that all sites on this path are occupied by type 1 at some time.

Figure 1: An illustration of a good box split into $B_{1}$ and $B_{2}$ with
respective depths $kr$ and $k^{2}$. Blue paths represent the highways
constructed in $B_{1}$ that connect the root of the box to depth $kr$. Red
paths represent the spines constructed in $B_{2}$ that extend certain highways
to depth $kr+k^{2}$ in a box as well as any edges incident to them.

###  3.2 Percolating structure in $B_{1}$

In this section we prove the existence of a percolating structure that
provides the highways needed for type 1 in $B_{1}$. This percolating structure
arises from partitioning $B_{1}$ into _sub-boxes_ where we wish to prove there
is the existence of paths for which there is strong control for the passage
times of type 1 and type 2.

We construct the sub-boxes as follows. Let $k$ be a large integer we set
later. For $z\in V(\mathbb{T}_{d})$, let $V^{\downarrow}_{k}(z)$ (resp.
$\partial V^{\downarrow}_{k}(z)$) denote the set of sites of graph-distance
less than or equal (resp. equal) to $k$ from $z$ such that the path to the
root from the site must pass through $z$.

######  Definition 3.1 (Good sub-box).

Let $z\in V(\mathbb{T}_{d})$ and consider the sub-box $V^{\downarrow}_{k}(z)$.
For $\varepsilon>0$, let
$\mathcal{H}_{\varepsilon}(z)=\mathcal{H}_{\varepsilon}^{(1)}(z)\cap\mathcal{H}_{\varepsilon}^{(\lambda)}(z)$,
where

$\displaystyle\mathcal{H}_{\varepsilon}^{(1)}(z)=\left\\{y\in\partial
V^{\downarrow}_{k}(z):T_{1}(z\rightarrow y)\leq(1-\varepsilon)k\right\\},$ (2)
$\displaystyle\mathcal{H}_{\varepsilon}^{(\lambda)}(z)=\left\\{y\in\partial
V^{\downarrow}_{k}(z):T_{d}(z\rightarrow
y)\geq(1-\varepsilon^{2})\tfrac{k}{\lambda}\right\\}.$ (3)

Define $V^{\downarrow}_{k}(z)$ to be $\varepsilon$-_good_ if
$\mathcal{H}_{\varepsilon}(z)$ contains at least two distinct elements and
$\varepsilon$-_bad_ otherwise.

Given $y\in\mathcal{H}_{\varepsilon}(z)$, the path from $z$ to $y$ is called a
_highway_.

######  Remark 3.2.

Note that in (3) in the definition of
$\mathcal{H}_{\varepsilon}^{(\lambda)}(z)$, we are using the _downward_ type 2
passage times.

With the notion of good sub-boxes to hand, we wish to prove that there is a
percolating structure of good sub-boxes up to depth $kr$. We first partition
$B_{1}$ into sub-boxes of depth $k$, so that each sub-box is independently
$\varepsilon$-good of every other sub-box. That is, the sub-boxes are of the
form

$V_{k}^{\downarrow}(z)\mbox{ for }z\in\partial V_{ik}\mbox{ with
}i\in\\{0,1,2,....,r-1\\}.$

A sub-box being good means that there are at least two paths from its root to
depth $k$ where there is a strong control on type 1 and type 2. In particular,
if $\lambda<1+\varepsilon$, then type 1 is faster along these paths since

$1-\varepsilon=\frac{1-\varepsilon^{2}}{1+\varepsilon}<\frac{1-\varepsilon^{2}}{\lambda}.$

In Figure 2, we see how highways in good sub-boxes can join, giving long
highways that allow for good control on type 1 and type 2.

To ensure that many such highways exist down to depth $kr$, we need to prove
that sub-boxes are good with high probability. This is the content of the
following lemma.

Figure 2: Triangles represent good sub-boxes and blue paths represent the
highways constructed within them.

######  Lemma 3.3.

There exists a constant $c>0$ such that the following holds. Fix $z\in
V(\mathbb{T}_{d})$. For all sufficiently small $\varepsilon>0$ and large
enough $k$,

$\mathbb{P}\left(V^{\downarrow}_{k}(z)\mbox{ is
}\varepsilon\mbox{-good}\right)\geq 1-\exp\left(-c\varepsilon^{4}k\right).$

To prove Lemma 3.3, we recall some results from the theory of branching random
walks, that can be viewed as an alternative representation of first passage
percolation on trees. The following result is due to Addario-Berry and Reed
[1, Theorem 3], although we emphasise their result holds in a much greater
generality and we state it only in what will be useful in our context.

######  Theorem 3.4.

For $n\geq 1$, let $M_{n}$ be the infimum over all passage time from $\sigma$
to $\partial V_{n}$, so that

$M_{n}=\inf_{y\in\partial V_{n}}T_{1}(o\to y).$

There exists a constant $\varepsilon_{0}\in(0,1)$ such that

$\mathbb{E}\left[M_{n}\right]\leq(1-\varepsilon_{0})n$

for all $n$ large enough. Moreover, there exist constants $C,\delta>0$ such
that for all $x\in\mathbb{R}$,

$\mathbb{P}\left(\left|M_{n}-\mathbb{E}\left[M_{n}\right]\right|\geq
x\right)\leq Ce^{-\delta x}.$

The result in Theorem 3.4 concerns only the fastest path, but we will require
that many paths can satisfy its conditions. This is the content of the
following lemma. Recall the definition of $\mathcal{H}_{\varepsilon}^{(1)}(z)$
in (2).

######  Lemma 3.5.

There exists a constant $c>0$ such that the following holds. Fix $z\in
V(\mathbb{T}_{d})$. For all $\varepsilon>0$ small enough, there exists
$k_{1}=k_{1}(\varepsilon)$ such that if $k>k_{1}$, then

$\mathbb{P}\left(\left|\mathcal{H}_{\varepsilon}^{(1)}(z)\right|\geq
d^{\varepsilon k}\right)\geq 1-e^{-c\varepsilon k}.$

###### Proof.

Firstly, we want to prove that the passage time over all paths from $z$ to
$\partial V_{\varepsilon k}^{\downarrow}(z)$ is bounded above by $C\varepsilon
k$ with sufficiently high probability, where $C$ is a large constant and
$\varepsilon$ is a small constant we set later. For a given site $y\in\partial
V_{\varepsilon k}^{\downarrow}(z)$ and a large enough constant $C$, we see
through a Chernoff bound argument (see Lemma A.1) that

$\mathbb{P}\left(T_{1}(z\to y)\geq C\varepsilon k\right)\leq
e^{-\tfrac{C\varepsilon k}{2}}.$

By taking the union bound over all $y\in\partial V_{\varepsilon
k}^{\downarrow}(z)$,

$\mathbb{P}\left(\sup_{y\in\partial V_{\varepsilon
k}^{\downarrow}(z)}T_{1}(\sigma\to y)\geq C\varepsilon k\right)\leq
d^{\varepsilon k}e^{-\tfrac{C\varepsilon k}{2}}.$

By Theorem 3.4, there exists a constant $\delta>0$, that does not depend on
$k$, such that for each $y\in\partial V_{\varepsilon k}^{\downarrow}(z)$, the
following holds. For all $x>0$

$\mathbb{P}\left(T_{1}(y\to\partial
V_{k}^{\downarrow}(z))\geq(1-\varepsilon_{0})(1-\varepsilon)k+x\right)\leq
e^{-\delta x},$

where $\varepsilon_{0}$ is as given in Theorem 3.4. By the union bound, for
all $x>0$

$\mathbb{P}\left(\bigcup_{y\in\partial V_{\varepsilon
k}^{\downarrow}(z)}\left\\{T_{1}(y\to\partial
V^{\downarrow}_{k}(z))\geq(1-\varepsilon_{0})(1-\varepsilon)k+x\right\\}\right)\leq
d^{\varepsilon k}e^{-\delta x}.$

If $x=2\varepsilon k\left(\log d\right)/\delta$, then the right-hand term
above can be bounded above by $e^{-\delta x/2}$. Hence

$\displaystyle\mathbb{P}\Bigg{(}$ $\displaystyle\bigcap_{y\in\partial
V_{\varepsilon k}^{\downarrow}(z)}\left\\{T_{1}(y\to\partial
V_{k}^{\downarrow}(z))<(1-\varepsilon_{0})(1-\varepsilon)k+\tfrac{2\varepsilon
k\log d}{\delta}\right\\}\cap\left\\{\sup_{y\in\partial V_{\varepsilon
k}^{\downarrow}(z)}T_{1}(z\to y)<C\varepsilon k\right\\}\Bigg{)}$
$\displaystyle\geq 1-e^{-\tfrac{\delta\varepsilon
k}{4}}-e^{-\tfrac{C\varepsilon k}{4}},$ $\displaystyle\geq 1-e^{-c\varepsilon
k},$

for some constant $c>0$ and all sufficiently large $k$. Moreover, the event
directly above implies the existence of at least $d^{\varepsilon k}$ sites in
$\partial V_{k}^{\downarrow}(z)$, so that if $y$ is one such site,

$\displaystyle T_{1}(z\to
y)\leq(1-\varepsilon_{0})(1-\varepsilon)k+\tfrac{2\varepsilon k\log
d}{\delta}+C\varepsilon k\leq(1-\varepsilon)k,$

where the final inequality holds so long as $\varepsilon$ satisfies

$\varepsilon\leq\frac{\varepsilon_{0}}{\varepsilon_{0}+C+\tfrac{2\log
d}{\delta}}.$

∎

With Lemma 3.5 established, we are now in a position to prove Lemma 3.3.

###### Proof of Lemma 3.3.

By Lemma 3.5, there exists a constant $c>0$ such that for a small enough
choice of $\varepsilon>0$ and then large enough choice of $k$,

$\mathbb{P}\left(\left|\mathcal{H}_{\varepsilon}^{(1)}(z)\right|\geq
2\right)\geq 1-e^{-c\varepsilon k}.$ (4)

Conditional on the event $\\{|\mathcal{H}_{\varepsilon}^{(1)}(z)|\geq 2\\}$,
let $y_{1},y_{2}\in\mathcal{H}_{\varepsilon}^{(1)}(z)$ be distinct sites. For
$i\in\\{1,2\\}$, let $E_{i}$ be the event

$E_{i}=\left\\{T_{d}(z\rightarrow
y_{i})\geq\left(1-\varepsilon^{2}\right)\tfrac{k}{\lambda}\right\\}.$

Since $T_{d}(z\to y_{i})$ is a sum of $k$ i.i.d. Exponential($\lambda$) random
variables, by a Chernoff bound argument (see Lemma A.1), we deduce there
exists a constant $c^{\prime}>0$ such that for all sufficiently large $k$,

$\mathbb{P}\left(E_{i}\right)\geq
1-\exp\left(-c^{\prime}\varepsilon^{4}k\right)\quad\mbox{for }i=1,2.$

By the union bound,

$\mathbb{P}\left(E_{1}\cap E_{2}\right)\geq
1-\mathbb{P}(E_{1}^{c})-\mathbb{P}(E_{2}^{c})\geq
1-2\exp\left(-c^{\prime}\varepsilon^{4}k\right).$ (5)

The result follows by (4) and (5) through the independence of the passage
times for type 1 and type 2. ∎

To ease the statements of results, henceforth we assume that $\varepsilon$ is
small enough and $k$ is large enough so Lemma 3.3 is satisfied.

Consider the box $B=V_{kr+k^{2}}$. Let $\mathcal{P}_{k}\subset\partial V_{kr}$
be the set of sites whose geodesic to $\sigma$ only passes through highways of
good sub-boxes. Our proof of Theorem 1.1 will rely on $\mathcal{P}_{k}$
containing sufficiently many elements, which is established in the following
lemma.

######  Lemma 3.6.

Fix $\alpha\in(1,2)$. If there exists a constant $c$ such that $r\leq k^{c}$,
then

$\lim_{k\to\infty}\mathbb{P}\left(\left|\mathcal{P}_{k}\right|>\alpha^{r}\right)=1.$

The proof of Lemma 3.6 is a consequence of Lemma 3.3 and the following
elementary result, as the sub-boxes in $B_{1}$ are independently good or bad.

######  Lemma 3.7.

Fix $\alpha\in(1,2)$ and $r\in\mathbb{N}$. Consider independent site
percolation of parameter $p=p(r)$ on the binary tree so that

$\lim_{r\to\infty}p^{r}=1.$

Let $N_{r}$ be the number of sites at depth $r$ connected to the origin by an
open path. Then

$\lim_{r\to\infty}\mathbb{P}\left(N_{r}>\alpha^{r}\right)=1.$

###### Proof.

By applying Markov’s inequality to the number of sites not connected to the
root by open sites, we deduce

$\mathbb{P}\left(2^{r}-N_{r}\geq
2^{r}-\alpha^{r}\right)\leq\frac{2^{r}-\mathbb{E}[N_{r}]}{2^{r}-\alpha^{r}}=\frac{1-p^{r}}{1-(\alpha/2)^{r}},$

and the result follows. ∎

###  3.3 Construction of spines

In Lemma 3.6 we proved in the box $B=V_{kr+k^{2}}$, there exists many sites at
depth $kr$ that are connected to the root of $B$ through highways with high
probability. The aim of this section is to extend some of these highways down
to depth $kr+k^{2}$ through what we will refer to as _spines_ , so that spines
provide a strong control on the type 1 and type 2 passage times.

Recall $\mathcal{P}_{k}$ is the set of sites $y$ in $\partial V_{kr}$ such
that the path from $\sigma$ to $y$ only passes through the highways in good
sub-boxes in $B_{1}$.

######  Definition 3.8 (Spine).

For each $z\in\mathcal{P}_{k}$, let $s(z)\in\partial
V^{\downarrow}_{k^{2}}(z)$ satisfy

$T_{1}(z\to s(z))=\inf_{y\in\partial
V^{\downarrow}_{k^{2}}(z)}T_{1}(z\rightarrow y).$

The path $s_{z}$ from $z$ to $s(z)$ is defined to be the _spine from_ $z$.

Given $z\in\mathcal{P}_{k}$, we want $s_{z}$ to satisfy the following
properties. Firstly, we want the type 1 passage time on $s_{z}$ to be bounded
above by $(1-\varepsilon)k^{2}$, so type 1 can readily spread to depth
$kr+k^{2}$. Secondly, we want every edge incident to a site in the spine
$s_{z}$ to have a type 2 passage time of at least $k^{3}$ with the exception
of edges only incident to $s(z)$. The edges with both endpoints contained on
$s_{z}$ will be measured with the downward passage times and edges with only
one endpoint on $s_{z}$ will be measured with the upward passage time. This
will give the required impediment to stop type 2 from blocking type 1 on
highways. Let $\mathcal{S}_{k}\subset\mathcal{P}_{k}$ be the set of sites that
satisfy these properties, so that
$\mathcal{S}_{k}=\mathcal{S}_{k}^{(1)}\cap\mathcal{S}_{k}^{(2)}\cap\mathcal{S}_{k}^{(3)}$,
where

$\displaystyle\mathcal{S}_{k}^{(1)}$
$\displaystyle=\left\\{z\in\mathcal{P}_{k}:T_{1}(s_{z})\leq(1-\varepsilon)k^{2}\right\\},$
$\displaystyle\mathcal{S}_{k}^{(2)}$
$\displaystyle=\left\\{z\in\mathcal{P}_{k}:\bigcap_{y\in s_{z}\backslash
s(z)}\bigcap\limits_{\begin{subarray}{c}y^{\prime}\sim y\\\ y^{\prime}\notin
s_{z}\end{subarray}}\left\\{T_{u}(yy^{\prime})\geq k^{3}\right\\}\right\\},$
$\displaystyle\mathcal{S}_{k}^{(3)}$
$\displaystyle=\left\\{z\in\mathcal{P}_{k}:\bigcap\limits_{\begin{subarray}{c}y,y^{\prime}\in
s_{z}\\\ y\sim y^{\prime}\end{subarray}}\left\\{T_{d}(yy^{\prime})\geq
k^{3}\right\\}\right\\}.$

It is clear the probability that a given site $z\in\mathcal{P}_{k}$ satisfies
the properties above is extremely small. We will recover that
$|\mathcal{S}_{k}|\geq 2$ with high probability so long as $\mathcal{P}_{k}$
contains sufficiently many elements. This can be achieved by setting
$r=k^{6}$, and henceforth, $r$ shall take this value. This is established
formally in the following lemma.

######  Lemma 3.9.

Let $r=k^{6}$ and assume $|\mathcal{P}_{k}|>\alpha^{r}$ for some
$\alpha\in(1,2)$. Then,

$\lim_{k\to\infty}\mathbb{P}\left(|\mathcal{S}_{k}|\geq 2\right)=1.$

###### Proof.

The collection of random variables
$\left\\{T_{1}(s_{z})\right\\}_{z\in\mathcal{P}_{k}}$ are independent, and by
Theorem 3.4, there exists a constant $\delta>0$ that does not depend on $k$,
such that for each $z\in\mathcal{P}_{k}$,

$\mathbb{P}\left(T_{1}(s_{z})\geq(1-\varepsilon)k^{2}\right)<e^{-\delta
k^{2}}.$

Moreover, since the probability that the type 2 passage time of an edge is at
least $k^{3}$ is $e^{-\lambda k^{3}}$, by independence of type 1 and type 2
passage times, for each $z\in\mathcal{P}_{k}$,

$\displaystyle\mathbb{P}$
$\displaystyle\left(\left\\{T_{1}(s_{z})<(1-\varepsilon)k^{2}\right\\}\cap\bigcap_{y\in
s_{z}\backslash s(z)}\bigcap\limits_{\begin{subarray}{c}y^{\prime}\sim y\\\
y^{\prime}\notin s_{z}\end{subarray}}\left\\{T_{u}(yy^{\prime})\geq
k^{3}\right\\}\cap\bigcap\limits_{\begin{subarray}{c}y,y^{\prime}\in s_{z}\\\
y\sim y^{\prime}\end{subarray}}\left\\{T_{d}(yy^{\prime})\geq
k^{3}\right\\}\right)$ $\displaystyle>\left(1-e^{-\delta
k^{2}}\right)e^{-\lambda k^{3}\cdot(d-1)k^{2}}$ $\displaystyle\geq
e^{-ck^{5}},$ (6)

for some constant $c=c(\lambda)>0$. From (6) we deduce the probability there
exist at least two such paths tends to 1 as $k$ goes to infinity as there are
more than $\alpha^{k^{6}}$ independent candidate paths. ∎

###  3.4 Controlling type 2 from conversions

In this section we consider how to control the spread of type 2 from
conversions. The idea is that we may set $\rho$ so small so that in good
boxes, no conversions take place until type 1 can spread far down the tree
with high probability. However, we must also control for conversions that
occur outside of good boxes. For example, type 1 may spread fast through a
good box and trigger an instantaneous conversion just outside it, that in turn
causes the spread of type 2. This spread may go back through the good box and
prevent highways from being occupied by type 1. The aim of this section is to
control how type 2 spreads from conversions outside of good boxes in such a
manner that type 1 is not impeded on highways.

Consider a box $B$ and suppose $|\mathcal{P}_{k}|>\alpha^{r}$ for some
$\alpha\in(1,2)$ and $|\mathcal{S}_{k}|\geq 2$, so that $\pi_{1}$ and
$\pi_{2}$ are highways of $B$ with respective spines $s_{1}$ and $s_{2}$. If
more highways and spines are available, we simply ignore them. The idea is
that type 2 must traverse a distance of at least $k^{2}$ to convert a site
outside $B$ and then occupy a site in $\pi_{1}\cup\pi_{2}$, and so by the time
type 1 occupies some $x\in\pi_{1}\cup\pi_{2}$, then it is able to propagate
downwards before being occupied by type 2. We make this notion rigorous below.

For $x\in\pi_{1}\cup\pi_{2}$, let $D(x)$ be the set of sites up to distance
$k^{2}$ down from $x$ that exclude all sites whose path to $x$ include an edge
on the highways $\pi_{1}$ and $\pi_{2}$ or the spines $s_{1}$ and $s_{2}$
(call this collection of excluded sites
$H_{k^{2}}(x;\pi_{1},\pi_{2},s_{1},s_{2})$), so

$D(x)=V^{\downarrow}_{k^{2}}(x)\backslash
H_{k^{2}}(x;\pi_{1},\pi_{2},s_{1},s_{2}).$

Let $\partial D(x)=\partial V_{k^{2}}(x)\cap D(x)$ and $D^{*}(x)$ be the event
that all paths through $D(x)$ _upwards_ have passage time at least $10k$, so
that

$D^{*}(x)=\bigcap_{y\in\partial D(x)}\left\\{T_{u}(y\to x)\geq 10k\right\\}.$
(7)

Intuitively, the event $D^{*}(x)$ guarantees that no upwards type 2 passage
time through $D(x)$ is fast enough to allow type 2 to catch type 1 on
$\pi_{1}\cup\pi_{2}$. It is important the event in (7) is independent of the
labelling sub-boxes as good or bad, as a good box is defined through the type
1 passage times and the _downward_ type 2 passage times.

The following lemma allows us to prove that $D^{*}(x)$ occurs for all
$x\in\pi_{1}\cup\pi_{2}$ with high probability, assuming the existence of
highways $\pi_{1},\pi_{2}$ and spines $s_{1},s_{2}$.

######  Lemma 3.10.

Let $r=k^{6}$. Assume $|\mathcal{P}_{k}|>\alpha^{r}$ for some $\alpha\in(1,2)$
and $|\mathcal{S}_{k}|\geq 2$. Let $\pi_{1}$ and $\pi_{2}$ be highways with
respective spines $s_{1}$ and $s_{2}$. There exists a constant $c>0$ such that

$\mathbb{P}\left(\bigcap_{x\in\pi_{1}\cup\pi_{2}}D^{*}(x)\right)\geq
1-e^{-ck^{2}\log k},$

for all $k$ large enough.

###### Proof.

If $\gamma$ is a path of length $k^{2}$ and $\theta>0$, then

$\mathbb{P}\left(T_{u}(\gamma)\leq
10k\right)=\mathbb{P}\left(\sum_{i=1}^{k^{2}}T_{u}(e_{i})\leq 10k\right)\leq
e^{10\theta k}\mathbb{E}\left[e^{-\theta T_{u}}\right]^{k^{2}},$

by a Chernoff bound argument, where $\left\\{e_{i}\right\\}_{i}$ is an
enumeration of the edges on the path $\gamma$ and $T_{u}$ is a copy of an
exponential random variable of rate $\lambda$. By setting $\theta=k/10$, we
deduce

$\mathbb{P}\left(T_{u}(\gamma)\leq
10k\right)\leq\left(\tfrac{\lambda}{\lambda+k/10}\right)^{k^{2}}e^{k^{2}}=e^{-ck^{2}\log
k},$

for some constant $c>0$. There are at most $d^{k^{2}}$ sites in $\partial
D(x)$ and hence, by the union bound,

$\mathbb{P}\left(\bigcup_{y\in\partial D(x)}\left\\{T_{u}(y\to x)\leq
10k\right\\}\right)\leq d^{k^{2}}e^{-ck^{2}\log k}\leq
e^{-\tfrac{c}{2}k^{2}\log k},$

for large enough $k$. Recall the depth of the percolating structure in a box
is $kr$ where $r=k^{6}$. By the union bound over all sites in
$\pi_{1}\cup\pi_{2}$,

$\mathbb{P}\left(\bigcup_{x\in\pi_{1}\cup\pi_{2}}\bigcup_{y\in\partial
D(x)}\left\\{T_{u}(y\to x)\leq 10k\right\\}\right)\leq
2k^{7}e^{-\tfrac{c}{2}k^{2}\log k},$

for all large enough $k$ and the result follows. ∎

###  3.5 Good boxes

In this section we put together the components constructed in the previous
sections to rigorously define good boxes. While the events discussed in
previous sections only concerned the box containing the origin, the events can
easily be translated to an arbitrary box when we partition $\mathbb{T}_{d}$
into boxes. Indeed, as we will see in this section, a box being good or bad
only depends on events measurable with respect to said box. Hence we will
retain the notation from previous sections without introducing ambiguity.

Fix a box $B$ and consider the following events. The first two events concern
the construction of highways down to depth $kr$, where we fix $r=k^{6}$ and
$\alpha\in(1,2)$, and at least two spines, so that

$\mathcal{G}_{1}=\left\\{\left|\mathcal{P}_{k}\right|>\alpha^{r}\right\\}\quad\mbox{and}\quad\mathcal{G}_{2}=\left\\{\left|\mathcal{S}_{k}\right|\geq
2\right\\}.$

Assuming that $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ hold, let $\pi_{1}$ and
$\pi_{2}$ be two highways to depth $kr$. We let $\mathcal{G}_{3}$ be the event
that all sites on $\pi_{1}\cup\pi_{2}$ satisfy the condition in (7), so that

$\mathcal{G}_{3}=\bigcap_{x\in\pi_{1}\cup\pi_{2}}D^{*}(x).$

The final ingredient is the event that no conversion occurs in the box until
at least time $3(kr+k^{2})$ has expired. This guarantees that type 1 is able
to traverse a large distance before type 2 originating from that box is able
to spread. Call this event $\mathcal{G}_{4}$, so that

$\mathcal{G}_{4}=\bigcap_{x\in B}\left\\{\mathcal{I}_{x}\geq
3(kr+k^{2})\right\\},$

where we recall $\mathcal{I}_{x}$ is the conversion time to type 2 at site
$x$.

We define a box $B$ to be _good_ if $\cap_{j=1}^{4}\mathcal{G}_{j}$ holds, and
_bad_ otherwise.

######  Lemma 3.11.

We have

$\lim_{k\to\infty}\lim_{\rho\downarrow 0}\mathbb{P}\left(B\emph{ is
good}\right)=1.$

###### Proof.

As the conversion times are independent of the passage times, we may express
the probability of a box being good as

$\mathbb{P}\left(B\mbox{ is
good}\right)=\mathbb{P}\left(\cap_{j=1}^{4}\mathcal{G}_{j}\right)=\mathbb{P}\left(\mathcal{G}_{1}\right)\mathbb{P}\left(\mathcal{G}_{2}|\mathcal{G}_{1}\right)\mathbb{P}\left(\mathcal{G}_{3}|\mathcal{G}_{1}\cap\mathcal{G}_{2}\right)\mathbb{P}\left(\mathcal{G}_{4}\right).$

As the event $\mathcal{G}_{3}$ only observes the upwards type 2 passage times,
then it is independent of $\mathcal{G}_{1}\cap\mathcal{G}_{2}$. The result
then follows by Lemma 3.6, Lemma 3.9 and Lemma 3.10.

∎

###  3.6 Proof of Theorem 1.1

In this section we prove Theorem 1.1. Before proceeding with the proof, we
introduce the following notion that will ease exposition. If a site $z$ is
occupied by type 2 because it converts to type 2 due to its own conversion
time expiring after being occupied by type 1, then we say $z$ is its own
_progenitor_. Otherwise, $z$ was occupied by type 2 by the spread of type 2
from a neighbouring site, $z^{\prime}$ say. If $z^{\prime}$ is its own
progenitor, then we say it is the progenitor for $z$ too. If $z^{\prime}$ is
not its own progenitor, then we can iterate this procedure until we find a
site that is the progenitor for $z$ and itself. In general, we write $p(z)$
for the progenitor for $z$ and $p_{z}$ as the path from $p(z)$ to $z$. Note a
progenitor must be occupied by type 1 at some time due to being occupied by
type 2 through its own conversion. As the passage times are exponentially
distributed, the progenitor is almost surely unique.

###### Proof of Theorem 1.1.

Let $\varepsilon$ be small enough and $k$ large enough so that Lemma 3.3
holds, and fix $\lambda<1+\varepsilon$. We construct an infinite path from
$\sigma$ that only passes through highways and spines of good boxes with
positive probability as follows. Consider the box up to depth $kr+k^{2}$ with
root $\sigma$. If this box is good, there exists at least two sites at depth
$kr+k^{2}$ that are connected to the root $\sigma$ via highways and spines of
this box. Considering these sites as roots of boxes to depth $2(kr+k^{2})$, by
observing if these boxes are good or bad, we deduce a lower bound on the
number of sites connected to $\sigma$ only through highways and spines of good
boxes. We can continue this procedure $n$ times to attempt to find sites at
depth $n(kr+k^{2})$ from the origin that only pass through the highways and
spines of good boxes. As boxes constructed this way are independently good or
bad, if we set $k$ large enough and then $\rho$ small enough so that

$\mathbb{P}\left(V_{kr+k^{2}}\mbox{ is good}\right)>1/2,$ (8)

this procedure does not terminate with positive probability. If this procedure
does not terminate, there exists an infinite path $\gamma$ from $\sigma$ that
only passes through the highways and spines of good boxes. We aim to prove the
existence of $\gamma$ implies type 1 survives, which would complete the proof.

Suppose for contradiction there exists an infinite path from $\sigma$ that
only passes through the highways and spines of good boxes, $\gamma$ say, and
type 1 does not occupy every site on $\gamma$. Then there must exist some good
box $B_{0}$ such that $\gamma$ contains a highway and spine of $B_{0}$, type 1
occupies some site in $B_{0}\cap\gamma$ but not every site in
$B_{0}\cap\gamma$.

Write $B_{0}\cap\gamma=\pi\cup s$ for the respective highway $\pi$ and spine
$s$ of $B_{0}$, and let $\sigma_{B_{0}}$ denote the root of $B_{0}$. Suppose
there exists $z\in\pi\cup s$ that is never occupied by type 1. Without losing
generality, we may assume $z$ is the closest such site to $\sigma_{B_{0}}$.
Recall $p(z)$ is the progenitor of $z$ and $p_{z}$ is the path from $p(z)$ to
$z$.

First consider the case $p(z)\in B_{0}$ (see Figure 3(a) for an illustration).
Then

$\tau_{2}(z)>\tau_{1}(p(z))+3(kr+k^{2})>\tau_{1}(\sigma_{B_{0}})+3(kr+k^{2}),$
(9)

where in the first inequality we recall $B_{0}$ is good and the second follows
as type 1 must occupy $\sigma_{B_{0}}$ before $p(z)$ by construction. As
$B_{0}$ is good, then

$T_{1}(\sigma_{B_{0}}\to z)<(1-\varepsilon)(kr+k^{2}).$ (10)

(a) $p(z)\in B_{0}$

(b) $p_{z}\cap\partial V_{kr+k^{2}}^{\downarrow}(\sigma_{B_{0}})\neq\emptyset$

(c) $p_{z}\cap\partial V_{kr+k^{2}}^{\downarrow}(\sigma_{B_{0}})=\emptyset$
and $p(z)\notin B_{0}$

Figure 3: The three cases for the nature of $p_{z}$, the path from $p(z)$ to
$z$. The red line is the path $p_{z}$. The top of the triangle is the root
$\sigma_{B_{0}}$ and the base represents the set $\partial
V_{kr+k^{2}}^{\downarrow}(\sigma_{B_{0}})$.

Comparing (9) and (10), we deduce

$\tau_{1}(z)<\tau_{1}(\sigma_{B_{0}})+(1-\varepsilon)(kr+k^{2})<\tau_{2}(z),$

which contradicts the assumption $z$ is never occupied by type 1. Hence
$p(z)\notin B_{0}$.

Now consider the case $p_{z}\cap\partial
V_{kr+k^{2}}^{\downarrow}(\sigma_{B_{0}})\neq\emptyset$ (see Figure 3(b)). Let
$z^{\prime}\in\pi\cup s$ be such that

$d(\sigma_{B_{0}},z^{\prime})=\min_{y\in B_{0}\cap\gamma\cap
p_{z}}d(\sigma_{B_{0}},y).$

It is immediate $d(\sigma_{B_{0}},z^{\prime})\leq d(\sigma_{B_{0}},z)$ through
the construction of the progenitor.

First consider the case $z\in\pi$. Then

$\tau_{2}(z)>\tau_{2}(z^{\prime})+\tfrac{(1-\varepsilon^{2})k}{\lambda}\left(\text{SB}(z,z^{\prime})-2\right)^{+},$

where SB$(z,z^{\prime})$ is the number of sub-boxes that intersect the path
from $z^{\prime}$ to $z$ and for $c\in\mathbb{R}$, we write
$(c)^{+}=\max\left\\{c,0\right\\}$. Recalling $\lambda<1+\varepsilon$, (7) and
that $B_{0}$ is good, we deduce

$\tau_{2}(z)>\tau_{1}(z^{\prime})+10k+(1-\varepsilon)\left(\text{SB}(z,z^{\prime})-2\right)^{+}k.$
(11)

Similarly, recalling $\pi$ is a highway and $B_{0}$ is good, we deduce

$T_{1}(z^{\prime}\to z)<(1-\varepsilon)\text{SB}(z,z^{\prime})k.$ (12)

By comparing (11) and with (12), we observe

$\tau_{1}(z)<\tau_{1}(z^{\prime})+(1-\varepsilon)\text{SB}(z,z^{\prime})k<\tau_{2}(z),$

which contradicts the assumption $z$ is never occupied by type 1.

Now consider the case $z\in s$ and $z\notin\pi$. By similar considerations to
the previous case,

$\tau_{2}(z)>\tau_{1}(z^{\prime})+(1-\varepsilon)\left(\text{SB}(z,z^{\prime})-2\right)^{+}k+k^{3},$
(13)

where the $k^{3}$ term is because at least one edge incident to the spine $s$
must be traversed for type 2 to propagate to $z$. Note that the function SB
only counts sub-boxes up to depth $kr$ in a box and not up to depth
$kr+k^{2}$. Hence

$T_{1}(z^{\prime}\to
z)<(1-\varepsilon)\text{SB}(z,z^{\prime})k+(1-\varepsilon)k^{2},$ (14)

where the $(1-\varepsilon)k^{2}$ term is from the construction of the spine.
By comparing (13) with (14), we have

$\tau_{1}(z)<\tau_{1}(z^{\prime})+(1-\varepsilon)\text{SB}(z,z^{\prime})k+(1-\varepsilon)k^{2}<\tau_{2}(z),$

which again gives a contradiction.

Finally, consider the case $p_{z}\cap\partial
V_{kr+k^{2}}^{\downarrow}(\sigma_{B_{0}})=\emptyset$ and $p(z)\notin B_{0}$
(see Figure 3(c)). Under this assumption, $p_{z}$ contains $\sigma_{B_{0}}$.
Moreover, as $p(z)\notin B_{0}$, $p_{z}$ also contains a site neighbouring
$\sigma_{B_{0}}$ in a spine of a good box that $\gamma$ passes through before
entering $B_{0}$. Call this site $u$. As $u$ is contained in a spine of a good
box, then

$\tau_{1}(\sigma_{B_{0}})<\tau_{1}(u)+(1-\varepsilon)k^{2},$ (15)

where we observe $\tau_{1}(u)<\infty$ as $\tau_{1}(\sigma_{B_{0}})<\infty$.
The downwards passage time from $u$ to $\sigma_{B_{0}}$ is at least $k^{3}$
and thus

$\tau_{2}(\sigma_{B_{0}})>\tau_{2}(u)+k^{3}>\tau_{1}(u)+k^{3}>\tau_{1}(\sigma_{B_{0}})+k^{3}-(1-\varepsilon)k^{2},$
(16)

where in the final inequality we use (15). If $z\in\pi$, by considering the
type 2 passage times from $\sigma_{B_{0}}$ to $z$ along the highways, we
deduce

$\tau_{2}(z)>\tau_{1}(\sigma_{B_{0}})+k^{3}-(1-\varepsilon)k^{2}+(1-\varepsilon)\left(\text{SB}(\sigma_{B_{0}},z)-2\right)^{+}k.$
(17)

By construction of the highway $\pi$, we have

$T_{1}(\sigma_{B_{0}}\to z)<(1-\varepsilon)\text{SB}(\sigma_{B_{0}},z)k.$ (18)

From (17) and (18), we have

$\tau_{1}(z)<\tau_{1}(\sigma_{B_{0}})+(1-\varepsilon)\text{SB}(\sigma_{B_{0}},z)k<\tau_{2}(z).$

However, this contradicts the assumption $z$ is never occupied by type 1. A
similar argument can be used if $z\in s$ and $z\notin\pi$ as in the previous
case. As we have considered all possible cases for $z$ and $p(z)$, the
contradiction is established and the proof is complete. ∎

##  4 Behaviour on $\mathbb{Z}^{d}$

In this section we consider the model on $\mathbb{Z}^{d}$. The model can be
constructed in the exact same way as before except we no longer distinguish
between upwards and downwards type 2 passage times. That is, we define
$\left\\{t_{1,e}\right\\}_{e\in E(\mathbb{Z}^{d})}$ and
$\left\\{t_{2,e}\right\\}_{e\in E(\mathbb{Z}^{d})}$ to be i.i.d. collections
of exponentially distributed random variables of rate 1 and rate $\lambda$ on
the edges of $\mathbb{Z}^{d}$, respectively.

Type 1 spreads according to the $\left\\{t_{1,e}\right\\}_{e\in
E(\mathbb{Z}^{d})}$ passage times to vacant sites. Once a site $z$ is occupied
by type 1, it attempts to convert to type 2 after waiting $\mathcal{I}_{z}$
time, where $\left\\{\mathcal{I}_{x}\right\\}_{x\in\mathbb{Z}^{d}}$ is an
i.i.d. collection of exponentially distributed random variables of rate
$\rho>0$. Type 2 spreads according to the $\left\\{t_{2,e}\right\\}_{e\in
E(\mathbb{Z}^{d})}$ passage times to vacant sites and sites occupied by type
1.

###  4.1 Proof of Theorem 1.3

In order to prove Theorem 1.3, we first recall some classical results for
first passage percolation on $\mathbb{Z}^{d}$.

Given sites $x,y\in\mathbb{Z}^{d}$, define the _passage time from $x$ to $y$_
as the random variable

$T_{1}(x,y)=\inf_{\gamma\in\Gamma_{x,y}}T_{1}(\gamma)$

where $T_{1}$ is as defined in (1) and $\Gamma_{x,y}$ is the set of all finite
paths from $x$ to $y$. For $t\geq 0$, let

$B(t)=\left\\{x\in\mathbb{Z}^{d}:T_{1}(0,x)\leq t\right\\}$

be the ball of radius $t$ centred at the origin under the metric induced by
the exponential passage times of rate 1. The idea of the shape theorem is
$B(t)$ converges to a deterministic shape once linearly rescaled in time. To
make sense of rescaling, let

$\tilde{B}(t)=\left\\{x+[-\tfrac{1}{2},\tfrac{1}{2})^{d}:T_{1}(0,x)\leq
t\right\\}$

be the set of sites in $B(t)$ considered as the centre of a unit cube in
$\mathbb{R}^{d}$.

######  Theorem 4.1 (Richardson [17]).

There exists a deterministic, convex, compact set
$\mathcal{B}_{1}\subset\mathbb{R}^{d}$ such that for all $\varepsilon>0$

$\mathbb{P}\left((1-\varepsilon)\mathcal{B}_{1}\subset\frac{\tilde{B}(t)}{t}\subset(1+\varepsilon)\mathcal{B}_{1}\emph{
for all large }t\right)=1.$

The set $\mathcal{B}_{1}$ is called the limit shape.

The proof of Theorem 4.1 relies on subadditivity arguments through Kingman’s
subadditive ergodic theorem [13]. Consequently, the exact limiting shape is
not known. Richardson’s shape theorem has been extended to more general
passage times by Cox and Durrett [7] and recent results about the limit shape
can be found in [2].

By time scaling, if one replaced rate 1 passage times with rate $\lambda$
passage times for some $\lambda>0$, then there would be a limit shape
$\mathcal{B}_{\lambda}$ such that
$\mathcal{B}_{\lambda}=\lambda\mathcal{B}_{1}$.

To prove Theorem 1.3, we first need to prove the set of sites ever occupied by
type 1 is contained in a first passage percolation process of rate strictly
less than one. This is the content of the following lemma. Let $\eta_{1}(t)$
(resp. $\eta_{2}(t)$) be the set of sites ever occupied by type 1 (resp. type
2) up to time $t$.

######  Lemma 4.2.

Consider the model on $\mathbb{Z}^{d}$ with $d\geq 2$ and $\rho,\lambda>0$.
There exists $\kappa=\kappa(\rho,d)>0$ such that for all $\varepsilon>0$,

$\eta_{1}(t)\subset(1+\varepsilon)(1-\kappa)t\mathcal{B}_{1}$

for all large enough $t$, almost surely.

###### Proof.

The proof is consequence of van den Berg and Kesten [3] (see [18, Proposition
6.4] for the result in terms of the limit shape). Their result states if a
distribution $F$ strictly dominates333We say a distribution $F$ _stricly
dominates_ a distribution $\tilde{F}$ if there exists a coupling between these
two distributions under which the random variable with distribution $F$ is
larger than the random variable with distribution $\tilde{F}$ with probability
$1$. another distribution $\tilde{F}$, then the limiting shape under $F$ is
strictly contained in the limiting shape under $\tilde{F}$, under some natural
conditions on $F$ and $\tilde{F}$. The idea is to prove type 1 is contained in
a process whose passage times strictly dominate a first passage percolation
process with exponential passage times of rate 1. However, there are subtle
dependencies arising because the spread of type 1 is facilitated by the
interplay of small type 1 passage times and large conversion times. We need to
prove we can construct the model in a manner that decouples these two sources
of randomness in order to apply a van den Berg–Kesten argument.

For $K>0$, let $u(K)$ be the probability an exponential random variable of
rate $\rho$ is at least $K$. Given $x\in\mathbb{Z}^{d}$, let $x$ be _marked_
with probability $1-u(K)$, and _unmarked_ with probability $u(K)$,
independently of every other site. Define an edge $e$ to be marked if both of
its endpoints are marked. The set of marked edges gives a 1-dependent
percolation process. By Liggett, Schonmann and Stacey [16], for any
$q\in(0,1)$, the set of marked edges stochastically dominates an i.i.d.
Bernoulli percolation process of parameter $q$ for a large enough choice of
$K$. The open edges according to this i.i.d. Bernoulli percolation process are
defined to be _semi-marked_.

For each edge $e$, determine whether $e$ is semi-marked and sample a candidate
type 1 passage time $\tilde{t}_{1,e}$ that is an exponential random variable
of rate 1, independently of every other edge. If $e$ is not semi-marked, we
let $f_{1,e}=\tilde{t}_{1,e}$. If $e$ is semi-marked, we set

$f_{1,e}=\begin{cases}\tilde{t}_{1,e}&\mbox{if }\tilde{t}_{1,e}\leq K,\\\
\infty&\mbox{if }\tilde{t}_{1,e}>K.\end{cases}$

Hence the evolution for type 1 is identical under
$\left\\{f_{1,e}\right\\}_{e}$ and $\left\\{\tilde{t}_{1,e}\right\\}_{e}$ by
this construction. The passage times given by $\left\\{f_{1,e}\right\\}_{e}$
are i.i.d. and are stochastically dominated by i.i.d. exponential passage
times of rate 1. Hence we may apply van den Berg–Kesten and the result follows
from Theorem 4.1. ∎

With Lemma 4.2 to hand, we are now in a position to prove Theorem 1.3.

###### Proof of Theorem 1.3.

Let $\kappa$ be as in Lemma 4.2 so for any $\varepsilon>0$,

$\eta_{1}(t)\subset(1+\varepsilon)(1-\kappa)t\mathcal{B}_{1},$

for all $t$ large enough, almost surely. By Theorem 4.1, if we only consider
the evolution of type 2 after the origin has been converted, for any
$\varepsilon>0$,

$\eta_{2}(t)\supset(1-\varepsilon)t\mathcal{B}_{\lambda}=(1-\varepsilon)\lambda
t\mathcal{B}_{1},$

for all $t$ large enough, almost surely. If $\lambda>1-\kappa$ and we fix
$\varepsilon$ small enough so

$(1+\varepsilon)(1-\kappa)<(1-\varepsilon)\lambda,$

we deduce type 1 dies out almost surely as $\eta_{1}(t)\subset\eta_{2}(t)$ for
all large enough $t$. ∎

###  4.2 Sidoravicius–Stauffer percolation

The proof of Theorem 1.5 relies on a coupling between our converting first
passage percolation model with a random competition process called
_Sidoravicius–Stauffer percolation_ (SSP). SSPs where introduced by Dauvergne
and Sly [8] as a streamlined version of FPPHE from [18]. The purpose of this
section is to define SSPs and outline an important encapsulation theorem by
Dauvergne and Sly.

An SSP on $\mathbb{Z}^{d}$ consists of two competing growth processes
$\mathfrak{R}(t)$ and $\mathfrak{B}(t)$ for $t\geq 0$, called red and blue for
clarity, respectively. Let $\overrightarrow{\text{E}}$ be the set of directed
edges on $\mathbb{Z}^{d}$, so that

$\overrightarrow{\text{E}}=\left\\{(u,v):(u,v)\in
E\left(\mathbb{Z}^{d}\right)\right\\}.$

To define an SSP we require the following.

  * •

Functions $X_{\mathfrak{R}}:\overrightarrow{\text{E}}\to[0,1]$ and
$X_{\mathfrak{B}}:\overrightarrow{\text{E}}\to[0,\infty)$ viewed as the clocks
defining the growth of the red and blue process, respectively.

  * •

A collection of blue seeds $\mathfrak{B}_{*}\subset\mathbb{Z}^{d}$.

  * •

A parameter $\kappa>1$ such that $X_{\mathfrak{B}}(u,v)\leq\kappa$ for all
$(u,v)\in\overrightarrow{\text{E}}$.

If $u$ and $v$ are both blue seeds, we take $X_{\mathfrak{B}}(u,v)=0$. That
is, the blue process spreads instantaneously through connected components of
blue seeds. For our purposes, the blue process spreads after waiting time
$\kappa$ through every other edge. Hence we will always consider the special
case where the blue process spreads according to the clocks given by

$X_{\mathfrak{B}}:\overrightarrow{\text{E}}\to[0,\kappa]\mbox{ with
}X_{\mathfrak{B}}(u,v)=\begin{cases}0&\mbox{if }u,v\mbox{ are blue seeds},\\\
\kappa&\mbox{otherwise}.\end{cases}$

At time $t=0$, the red process only occupies the origin while the blue process
is dormant in seeds. SSPs evolve in time through the following dynamics. Given
a site $u$, let $T(u)$ be the earliest time the red or blue process occupied
$u$ and $C(u)\in\\{\mathfrak{R},\mathfrak{B}\\}$ denote the colour of $u$ once
it is occupied. Given an edge $(u,v)$, at time $T(u)+X_{C(u)}(u,v)$, the edge
$(u,v)$ will ring and the process evolves in the following manner. If
$T(v)<T(u)+X_{C(u)}(u,v)$, then the occupation is suppressed as $y$ has
already been coloured by an invasion from another edge. Otherwise, we colour
$v$ according to the following rules:

  * •

If $C(u)=\mathfrak{R}$ and $v\in\mathfrak{B}_{*}$, then $C(v)=\mathfrak{B}$.

  * •

If $C(u)=\mathfrak{R}$ and $v\notin\mathfrak{B}_{*}$, then
$C(v)=\mathfrak{R}$.

  * •

If $C(u)=\mathfrak{B}$, then $C(v)=\mathfrak{B}$.

The construction of SSPs in [8] allows for the red process to invade through
blue sites in some circumstances due to caveats in their application. They
couple SSPs with a spread of infection model and blue regions are only where
they cannot guarantee the infection is moving fast enough. For our
application, we do not need to consider these cases and so the above
construction will suffice.

We define the red (resp. blue) process to _survive_ if there is an infinite
connected region of red (resp. blue) sites in the limit as $t\to\infty$.
Otherwise we say the red (resp. blue) process _dies out_.

The following is an encapsulation result of Dauvergne and Sly [8, Theorem
2.14], where the red process survives and encapsulates all blue regions with
positive probability, so long as $\kappa$ is large enough (so the blue process
is substantially slower than the red process) and blue seeds are
stochastically dominated by an i.i.d. Bernoulli process of small enough
parameter.

######  Theorem 4.3.

Consider a random SSP $(\mathfrak{R},\mathfrak{B})$ on $\mathbb{Z}^{d}$ driven
by potentially random clocks $X_{\mathfrak{R}},X_{\mathfrak{B}}$, a collection
of blue seeds $\mathfrak{B}_{*}$ and a constant parameter $\kappa>4000$.
Suppose additionally $\mathfrak{B}_{*}$ is stochastically dominated by an
i.i.d. Bernoulli process of parameter $p>0$. There exists a universal constant
$c>0$ such that the probability the red process survives and the blue process
dies out is at least $1-cp$.

The statement of Theorem 4.3 suffices for our purposes and is provided in a
more detailed manner in [8].

###  4.3 Proof of Theorem 1.5

In this section we prove Theorem 1.5 and in doing so, partially answer
Conjecture 1.4. We first prove $\rho_{\ell}>0$ through a coupling with an
appropriate SSP. To facilitate this coupling, we need to provide an
alternative construction of the model on $\mathbb{Z}^{d}$.

Type 1 evolves according the passage times $\left\\{t_{1,e}\right\\}_{e\in
E(\mathbb{Z}^{d})}$ and converts to type 2 according to the conversion times
$\left\\{\mathcal{I}_{x}\right\\}_{x\in\mathbb{Z}^{d}}$, as before. When a
site $x$ is occupied by type 2, it attempts to spread to neighbouring sites,
according to the passage times $\left\\{t_{2,e}\right\\}_{e\in
E(\mathbb{Z}^{d})}$. That is, if $x$ is first occupied by type 1 at time $s$
and $y\sim x$ is vacant at time $s+t_{2,xy}$, then $y$ is occupied by type 2
at time $s+t_{2,xy}$. If $y$ is occupied by type 1 at some time
$s^{\prime}\in[s,s+t_{2,xy})$, type 2 now attempts to occupy $x$ at time
$s^{\prime}+t_{3,xy}$ where $\left\\{t_{3,e}\right\\}_{e\in
E(\mathbb{Z}^{d})}$ is an i.i.d. collection of exponential random variables of
rate $\lambda$. By the memoryless property of the exponential distribution,
this construction is equivalent to how the model is defined before.

###### Proof of Theorem 1.5: $\rho_{\ell}>0$..

Let $C$ be a large constant we fix later. Define a site $x$ to be a _type
2-seed_ if at least one of the following holds.

  * •

There exists $y\sim x$ such that $t_{1,xy}\geq C$.

  * •

There exists $y\sim x$ such that $t_{2,xy}<C^{2}$.

  * •

There exists $y\sim x$ such that $t_{3,xy}<C^{2}$.

  * •

The conversion time at $x$ satisfies $\mathcal{I}_{x}<C^{2}$.

We now construct the appropriate SSP required for the coupling argument. The
blue seeds for the SSP are given by the type 2-seeds above. Define the clocks
$X_{\mathfrak{R}}$ and $X_{\mathfrak{B}}$ governing the spread of the red and
blue process as follows:

$X_{\mathfrak{R}}:\overrightarrow{\text{E}}\to[0,C]\mbox{ with
}X_{\mathfrak{R}}(x,y)=\min\\{t_{1,xy},C\\},$

and

$X_{\mathfrak{B}}:\overrightarrow{\text{E}}\to[0,C^{2}]\mbox{ with
}X_{\mathfrak{B}}(x,y)=\begin{cases}0&\mbox{if }x,y\mbox{ are blue seeds},\\\
C^{2}&\mbox{otherwise}.\end{cases}$

Note the change from $X_{\mathfrak{R}}$ being bounded by 1 to being bounded by
$C$ amounts to a time change and does not alter any arguments. The only edges
where the red clock is equal to $C$ must have a type 2-seed at each endpoint
and thus play no role in the evolution of the red process.

We deduce if a site $x$ is not a type 2-seed and occupied by type 1, it is
able to attempt to spread type 1 to all of its neighbours. For example, if
$y\sim x$, then

$t_{1,xy}<\min_{z\sim x}\left\\{t_{2,xz},t_{3,xz},\mathcal{I}_{x}\right\\}$

via the construction of type 2-seeds and so the propagation of type 1 from $x$
cannot be blocked by the spread of type 2 or conversions. This is precisely
why we needed to define the passage times $\left\\{t_{3,e}\right\\}_{e\in
E(\mathbb{Z}^{d})}$, so the spread of type 2 from a neighbouring site does not
block the spread of type 1.

If there where no type 2-seeds, it is immediate type 1 survives as type 2 does
not have the potential to block its spread. Through this construction, type 2
only can block type 1 through the spread from type 2-seeds. Sites that can
potentially be blocked from type 1 through type 2-seeds are then the blue
process while the remaining sites are red. Hence, proving the red process
survives implies type 1 survives.

It is easy to verify this construction yields a valid SSP in the language from
[8]. Moreover, by setting $C$ large enough and then $\lambda$ and $\rho$ small
enough, the probability a site is a type 2-seed can be made arbitrarily small.
The process of labelling sites as type 2-seeds defines a 1-dependent
percolation process and so can be constructed to be stochastically dominated
by an i.i.d. Bernoulli process of parameter $p$, for any $p\in(0,1)$ by
Liggett, Schonmann and Stacey [16]. This observation allows us to deduce from
Theorem 4.3 that for $\lambda$ and $\rho$ small enough, the red process
survives and occupies infinitely many sites, and all connected components of
the blue process are finite, with positive probability. ∎

The proof that $\rho_{u}$ exists and is finite requires less machinery.

###### Proof of Theorem 1.5: $\rho_{u}<\infty$.

Define a site $x$ to be _closed_ if the minimum type 1 passage time on an edge
incident to $x$ is greater than the time it takes $x$ to convert once occupied
by type 1, so that

$\min_{y\sim x}t_{1,xy}>\mathcal{I}_{x}.$

Note that closed sites cannot pass type 1 to any of their neighbors. The
process of labelling sites closed is a 1-dependent percolation process with

$\lim_{\rho\to\infty}\mathbb{P}\left(x\mbox{ is closed}\right)=1.$

Through Liggett, Schonmann and Stacey [16], by setting $\rho$ large enough, we
deduce closed sites stochastically dominate a supercritical i.i.d. Bernoulli
percolation process. Hence, for all large enough $\rho$, type 1 dies out
almost surely as the origin is encapsulated by closed sites. ∎

##  Appendix A Appendix: Standard large deviation results

######  Lemma A.1 (Chernoff bounds for Poisson random variables).

Let $P$ be a Poisson random variable of mean $\mu$. For any
$\varepsilon\in(0,1)$,

$\mathbb{P}\left(P<(1-\varepsilon)\mu\right)<\exp\left\\{-\mu\varepsilon^{2}/2\right\\}$

and

$\mathbb{P}\left(P>(1+\varepsilon)\mu\right)<\exp\left\\{-\mu\varepsilon^{2}/4\right\\}.$

For any $C>0$ and $\theta\in\mathbb{R}$, we have

$\mathbb{P}\left(P>C\mu\right)\leq\exp\left\\{-\mu\left(1-e^{\theta}+\theta
C\right)\right\\}.$

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# Effect of Backing on Neutron Spectra for Low Energy Quasi-Mono-energetic
p+7Li Reaction.

H. Kumawat1111author. Email address<EMAIL_ADDRESS>1Nuclear Physics
Division, BARC, Mumbai-400085, India

###### Abstract

$\underline{\textbf{MO}}$nte-carlo $\underline{\textbf{N}}$ucleon transport
$\underline{\textbf{C}}$ode (MONC) for nucleon transport is extended for below
20MeV proton transport using ENDF and EXFOR data base. It is used to simulate
p+7Li reaction upto 20MeV proton energies and produced neutron spectra are
reported here. The simulated results are compared with calculated values from
other available codes like PINO, EPEN, SimLiT and experimental data. The
spectra reported here can be used to get the neutron cross-section for the
quasi-mono-energetic neutron reaction and will help to subtract the low energy
contribution. The neutron spectra also useful as this reaction is used for
accelerator based Boron Neutron Capture Therapy. The backing materials are
used to fully stop the proton beam hence contribution from the neutrons from
backing materials is estimated. It is found that Tantalum is good backing
material below $\sim$8 MeV and Carbon is better at higher energies.

Monte Carlo, Quasi-mono-energetic Neutron Source, Boron Neutron Capture
Therapy

##  I Introduction

Measurement of neutron cross-section is an important research activity for
it’s application in nuclear reactors, cancer therapy, neutron dosimetry,
nuclear astrophysics etc Bowman et al. (1992); Carminti et al. (1993); Tolstov
(1989); Bharti and Singh (2018); Lederer et al. (2016); Reifarth et al.
(2014); Couture et al. (2021); Koehler and O’Brien (1989); Dillmann et al.
(2023). The 7Li(p, n)X reaction is used as quasi mono-energetic neutron source
to measure cross-sections and also a possible accelerator based source of
neutrons for Boron Neutron Capture Therapy (BNCT) Wang et al. (2023); Nakamura
et al. (2019); Kononov et al. (2003). The threshold for the 7Li(p, n)7Beg
reaction is $\sim$1.88 MeV and cross-section rise rapidly near the threshold
energy. For protons above 2.37 MeV, production of another group of neutrons
starts due to inelastic state of 7Be at 429 keV. Neutron production threshold
for three-body breakup channel 7Li(p, n+3He)$\alpha$ is 3.7 MeV which gives a
broad neutron energy distribution. Thus, 7Li(p, n)X reaction can be considered
quasi-mono-energetic near threshold but it has a tail above 4 MeV of proton
energies. It can be still used as a quasi-mono-energetic source for neutron
threshold reactions where low energy tail does not contribute. The
contribution of tail should be carefully subtracted for neutron capture
reactions where low energy contributes more in neutron activation technique in
cross-section measurement.

Additionally, the spread in the neutron spectra occurs due to thickness of the
Lithium target (up to 100 $\mu$m), used by various experimentalists and the
corresponding neutron energy spread may be up to 500keV. Some experimentalists
use cadmium foil to cut very low energy neutron contribution but it is
important to quote the energy spectra for a particular measurement. Recently,
several experiments have been conducted at BARC-FOTIA Lawriniang et al.
(2015); Vansola et al. (2015); Ghosh et al. (2016); Bakshi et al. (2020) and
BARC-TIFR Singh et al. (2023a); Mukherjee et al. (2019); Singh, R. K. et al.
(2021); Singh et al. (2023b); Soni et al. (2020); Singh et al. (2022)
facilities using this reaction. MONC Kumawat and Venkata (2020, 2013); Kumawat
et al. (2016a) code is used for Monte-carlo simulations for Lithium target of
thickness 4mg/cm2 which is typical thickness used at many experimental
facilities in Mumbai, India. The contribution of low energy tail and second
peak should be considered while quoting cross-section for a single energy and
in best practices it should be subtracted as mentioned in Ref. Smith et al.
(2005) or by similar methods. The neutron flux monitor reaction should be
sensitive in the similar energy range as that of reaction of measurement. The
calculations are also performed using code PINO Reifarth et al. (2009) which
includes only 7Li(p, n0)7Beg and 7Li(p, n1)7Be∗ reactions hence valid in low
energy region ($<$ 7.0 MeV). The simulated values are also compared with
available experimental data and calculated values from literature by SimLiT
and EPEN Pachuau, Rebecca et al. (2017) at 3.5 MeV.

At higher energies, the proton beam has to stop in some other material
otherwise the quasi-mono-energeticity does not remain valid. Usually Tantalum
and Carbon are used as the backing materials. The neutrons produced by the
backing material should be estimated to get the total neutron spectrum.

The outline of this paper is as follows. In Sec. II we present brief
description of MONC. Section III contains simulation results. Conclusions are
given in Sec. IV.

##  II Brief description of MONC

Monte Carlo program MONC incorporates Intra-nuclear Cascade, Pre-equilibrium,
Evaporation and Fission models to simulate spallation reaction mechanism for
thin and thick targets. Modeling details of Intra-nuclear cascade, Pre-
equilibrium particle emission are described in detail in Ref. Barashenkov and
Toneev (1972); Barashenkov (2000). Fission barrier, level density parameter
and inverse cross sections for pre-equilibrium/evaporation/fission model are
given in detail in Ref. Kumawat (2004); Kumawat and Barashenkov (2005).

Benchmark of spallation models for experimental values of neutron, charged
particles, and pions double differential production cross-sections, particle
multiplicities, spallation residues and excitation functions was organized by
IAEA and is given in Ref. Kumawat (March 2010). We have used the predecessor
of this code named CASCADE.04 to calculate these quantities in the IAEA
benchmark. Heat Deposition algorithm for thick spallation targets and thin
films was modified and benchmarked as mentioned in Ref. Kumawat et al. (2008).
The code was further developed for the Neutron shielding and dosimetry
applications and published Kumawat et al. (2009). Energy loss of the charge
particle is calculated during the transport in the thick target.

MONC realizes the particle transport in three stages: 1) sampling of particle
(ion) mean free path in the medium taking into account the energy loss of a
charged particle and a possible decay of non-stable particles ($\pi^{0}$,
$\pi^{\pm}$). All $\pi^{0}$-mesons are considered to decay into
$\gamma$-quanta at the point of their creation. The ionization losses of $\pi$
\- mesons, protons and light ions are calculated by Sternheimer’s method
Sternheimer (1971) using well established Bathe formula for the average
ionization loss calculations with proper density effects. Here, it is
important to mention that the density effect shows reduction in ionization
loss for fast charged particles due to dielectric polarization of the medium.
In the lower energy region ($<$ 2.0 MeV) Lindhard’s approach Lindhard et al.
(1963) is used and a semi-phenomenological procedure Barashenkov et al. (1993)
is applied for the heavy ions. While doing the practical simulation one has to
calculate the ionization and nuclear interaction ranges and then uses the
formulation to deposit heat.

2) Simulation of the particle interaction with a nucleus is considered along
its path. In case of inelastic interaction the MONC code considers three
stages of reaction for calculation: a) intranuclear cascade originally
developed at Dubna: In this part of the calculation, primary particles can be
re-scattered and they may produce secondary particles several times prior to
absorption or escape from the target. Modeling of intra-nuclear cascades
Barashenkov and Toneev (1972); Barashenkov (2000) is in general rather closer
to the methods used in other transport codes. Cross-sections of the hadron-
nucleus collisions are calculated based on the compilations of the
experimental data Barashenkov (1993); Barashenkov and Polanski (1994). To
calculate the nucleus-nucleus cross-sections we used analytical approximations
with parameters defined in Ref. Barashenkov and Kumawat (2003). b) Pre-
equilibrium stage: In this part of the reaction, relaxation of the nuclear
excitation is treated according to the exciton model of the pre-equilibrium
decay. The relaxation is calculated by the method based on the Blann’s model
Mashnik and Toneev (1974); Blann et al. (1976). Proton, neutron, deuterium,
tritium, 3He and 4He are considered as emitted particles in the pre-
equilibrium and in the subsequent equilibrium stage. c) Equilibrium stage:
This part considers the particle evaporation/fission of the thermally
equilibrated nucleus.

Figure 1: 7Li(p,x)Y cross-section from ENDF VIII.0 as used in the MONC. 7Li(p,
n+3He)$\alpha$ reaction cross-section is taken from Ref. Scott (1971)

3) Low energy neutron transport code is developed recently. A package has been
developed for reading pointwise cross sections for neutron in ACE (A Compact
ENDF) format. ENDF data file processing and generating point data at different
temperatures has also been developed Kumawat et al. (2016b). The delayed
neutrons are treated exclusively with their energy spectra for which data are
available. Spontaneous and induced fission fragment yield are read from ENDF
fission yield libraries. The free gas thermal treatment of the neutron
interaction for below 4eV can be used for compound and crystal material or
Thermal scattering law can be used if available in ENDF file. Probability
table method is used in the un-resolved energy region. Low energy ($<$ 20 MeV)
proton data are used to simulate the reaction mechanism and outgoing particles
energy and angular distributions. No cross-section data are given for 7Li(p,
n+3He)$\alpha$ reaction in the ENDF file so it is taken from Ref. Scott (1971)
while energy and angular distributions are calculated using 3-body kinematics.
The cross-sections used in the present simulations are given in Fig. 1 for 7Li
nucleus. There is a thick tantalum or carbon sheet placed at the end of
Lithium target to stop the proton beam. The cross-section for Tantalum and
Carbon are taken from Evaluated neutron data libraries and EXFOR experimental
database. The energy and angular distributions are calculated using 2, 3-body
kinematics.

Figure 2: Proton induced neutron spectrum from 7Li(p,n)7Be reaction.
Experimental data are taken from Ref. Lederer et al. (2012); Feinberg et al.
(2012); Ratynski and Käppeler (1988) and calculations are performed using
MONC.

##  III  Simulation and Results

Monte-carlo simulations are carried out for 4mg/cm2 thick 7Li target which is
92.41% in the natural Lithium and contributes most for neutron production in
the natural Lithium target. Proton energies considered are up to 21MeV (Namely
6, 10, 15 and 21 MeV). Experimental data near threshold energy at 1.912 MeV
Lederer et al. (2012); Feinberg et al. (2012); Ratynski and Käppeler (1988)
are compared in Fig. 2. Angular distributions are given in the ENDF library
and corresponding energy is calculated using 2-body kinematics 2dk (2023). The
calculated spectrum from MONC has overall agreement with slight
underestimation at peak position of the energy spectrum.

Figure 3: Proton (Ep=3.5 MeV) induced neutron spectrum from 7Li(p,n)X reaction
for 0∘-5∘ of neutrons simulated using MONC. Experimental data are taken from
Ref. Lefevre and Din (1969) Figure 4: Proton (Ep=3.5 MeV) induced neutron
spectrum from 7Li(p,n)X reaction for neutrons simulated using MONC and PINO.
Values for EPEN/SimLiT are taken from Ref. Pachuau, Rebecca et al. (2017)
Figure 5: Proton (Ep=3MeV) induced neutron spectrum from 7Li(p,n)X reaction
for 0∘-10∘ of neutrons simulated using MONC.

In case of thick Lithium target, experimental data at 3.45 MeV and 0∘ are
taken from Ref. Lefevre and Din (1969) and simulations are done for
$\theta_{n}$ = 0-5∘ using MONC. A comparison is shown in Fig. 3. There is an
agreement from peak energy up to highest neutron energy. MONC overestimate the
neutron spectrum below the peak neutron energy.

Simulation was performed for 38$\mu$m thick target where EPEN and SimLiT
Pachuau, Rebecca et al. (2017) calculated values were also available in the
literature. The calculations are also performed using PINO code Reifarth et
al. (2009) for comparison which is available on web portal. The published
values from EPEN/SimLiT Pachuau, Rebecca et al. (2017) are in good agreement
with PINO and MONC calculated values for the first group of the neutron
energies (see Fig. 4) (ground state transition to 7Be) but the second group of
neutron corresponding to the excited state transition to 7Be is underestimated
by PINO. The data for second group of neutrons by EPEN is not given in this
publication hence could not be compared.

Figure 6: Proton (Ep=21, 15, 10 and 6 MeV) induced neutron spectrum from
7Li(p,n)X, 181Ta(p,n)X and natC(p,n)X reaction for 0∘-10∘ of neutrons
simulated using MONC.

Neutron spectra were simulated at 6, 10, 15 and 21 MeV proton energies. The
Lithium target thickness was taken as 4mg/cm2. Tantalum and Natural Carbon
were taken as backing material to stop the protons. The spectra for neutron
angle $\theta_{n}$ 0-10∘ are given in Fig. 6 where usually samples are kept to
measure neutron cross-sections. The Lithium spectra show a third group of
neutrons from 6 to 20 MeV proton energies which is coming from three body
breakup reaction as mentioned above. The neutrons from natural carbon target
is contributed by 13C(p, n) reaction which has 1.1% abundance. The barrier for
this reaction is $\sim$1.15 MeV and threshold is 3.24 MeV. The neutron
production for Tantalum is through 181Ta(p, n), 181Ta(p, 2n) and 181Ta(p, 3n)
reactions with thresholds of 0.99 MeV, 7.70 MeV and 16.16 MeV, respectively.
It is clear from the spectra that large contribution of neutrons is produced
from Tantalum at higher energies. Carbon target gives less contribution by
factor of two orders of magnitudes compared to that profuced from 7Li target.

##  IV  Conclusion

The Monte Carlo code MONC has been developed for proton induced reactions at
low energies. The 7Li(p, n)X reaction which is widely used for neutron cross-
section measurement and a potential candidate for Boron Neutron Capture
Therapy, is investigated. The simulated neutron spectra are compared with the
experimental data and calculated values from PINO and EPEN/SimLiT codes and
the results are in good agreement for the first group 7Li(p,n)7Beg at least.
There is a disagreement for the second group of neutrons through 7Li(p,n)7Be∗
reaction. PINO shows very small contribution from second group of neutrons
coming from excited 7Be state at 429 keV. Ratio of these two groups are
measured at different angle Bevington et al. (1961) which shows a relative
contribution of 10-15% around 0∘-10∘ neutron emission angles and the ratio
from MONC are of similar magnitudes for these group of neutrons. The neutron
spectra at forward angles are given here because the measurements are done in
the forward direction using neutron activation technique. The simulated
neutron spectra are useful for the experimental measurements using neutron
activation analysis although we use exact geometry of the arrangement of
samples, sample holders, monitor foils etc. Tawade et al. (2022) for our
measurements and Monte carlo code is best suited for that compared to
deterministic codes. The contribution to the neutron spectra by backing
material is significant by Tantalum at higher energies and negligible below
$\sim$8 MeV. The contribution from Natural carbon as backing material is less
by two orders of magnitudes. The spectra from Carbon does not interfere with
the major quasi-energetic peak while tantalum gives neutrons beyond this peak.
Hence, Natural Carbon is better as backing material compared to Tantalum at
higher energies.

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