# Growth properties of the $\infty$-parent spatial Lambda-Fleming Viot process

Apolline Louvet  MAP5, Université Paris Cité, CNRS, 45 rue des Saints-Pères,
75006 Paris, France  CMAP, Ecole Polytechnique, CNRS, Institut Polytechnique
de Paris, Route de Saclay, 91128 Palaiseau Cedex, France  Amandine Véber
MAP5, Université Paris Cité, CNRS, 45 rue des Saints-Pères, 75006 Paris,
France

###### Abstract

The $\infty$-parent spatial Lambda-Fleming Viot process, or $\infty$-parent
SLFV, is a model for spatially expanding populations in which empty areas are
filled with "ghost" individuals. The interest of this process lies in the fact
that it is akin to a continuous-space version of the classical Eden growth
model, while being associated to a dual process encoding ancestry and allowing
one to study the evolution of the genetic diversity in such a population.

In this article, we focus on the growth properties of the $\infty$-parent SLFV
in two dimensions. To do so, we first define the quantity that we shall use to
quantify the speed of growth of the area covered with the subpopulation of
real individuals. Using the associated dual process and a comparison with a
first-passage percolation problem, we show that the growth of the “occupied"
region in the $\infty$-parent SLFV is linear in time. We use numerical
simulations to approximate the growth speed, and conjecture that the actual
speed is higher than the speed expected from simple first-moment calculations
due to the characteristic front dynamics.

We then study a toy model of two interacting growing piles of cubes in order
to understand how the growth dynamics at the front edge can increase the
global growth speed of the "occupied" region. We obtain an explicit formula
for this speed of growth in our toy model, using the invariant distribution of
a discretised version of the model. This study is of interest on its own
right, and its implications are not restricted to the case of the
$\infty$-parent SLFV.

rowth properties of the $\infty$-parent SLFV

Keywords: spatial Lambda-Fleming Viot process, range expansion, duality,
percolation, stochastic growth process, sub-additivity, speed of growth

MSC 2020 Subject Classification: Primary: 60G55, 60G57, 82C43, 92D25
Secondary: 60J25, 60G50, 60K20, 82C41

###### Contents

  1. 1 Introduction
  2. 2 The $\infty$-parent spatial Lambda-Fleming Viot process with elliptical reproduction events
    1. 2.1 Definition of the process
    2. 2.2 Dual process and duality relation
    3. 2.3 Speed of growth of the occupied region in the $\infty$-parent SLFV: Definition and main result
    4. 2.4 Numerical simulations
  3. 3 Lower bound on the speed of growth
    1. 3.1 Sub-additivity
    2. 3.2 Definition of the express chain
    3. 3.3 Properties of the express chain
  4. 4 Upper bound on the speed of growth
    1. 4.1 A first-passage percolation problem
    2. 4.2 Discretisation of the $\infty$-parent ancestral process
    3. 4.3 Comparison to the first-passage percolation problem
  5. 5 The two column growth process
    1. 5.1 The two column growth process: Definition and properties
    2. 5.2 The discretised two columns growth process
    3. 5.3 Invariant distribution of the discretised 2-CGP
  6. A Geometrical properties of ellipses

##  1 Introduction

The $\infty$-parent spatial $\Lambda$-Fleming Viot process, or $\infty$-parent
SLFV, was introduced in [32] as a model for expanding populations in
$\mathbb{R}^{2}$. Its main feature is the use of "ghost" individuals
(thereafter referred to as "type $0$" individuals) to fill empty areas,
adapting ideas from [13, 23]. In [13, 23, 32], ghost individuals can reproduce
as well, modelling stochastic fluctuations in population sizes, but with a
selective disadvantage against real individuals (thereafter referred to as
"type $1$" individuals), ensuring that population expansions can indeed occur.
In [32], it is shown that the $\infty$-parent SLFV corresponds to the limit of
an SLFV with strong selection when the selective advantage of type $1$
individuals over type $0$ individuals becomes infinitely strong. Therefore, in
the limiting regime, we no longer observe local extinctions due to the
stochasticity in reproduction at the front, where densities of type $1$
individuals are lower.

One of the main interests of the $\infty$-parent SLFV lies in its dual
interpretation, both as a stochastic growth model akin to models from
percolation theory, and as a population genetics process with a spatial
structure. Indeed, if we consider the area occupied by type $1$ individuals,
then its growth dynamics corresponds to a continuous space version of the Eden
model [14]. Moreover, and contrary to the Eden model, the $\infty$-parent SLFV
is equipped with tools allowing one to investigate genetic diversity patterns,
assuming that real individuals are further subdivided into different types
(for instance using the concept of "tracers", see [13, 23]). Therefore, it is
potentially suited to study how the characteristic genetic diversity patterns
observed in real expanding populations [21, 22, 24] arise. In this paper, we
shall only focus on the region covered by real individuals, without specifying
different subtypes of real (or type $1$) individuals. However, understanding
the family structure in this model is a first step towards the understanding
of how genetic diversity evolves in a multitype population.

The $\infty$-parent SLFV process belongs to the family of spatial
$\Lambda$-Fleming Viot processes, which are characterised by a reproduction
dynamics driven by an exogeneous Poisson point process of reproduction events
rather than by individual reproduction. Informally, the $\infty$-parent SLFV
is constructed as follows. We consider that at any time $t$, the area occupied
by type $1$ individuals is represented by a measurable set
$\mathfrak{E}_{t}\subseteq\mathbb{R}^{d}$. Reproduction events are given by a
Poisson point process $\Pi$ on $\mathbb{R}_{+}\times\mathbb{R}^{2}$ with
intensity measure $\alpha dt\otimes dx$, $\alpha>0$, which encodes the
sequence of times and centres of the reproduction events. For the moment, to
ease the exposition, we consider that all reproduction events have the same
shape: a ball with fixed radius $\mathcal{R}>0$. In Section 2, we shall
consider the more general case of ellipses with random bounded parameters.
Then, for all $(t,z)\in\Pi$, if $\mathcal{B}_{\mathcal{R}}(z)$ is the ball
with centre $z$ and radius $\mathcal{R}$ and if
$\mathrm{Vol}(\mathfrak{E}_{t-}\cap\mathcal{B}_{\mathcal{R}}(z))\neq 0$, we
set $\mathfrak{E}_{t}=\mathfrak{E}_{t-}\cup\mathcal{B}_{\mathcal{R}}(z)$, and
we do nothing otherwise. In other words, whenever a reproduction event occurs,
if the corresponding area contains some real individuals, then one of them
reproduces and completely fills the area with its descendants. When we deal
with genealogies, we shall assume that the reproducing individual is chosen
uniformly at random among the real individuals occupying the ball just before
the event (though in fact we shall not need to specify an ancestor in our
approach below).

Formulated this way, the $\infty$-parent SLFV is very similar to models from
continuous first-passage percolation, and in particular to the model
introduced by Deijfen in [10], in which reproduction occurs when
$z\in\mathfrak{E}_{t-}$. Contrary to the $\infty$-parent SLFV, this process
does not jump when $z\notin\mathfrak{E}_{t-}$ but
$\mathrm{Vol}(\mathfrak{E}_{t-}\cap\mathcal{B}_{\mathcal{R}}(z))>0$, and so we
expect its growth speed to be different from that of the $\infty$-parent SLFV
(see [10, 11]). Moreover, and perhaps more importantly, the $\infty$-parent
SLFV is defined for more general initial conditions, which are not restricted
to $\mathrm{Vol}(\mathfrak{E}_{0})<+\infty$. In particular, this means that
the definition we just provided is only informal: when
$\mathrm{Vol}(\mathfrak{E}_{0})=+\infty$, an infinite number of reproduction
events intersect the occupied area in any time interval, and the jump rate of
the process is infinite. Rigorously, the $\infty$-parent SLFV is a measure-
valued process, which can be defined as the limit of a sequence of SLFVs with
strong selection, or (under stricter conditions which will be satisfied in
this article) as the unique solution to a well-posed martingale problem [32].
If $\mathrm{Vol(\mathfrak{E}_{0})}<+\infty$, it can also be defined as a
process taking its values in the class of measurable subsets of
$\mathbb{R}^{d}$. In what follows, we shall take $\mathfrak{E}_{0}$ such that
$\mathrm{Vol}(\mathfrak{E}_{0})=+\infty$ and we shall use the characterisation
of the process as the solution to a well-posed martingale problem (and in fact
the dual process offered by this approach will be the key ingredient of most
of our proofs).

While most of the studies on SLFVs with selection were carried out in a weak
selection regime (see, _e.g._ , [16, 17, 18] for selection against one type,
[20] for general forms of selection, [6, 8, 29] for fluctuating selection and
[15] for selection against heterozygosity), the $\infty$-parent SLFV
corresponds to a strong selection limit, motivated by the interpretation of
ghost individuals as modelling empty areas. In particular, and contrary to
previous works on SLFVs with selection, the results we obtain do not require
to rescale time, nor space. Moreover, in most SLFVs (as well as in the
original $\infty$-parent SLFV, see [32]), the affected areas are balls, whose
radii can be fixed or random. However, biological experiments suggest that the
shape of the areas impacted by reproduction events influences the observed
genetic diversity patterns [22]. Moreover, this excludes the case of a
preferential expansion direction, for example towards a resource by a
phenomenon similar to chemotaxis. Therefore, in this article, we consider a
variant of the $\infty$-parent SLFV in which the area affected by a
reproduction event can be shaped like an ellipse rather than a ball.

In this article, we are interested in the growth properties of the occupied
area in the $\infty$-parent SLFV and we focus on the following questions: Is
the growth linear in time ? To what extent is the speed of growth affected by
the reproduction dynamics at the front edge ? How is it affected by the shape
of the reproduction events ? In particular, we would like to compare the
growth properties of the occupied region in the $\infty$-parent SLFV to those
of first-passage percolation models, whose most well-known representative is
probably the Eden growth model. In order to do so, we first give a brief
overview of what is known of their growth properties. First-passage
percolation processes are stochastic growth models originally defined on a
lattice (generally $\mathbb{Z}^{d},\,d\geq 2$). In these models, each vertex
is either occupied, or empty. If vertex $x\in\mathbb{Z}^{d}$ becomes occupied
at time $t$, and if it is connected to a vertex $y\in\mathbb{Z}^{d}$ by an
edge (denoted $e$), then vertex $y$ becomes occupied at time $t+\tau_{e}$,
where $\tau_{e}$ is independent from one edge to another (but not necessarily
identically distributed). Whether the expansion is linear in time depends on
the distribution of the time needed to pass through any given edge of the grid
(see, _e.g._ , [4]) and whether one considers short-range percolation, such as
nearest neighbour percolation, or long-range percolation, in which two
vertices $x,y\in\mathbb{Z}^{d}$ are connected by an edge no matter the
distance between them [7, 9, 34]. In particular, when edge passing times are
distributed as in the Eden growth model, growth is linear in time for short-
range percolation and potentially faster for long-range percolation, depending
on the relation between the distribution of $\tau_{e}$ and the distance
between the two vertices it connects [4, 7]. See, _e.g._ , [10, 12, 26] for
examples of extensions of first-passage percolation models to a continuous
setting.

When the growth is linear in time, in general it is possible to obtain lower
and upper bounds on the speed of growth (see, _e.g._ , [2, 38]), or to use
simulations to approximate it [1]. For other stochastic growth models, such as
the corner growth model [36], which belongs to the family of last-passage
percolation models, it is possible to obtain an explicit speed of growth for
specific passage time distributions [35]. As many other growth models, the
Eden model is conjectured to belong to the universality class of the Kardar-
Parisi-Zhang (KPZ) equation [28]. This equation generates rough fronts, whose
characteristics are similar to the ones of fronts observed in some expanding
biological populations (see, _e.g._ , [27]). Such a conjecture is notably
difficult to establish; to our knowledge, it has only been demonstrated in the
case of the solid-on-solid growth model in [5].

The main result of this article is Theorem 2.11 in Section 2.3, and states
that when the area affected by a reproduction event has bounded shape
parameters, the growth of the area occupied by type $1$ individuals is linear
in time. In particular, this means that compared to the model of Deijfen [10],
the increased reproduction rate at the front and the unbounded initial
condition do not lead to superlinear growth. To prove this result, we first
use sub-additivity arguments to show that the growth is at least linear in
time. Next, we use a comparison with a short-range percolation model to show
that the growth is also at most linear in time, which allows us to conclude.
Our proof also yields an explicit lower bound on the speed of growth.

In Section 2.4, we use numerical simulations to approximate the growth speed
of the area occupied by type $1$ individuals, and we compare it to the lower
bound we obtained. This bound was initially conjectured to be equal to (or at
least a good approximation for) the growth speed, but the simulations show
that the actual speed is significantly higher. Moreover, they suggest that the
growth is driven by "spikes" which occur at the front and then thicken in all
directions.

In order to understand how the spike phenomenon can make the front of an
expanding population advance faster, we consider a simple toy model in Section
5, composed of two interacting growing piles of cubes. Its main interest is
that it is possible to obtain an explicit expression for its speed of growth.
In this simplified model, new cubes can also grow sideways, at a rate which is
proportional to the height difference between the two piles. We show that the
interaction between the two piles leads to an increase in the speed of growth
of each pile by a factor of $\sim 1.46$ compared to the rate at which each
pile grows by itself independently of the other pile, which is in line with
numerical simulations. The proof relies on a coupling with a discretised
version of the process, for which it is possible to compute the invariant
distribution.

More precisely, the paper is structured as follows. In Section 2, we define
the variant of the $\infty$-parent SLFV with elliptical reproduction events
rigorously, introduce its dual, and define what we mean by the "speed of
growth" of the occupied region in the process. We also state the main result
of the paper, Theorem 2.11, and analyse numerical simulations of the
$\infty$-parent SLFV in Section 2.4. The proof of Theorem 2.11 spans two
sections. In Section 3, we use the dual process to show that the growth is at
least linear in time, and provide a lower bound on the speed of growth of the
process. In Section 4, we use a comparison with a first-passage percolation
process (also based on the dual process) to show that the growth is at most
linear in time. Combining the results from both sections yields Theorem 2.11.
In Section 5, we study the toy model of interacting growing piles of cubes,
and obtain an explicit expression for its speed of growth. This result is of
interest in its own right, but also explains the discrepancy between the
numerical approximation for the speed of growth of the occupied area in the
$\infty$-parent SLFV and the value initially conjectured.

##  2 The $\infty$-parent spatial Lambda-Fleming Viot process with elliptical
reproduction events

In this section, we rigorously define the $\infty$-parent SLFV, in the version
we use in this article. Indeed, the process was originally defined with
reproduction events occurring in balls, while here we consider that they
rather occur in ellipses, as it is straightforward to generalise the
construction in [32] to our case. Then, we introduce the dual process of
potential ancestors associated to the $\infty$-parent SLFV. We conclude by
formalising what we mean by the speed of growth of the occupied area, and by
explaining how to study it using the dual process.

###  2.1 Definition of the process

All the random objects we consider in this section are defined over some
probability space $(\Omega,\mathcal{F},\mathbb{P})$. We use the notation
$\mathbb{N}=\\{0,1,2,...\\}$. In all that follows, let $\tilde{\mu}$ be a
finite measure on $(0,+\infty)^{2}\times(-\pi/2,\pi/2)$ such that there exists
$\mathcal{R}>0$ satisfying

$\tilde{\mu}\Big{(}\big{(}(0,\mathcal{R}]\times(0,\mathcal{R}]\big{)}^{c}\times(-\pi/2,\pi/2)\Big{)}=0.$

Let $\mathcal{R}_{\tilde{\mu}}$ be the smallest $\mathcal{R}>0$ such that this
condition is satisfied. We also set

$S_{\tilde{\mu}}=(0,\mathcal{R}_{\tilde{\mu}}]\times(0,\mathcal{R}_{\tilde{\mu}}]\times(-\pi/2,\pi/2).$

#### Ellipses.

We first set the notation regarding ellipses.

######  Definition 2.1.

Let $z_{c}=(x_{c},y_{c})\in\mathbb{R}^{2}$, $(a,b)\in(0,+\infty)^{2}$ and
$\gamma\in(-\pi/2,\pi/2)$. The ellipse with centre $z_{c}$ and parameters
$(a,b,\gamma)$, denoted by $\mathfrak{B}_{a,b,\gamma}(z_{c})$, is defined by:

$\mathfrak{B}_{a,b,\gamma}(z_{c})=\left\\{\begin{pmatrix}x_{c}\\\
y_{c}\end{pmatrix}+A_{\gamma}\begin{pmatrix}ar\cos(\theta)\\\
br\sin(\theta)\end{pmatrix}:r\in[0,1],\theta\in[0,2\pi)\right\\}$

where

$A_{\gamma}=\begin{pmatrix}\cos(\gamma)&-\sin(\gamma)\\\
\sin(\gamma)&\cos(\gamma)\end{pmatrix}.$

See Figure 1 for an illustration. We denote the volume of an ellipse with
parameters $(a,b,\gamma)$ by
$V_{a,b,\gamma}:=\text{Vol}(\mathfrak{B}_{a,b,\gamma}(0))$.

Figure 1: Ellipse with parameters $(a,b,\gamma)$.

For all $(a,b)\in(0,+\infty)^{2}$ and $\gamma\in(-\pi/2,\pi/2)$, if $f$ is an
element of the space $C_{c}(\mathbb{R}^{2})$ of all continuous and compactly
supported functions $\mathbb{R}^{2}\to\mathbb{R}$, let

$\text{Supp}^{a,b,\gamma}(f):=\left\\{z\in\mathbb{R}^{2}:\text{Vol}\left(\mathfrak{B}_{a,b,\gamma}(z)\cap\text{Supp}(f)\right)\neq
0\right\\},$

where $\text{Supp}(f)$ stands for the support of $f$. The set
$\text{Supp}^{a,b,\gamma}(f)$ can be interpreted as the set of all potential
centres $z\in\mathbb{R}^{2}$ for ellipses with parameters $(a,b,\gamma)$
overlapping the support of $f$. For all $z\in\mathbb{R}^{2}$ and
$\omega:\mathbb{R}^{2}\to\\{0,1\\}$ Lebesgue-measurable, let
$\Theta_{z}^{a,b,\gamma}(\omega):\mathbb{R}^{2}\to\\{0,1\\}$ be the function
defined by

$\Theta_{z}^{a,b,\gamma}(\omega):=\mathds{1}_{\\{\mathfrak{B}_{a,b,\gamma}(z)^{c}\\}}\times\omega.$

If $\omega$ represents the density of type $0$ (or ghost) individuals, then
$\Theta_{z}^{a,b,\gamma}(\omega)$ corresponds to filling the ellipse
$\mathfrak{B}_{a,b,\gamma}(z)$ with type $1$ individuals, without affecting
the rest of $\mathbb{R}^{2}$.

#### State space.

We now introduce the state space over which the $\infty$-parent SLFV is
defined. Let $\widetilde{\mathcal{M}}_{\lambda}$ be the space of all measures
on $\mathbb{R}^{2}\times\\{0,1\\}$ whose marginal distribution over
$\mathbb{R}^{2}$ is Lebesgue measure. In other words,
$\widetilde{\mathcal{M}}_{\lambda}$ is the space of all measures $M$ on
$\mathbb{R}^{2}\times\\{0,1\\}$ such that the following property is satisfied:

$\forall f\in
C_{c}(\mathbb{R}^{2}),\int_{\mathbb{R}^{2}\times\\{0,1\\}}f(z)M(dz,dk)=\int_{\mathbb{R}^{2}}f(z)dz.$

By a standard decomposition theorem, for all
$M\in\widetilde{\mathcal{M}}_{\lambda}$, there exists
$\omega:\mathbb{R}^{2}\to[0,1]$ measurable such that

$M(dz,dk)=(\omega(z)\delta_{0}(dk)+(1-\omega(z))\delta_{1}(dk))dx.$ (2.1)

Let $\mathcal{M}_{\lambda}$ be the set of all measures
$M\in\widetilde{\mathcal{M}}_{\lambda}$ such that there exists
$\omega:\mathbb{R}^{2}\to\\{0,1\\}$ (instead of $[0,1]$) satisfying (2.1). For
all $M\in\mathcal{M}_{\lambda}$, we refer to any measurable function
$\omega:\mathbb{R}^{2}\to\\{0,1\\}$ satisfying (2.1) as a density of $M$, and
denote it by $\omega_{M}$. Note that $\omega_{M}$ is not unique, but two
densities will only differ at a Lebesgue-null set of points. Therefore, since
we shall only consider integrals of continuous functions with respect to the
measures describing the current state of the population, the choice of the
density $\omega_{M}$ used in the analysis below will not matter. Moreover,
$\omega_{M}$ represents the density in type $0$ (or ghost) individuals. This
choice is motivated by the duality formula proved in [32], which we recall in
Section 2.2.

We endow $\mathcal{M}_{\lambda}$ with the topology of vague convergence. Let
$D_{\mathcal{M}_{\lambda}}[0,+\infty)$ be the space of all càdlàg
$\mathcal{M}_{\lambda}$-valued paths, endowed with the standard Skorokhod
topology.

######  Remark 2.2.

Assume that the state of the population at time $t$ is encoded by
$M\in\mathcal{M}_{\lambda}$. Since the density $\omega_{M}$ is not uniquely
defined, for any given location $z\in\mathbb{R}^{2}$, $\omega_{M}(z)$ is not
uniquely defined. Hence, it cannot be interpreted as the type of the
individuals living there. Note however that just below Definition 2.9, we
shall introduce a way to uniquely assign a type to any given location, which
does not depend on the choice of the density.

#### Martingale problem.

When reproduction events have bounded shape parameters, it is possible to
define the $\infty$-parent SLFV as the solution to a well-posed martingale
problem. In order to do so, we need to introduce some more notation. Let
$C^{1}(\mathbb{R})$ be the space of all continuously differentiable functions
$F:\mathbb{R}\to\mathbb{R}$. For all $f\in C_{c}(\mathbb{R}^{2})$ and $F\in
C^{1}(\mathbb{R})$, if $\omega:\mathbb{R}^{2}\to\\{0,1\\}$ is measurable, we
set

$\langle\omega,f\rangle:=\int_{\mathbb{R}^{2}}f(z)\omega(z)dz.$

Moreover, we define the function
$\Psi_{F,f}:M\in\mathcal{M}_{\lambda}\to\Psi_{F,f}(M)\in\mathbb{R}$ by

$\forall
M\in\mathcal{M}_{\lambda},\Psi_{F,f}(M):=F\left(\langle\omega_{M},f\rangle\right).$

The functions $\Psi_{F,f}$ with $F\in C^{1}(\mathbb{R})$ and $f\in
C_{c}(\mathbb{R}^{2})$ will be used as the set of functions on which the
martingale problem characterising the process of interest is defined. This
martingale problem is associated to the operator
$\mathcal{L}_{\tilde{\mu}}^{\infty}$ defined as follows. For all
$M\in\mathcal{M}_{\lambda}$,

$\displaystyle\mathcal{L}_{\tilde{\mu}}^{\infty}\Psi_{F,f}(M):=\int_{S_{\tilde{\mu}}}\int_{\text{Supp}^{a,b,\gamma}(f)}$
$\displaystyle\left(1-\delta_{0}\left(\int_{\mathfrak{B}_{a,b,\gamma}(z)}\left(1-\omega_{M}(z^{\prime})\right)dz^{\prime}\right)\right)$
$\displaystyle\times\left(F\left(\langle\Theta_{z}^{a,b,\gamma}(\omega_{M}),f\rangle\right)-F\left(\langle\omega_{M},f\rangle\right)\right)dz\tilde{\mu}(da,db,d\gamma).$

This operator encodes exactly the dynamics described informally in the
Introduction. Indeed, reproduction events of centre $z$ and parameters
$(a,b,\gamma)$ happen at rate $dz\otimes\tilde{\mu}(da,db,d\gamma)$. Whenever
a reproduction event occurs, then:

  1. 1.

If the affected area $\mathfrak{B}_{a,b,\gamma}(z)$ contains a positive
fraction of (real) type $1$ individuals (_i.e._ , if
$\int_{\mathfrak{B}_{a,b,\gamma}(z)}\left(1-\omega_{M}(z^{\prime})\right)dz^{\prime}\neq
0$), then it is filled with type $1$ individuals, as encoded by the action of
$\Theta_{z}^{a,b,\gamma}$ on $\omega_{M}$.

  2. 2.

Otherwise, it stays filled with (ghost) type $0$ individuals.

We then have the following characterisation of the $\infty$-parent SLFV
process with elliptical reproduction events.

######  Theorem 2.3.

For all $M^{0}\in\mathcal{M}_{\lambda}$, there exists a unique
$D_{\mathcal{M}_{\lambda}}[0,+\infty)$-valued process $(M_{t})_{t\geq 0}$ such
that $M_{0}=M^{0}$ and, for all $F\in C^{1}(\mathbb{R})$ and $f\in
C_{c}(\mathbb{R}^{2})$,

$\left(\Psi_{F,f}(M_{t})-\Psi_{F,f}(M_{0})-\int_{0}^{t}\mathcal{L}_{\tilde{\mu}}^{\infty}\Psi_{F,f}(M_{s})ds\right)_{t\geq
0}$

is a martingale.

Moreover, this process is Markovian.

The proof of this theorem is a direct generalisation of the proofs of the
second part of Theorem 2.14 and of Lemma 3.8 in [32], and so we omit it.

######  Definition 2.4.

Let $M^{0}\in\mathcal{M}_{\lambda}$. The $\infty$-parent spatial
$\Lambda$-Fleming Viot process with elliptical reproduction events (or
$\infty$-parent SLFV) with initial condition $M^{0}$ associated to
$\tilde{\mu}$ is the unique solution to the martingale problem
$(\mathcal{L}_{\tilde{\mu}}^{\infty},\delta_{M^{0}})$ stated in Theorem 2.3.

######  Remark 2.5.

From a population genetics viewpoint, the $\infty$-parent SLFV can be seen as
modelling the spread of an extremely advantageous gene, corresponding to the
"real" type. For weaker strengths of selection, the most classical tool to
study this question is the Fisher-KPP equation [19, 30], defined as follows.
If the proportion of individuals of the favoured type at the spatial location
$x\in\mathbb{R}^{d},\,d\geq 1$, and at time $t\geq 0$ is given by
$p(t,x)\in[0,1]$, then $p(t,x)$ evolves according to the Fisher-KPP equation
if it is a weak solution to the equation

$\frac{\partial p}{\partial t}(t,x)=\frac{m}{2}\Delta
p(t,x)dt+s_{0}p(t,x)(1-p(t,x)),\quad\forall x\in\mathbb{R}^{d},\forall t\geq
0,$ (2.2)

where $m\geq 0$ and $s_{0}\geq 0$ are respectively the diffusion and selection
parameters.

In dimension $1$, it is possible to add some stochasticity through a Wright-
Fisher noise term. This cannot be done in higher dimensions. Equation (2.2)
and its stochastic counterpart in dimension $1$ both admit travelling wave
solutions (see, _e.g._ , [33]), corresponding to a linear speed of
spread/growth. However, the spatio-temporal scales at which the spread of a
weakly advantageous allele is modelled by the Fisher-KPP equation are very
different from the ones in which the spread of an extremely advantageous gene
is visible. The approaches and results based on the model developped here and
on the Fisher-KPP equation are thus difficult to compare.

The interest of the $\infty$-parent SLFV as a model for expanding populations
lies in the fact that it is associated to a dual process of potential
ancestors, which can be used to study the properties of the $\infty$-parent
SLFV. We now define this dual process, and state the duality relation.

###  2.2 Dual process and duality relation

The dual process introduced in this section is defined on a different
probability space $(\mathbf{\Omega},\mathbf{\mathcal{F}},\mathbf{P})$.

#### State space.

Let $\mathcal{E}^{c}$ be the set of all subsets of $\mathbb{R}^{2}$ which are
Lebesgue-measurable, connected, and whose Lebesgue measure is finite and non-
zero. The state space we consider for the dual process is then the set
$\mathcal{E}^{cf}$ of all finite unions of elements of $\mathcal{E}^{c}$.

#### Definition of the dual process.

We can now define the dual process, called the $\infty$-parent ancestral
process.

######  Definition 2.6.

Let $\overleftarrow{\Pi}$ be a Poisson point process with intensity measure
$dt\otimes dz\otimes\tilde{\mu}(da,db,d\gamma)$ on
$\mathbb{R}_{+}\times\mathbb{R}^{2}\times S_{\tilde{\mu}}$, defined on the
probability space $(\mathbf{\Omega},\mathbf{\mathcal{F}},\mathbf{P})$, and let
$E^{0}\in\mathcal{E}^{cf}$. The $\mathcal{E}^{cf}$-valued $\infty$-parent
ancestral process $(E_{t}^{\infty})_{t\geq 0}$ with initial condition $E^{0}$
associated to $\tilde{\mu}$ is defined as follows.

First, we set $E_{0}^{\infty}=E^{0}$. Then, for all
$(t,z,a,b,\gamma)\in\overleftarrow{\Pi}$, if
$E_{t-}^{\infty}\cap\mathfrak{B}_{a,b,\gamma}(z)$ has non-zero Lebesgue
measure, we set

$E_{t}^{\infty}=E_{t-}^{\infty}\cup\mathfrak{B}_{a,b,\gamma}(z).$

######  Lemma 2.7.

The process $(E_{t}^{\infty})_{t\geq 0}$ introduced in Definition 2.6 is well-
defined and Markovian.

###### Proof.

Recall that for all $(t,z,a,b,\gamma)\in\overleftarrow{\Pi}$, we have
$(a,b)\in(0,\mathcal{R}_{\tilde{\mu}}]^{2}$. Therefore,

$\mathfrak{B}_{a,b,\gamma}(z)\subseteq\mathcal{B}_{\mathcal{R}_{\tilde{\mu}}}(z)\text{
a.s.,}$

where $\mathcal{B}_{\mathcal{R}_{\tilde{\mu}}}(z)$ is the ball of radius
$\mathcal{R}_{\tilde{\mu}}$ centred at $z$, and so we can bound the jump rate
of $(E_{t}^{\infty})_{t\geq 0}$ from above by that of a $\infty$-parent
ancestral process with the same initial condition and associated to
$\tilde{\mu}(S_{\tilde{\mu}})\delta_{(\mathcal{R}_{\tilde{\mu}},\mathcal{R}_{\tilde{\mu}},0)}(da,db,d\gamma)$,
which is finite (see Section $4$ in [32]). ∎

#### Duality relation.

For all $M\in\mathcal{M}_{\lambda}$ and $E\in\mathcal{E}^{cf}$, we set

$\widetilde{D}(M,E):=\delta_{0}\left(\int_{E}\left(1-\omega_{M}(z)\right)dz\right).$

Intuitively, $\widetilde{D}(M,E)=0$ if the area $E$ contains a positive
fraction of real individuals when the population state is $M$, and
$\widetilde{D}(M,E)=1$ if the area is empty.

The $\infty$-parent SLFV and the $\infty$-parent ancestral process then
satisfy the following duality relation, whose proof is similar to the one in
[32].

######  Proposition 2.8.

Let $M^{0}\in\mathcal{M}_{\lambda}$, and let $(M_{t}^{\infty})_{t\geq 0}$ be
the unique solution to the martingale problem associated to
$(\mathcal{L}_{\tilde{\mu}}^{\infty},\delta_{M^{0}})$. Let
$E^{0}\in\mathcal{E}^{cf}$, and let $(E_{t}^{\infty})_{t\geq 0}$ be the
$\infty$-parent ancestral process with initial condition $E^{0}$ associated to
$\tilde{\mu}$. Then, for all $t\geq 0$,

$\mathbb{E}_{M^{0}}\left[\widetilde{D}(M_{t}^{\infty},E^{0})\right]=\mathbf{E}_{E^{0}}\left[\widetilde{D}(M^{0},E_{t}^{\infty})\right],$

or equivalently,

$\mathbb{E}_{M^{0}}\left[\delta_{0}\left(\int_{E^{0}}\left(1-\omega_{M_{t}^{\infty}}(z)\right)dz\right)\right]=\mathbf{E}_{E^{0}}\left[\delta_{0}\left(\int_{E_{t}^{\infty}}\left(1-\omega_{M^{0}}(z)\right)dz\right)\right].$

This duality relation can be interpreted as follows. Whenever a reproduction
event affects one area, all the individuals in the area can be considered as
potential parents. If $\overleftarrow{\Pi}$ encodes the reproduction events
affecting the population when going backwards in time, then $E_{t}^{\infty}$
encodes the locations of the potential ancestors at time $0$ of the
individuals living in $E^{0}$ at time $t$. The duality relation then states
that a given area $E^{0}$ contains only ghost individuals at time $t$ if, and
only if all their potential ancestors at time $0$ (located in
$E_{t}^{\infty}$) are ghost individuals.

###  2.3 Speed of growth of the occupied region in the $\infty$-parent SLFV:
Definition and main result

Let $HP^{0}$ stand for the half-plane

$HP^{0}:=\big{\\{}(x,y)\in\mathbb{R}^{2}:x\geq 0\big{\\}}.$

In order to show that the growth of the region occupied by type $1$
individuals in the $\infty$-parent SLFV is linear in time, we consider that
initially, type $1$ individuals cover the half-plane

$\overline{HP}^{0}:=\big{\\{}(x,y)\in\mathbb{R}^{2}:x<0\big{\\}}.$

That is, we take the measure $M^{HP}(dz):=\omega^{HP}(z)dz$ as an initial
condition for the $\infty$-parent SLFV, where

$\forall z\in\mathbb{R}^{2},\quad\omega^{HP}(z)=\mathds{1}_{HP^{0}}(z).$

Let $(M_{t}^{HP})_{t\geq 0}$ be the $\infty$-parent SLFV with initial
condition $M^{HP}$ associated to $\tilde{\mu}$. Moreover, for all $t\geq 0$,
let $\omega_{t}^{HP}$ be a density of $M_{t}^{HP}$.

######  Definition 2.9.

For all $x\in\mathbb{R}$, let

$\overrightarrow{\tau}\\!_{x}:=\min\left\\{t\geq 0:\lim\limits_{\epsilon\to
0}V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz<1\right\\},$

where we recall that $\mathcal{B}_{\epsilon}((x,0))$ is the ball with centre
$(x,0)$ and radius $\epsilon$, and $V_{\epsilon}$ is the volume of
$\mathcal{B}_{\epsilon}((x,0))$.

Informally, $\overrightarrow{\tau}\\!_{x}$ is the first time at which the
location $(x,0)$ is reached by type $1$ individuals. Since densities are only
defined up to a Lebesgue null set, we consider that $(x,0)$ is occupied by
type $1$ individuals if, and only if all neighbourhoods of $(x,0)$ contain a
non-zero fraction of type $1$ individuals.

The following result means that once the location $(x,0)$ is occupied by type
$1$ individuals, it cannot become empty again. However, notice that the
function $x\to\overrightarrow{\tau}\\!_{x}$ is not necessarily increasing. See
Figure 2 for an illustration.

######  Lemma 2.10.

Let $x\in\mathbb{R}$. Then, for all $t\geq 0$,

$\mathbb{P}\left(\lim\limits_{\epsilon\to
0}V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz<1\Big{|}\overrightarrow{\tau}\\!_{x}\leq
t\right)=1.$

This property is obvious from the set-valued informal description of the
$\infty$-parent SLFV given in the Introduction. However, because this
description does not hold when the initial area occupied by type 1 individuals
has infinite volume, we have to work a bit more to show this property by using
the martingale problem characterisation.

###### Proof.

We first show that for all $\epsilon>0$,

$t\to\mathbb{P}\left(V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz<1\right)$

is a non-decreasing function.

In order to do so, let $\epsilon>0$, and let $0\leq t\leq t^{\prime}$. Then,
by Proposition 2.8,

$\displaystyle\mathbb{P}\left(V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz<1\right)$
$\displaystyle=\mathbb{P}\left(\delta_{0}\left(\int_{\mathcal{B}_{\epsilon}((x,0))}(1-\omega_{t}^{HP}(z))dz\right)=0\right)$
$\displaystyle=1-\mathbb{E}\left[\delta_{0}\left(\int_{\mathcal{B}_{\epsilon}((x,0))}(1-\omega_{t}^{HP}(z))dz\right)\right]$
$\displaystyle=1-\mathbf{E}\left[\delta_{0}\left(\int_{E_{t}^{\infty}}(1-\omega_{M^{0}}(z))dz\right)\right],$

where $(E_{t}^{\infty})_{t\geq 0}$ is the $\infty$-parent ancestral process
with initial condition $\mathcal{B}_{\epsilon}((x,0))$ and intensity
$\tilde{\mu}$. Similarly,

$\mathbb{P}\left(V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t^{\prime}}^{HP}(z)dz<1\right)=1-\mathbf{E}\left[\delta_{0}\left(\int_{E_{t^{\prime}}^{\infty}}(1-\omega_{M^{0}}(z))dz\right)\right].$

Moreover, the sequence $(E_{t}^{\infty})_{t\geq 0}$ is increasing for the
inclusion and since $\omega_{M^{0}}$ is $\\{0,1\\}$-valued, we have

$\mathbf{E}\left[\delta_{0}\left(\int_{E_{t}^{\infty}}(1-\omega_{M^{0}}(z))dz\right)\right]\geq\mathbf{E}\left[\delta_{0}\left(\int_{E_{t^{\prime}}^{\infty}}(1-\omega_{M^{0}}(z))dz\right)\right].$

We conclude that

$\mathbb{P}\left(V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz<1\right)\leq\mathbb{P}\left(V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t^{\prime}}^{HP}(z)dz<1\right).$

Then, let $t\geq 0$, and let $(\epsilon_{n})_{n\in\mathbb{N}}$ be a sequence
decreasing to $0$. Since

$\lim\limits_{\epsilon\to
0}V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz$

exists, we have

$\mathbb{P}\left(\left.\lim\limits_{\epsilon\to
0}V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz<1\right|\overrightarrow{\tau}\\!_{x}\leq
t\right)=\mathbb{P}\left(\left.\lim\limits_{n\to+\infty}V_{\epsilon_{n}}^{-1}\int_{\mathcal{B}_{\epsilon_{n}}((x,0))}\omega_{t}^{HP}(z)dz<1\right|\overrightarrow{\tau}\\!_{x}\leq
t\right).$

Noticing that the sequence
$(\mathcal{B}_{\epsilon_{n}}((x,0)))_{n\in\mathbb{N}}$ is decreasing for the
inclusion, we have that if

$V_{\epsilon_{n}}^{-1}\int_{\mathcal{B}_{\epsilon_{n}}((x,0))}\omega_{t}^{HP}(z)dz=1$

for some integer $n$, then this is also the case for all $N\geq n$. Therefore,

$\displaystyle\mathbb{P}\left(\left.\lim\limits_{n\to+\infty}V_{\epsilon_{n}}^{-1}\int_{\mathcal{B}_{\epsilon_{n}}((x,0))}\omega_{t}^{HP}(z)dz=1\right|\overrightarrow{\tau}\\!_{x}\leq
t\right)$ $\displaystyle=\mathbb{P}\left(\left.\exists
n\in\mathbb{N},V_{\epsilon_{n}}^{-1}\int_{\mathcal{B}_{\epsilon_{n}}((x,0))}\omega_{t}^{HP}(z)dz=1\right|\overrightarrow{\tau}\\!_{x}\leq
t\right)$
$\displaystyle\leq\sum_{n\in\mathbb{N}}\mathbb{P}\left(\left.V_{\epsilon_{n}}^{-1}\int_{\mathcal{B}_{\epsilon_{n}}((x,0))}\omega_{t}^{HP}(z)dz=1\right|\overrightarrow{\tau}\\!_{x}\leq
t\right)$
$\displaystyle\leq\sum_{n\in\mathbb{N}}\mathbb{P}\left(\left.V_{\epsilon_{n}}^{-1}\int_{\mathcal{B}_{\epsilon_{n}}((x,0))}\omega_{\overrightarrow{\tau}\\!_{x}}^{HP}(z)dz=1\right|\overrightarrow{\tau}\\!_{x}\leq
t\right)$ $\displaystyle=0$

by definition of $\overrightarrow{\tau}\\!_{x}$. Here we have used the first
part of the proof to pass from the second line to the third line, which allows
us to conclude. ∎

Figure 2:  State of a population evolving according to the $\infty$-parent
SLFV at time $t$. Initially, type $1$ individuals cover the half-plane
$\overline{HP}^{0}$, corresponding to the hatched area. Each grey ellipse
represents a reproduction event occurring during the time interval $[0,t]$
which overlaps an area initially empty, and resulting in the corresponding
area being completely filled with type $1$ individuals. Here
$\overrightarrow{\tau}\\!_{1}>t$ but $\overrightarrow{\tau}\\!_{2}\leq t$.

The goal of the article is to show the following result, which tells us that
the growth of the occupied area in the $\infty$-parent SLFV is linear in time.

######  Theorem 2.11.

There exists $\nu>0$ such that

$\lim\limits_{x\to+\infty}\frac{\mathbb{E}\left[\overrightarrow{\tau}\\!_{x}\right]}{x}=\nu.$

We can then interpret $\nu^{-1}$ as the limiting speed of growth of the
process. As explained in the Introduction, the proof of Theorem 2.11 spans
Sections 3 and 4 and is concluded at the end of Section 4.3.

In order to show Theorem 2.11, we make use of the dual process associated to
the $\infty$-parent SLFV, and define an equivalent of
$\overrightarrow{\tau}\\!_{x}$ for the $\infty$-parent ancestral process. In
order to do so, we first introduce a slight generalisation of the
$\infty$-parent ancestral process which allows to take single points as
initial conditions. This process is defined on the state space

$\widetilde{\mathcal{E}}^{cf}:=\mathcal{E}\cup\big{\\{}\\{z\\}:z\in\mathbb{R}^{2}\big{\\}}.$

######  Definition 2.12.

Let $\overleftarrow{\Pi}$ be a Poisson point process with intensity measure
$dt\otimes dz\otimes\tilde{\mu}(da,db,d\gamma)$ on
$\mathbb{R}_{+}\times\mathbb{R}^{2}\times S_{\tilde{\mu}}$, defined on the
probability space $(\mathbf{\Omega},\mathbf{\mathcal{F}},\mathbf{P})$, and let
$z_{0}\in\mathbb{R}^{2}$. The $\widetilde{\mathcal{E}}^{cf}$-valued
$\infty$-parent ancestral process $(E_{t}^{\infty,z_{0}})_{t\geq 0}$ with
initial condition $\\{z_{0}\\}$ and associated to $\tilde{\mu}$ is defined as
follows.

First, we set $E_{0}^{\infty,z_{0}}=\\{z_{0}\\}$. Then, for all
$(t,z,a,b,\gamma)\in\overleftarrow{\Pi}$:

  * •

If $E_{t-}^{\infty,z_{0}}=\\{z_{0}\\}$ and
$z_{0}\in\mathfrak{B}_{a,b,\gamma}(z)$, we set
$E_{t}^{\infty,z_{0}}=\mathfrak{B}_{a,b,\gamma}(z)$.

  * •

If $E_{t-}^{\infty,z_{0}}\neq\\{z_{0}\\}$ and
$E_{t-}^{\infty,z_{0}}\cap\mathfrak{B}_{a,b,\gamma}(z)$ has non-zero Lebesgue
measure, we set

$E_{t}^{\infty,z_{0}}=E_{t-}^{\infty,z_{0}}\cup\mathfrak{B}_{a,b,\gamma}(z).$

Using the same argument as for the initial $\infty$-parent ancestral process,
$(E_{t}^{\infty,z_{0}})_{t\geq 0}$ is well-defined and Markovian. Moreover,
once it jumps for the first time, its behaviour is identical to that of the
original $\infty$-parent ancestral process.

In all that follows, let $\overleftarrow{\Pi}$ be a Poisson point process as
in Definition 2.12. For all $x>0$ and all $\epsilon>0$, let
$(E_{t}^{\epsilon,x})_{t\geq 0}$ be a sequence of $\infty$-parent ancestral
processes with initial condition $\mathcal{B}_{\epsilon}((x,0))$ associated to
$\tilde{\mu}$, all constructed using the same underlying Poisson point process
$\overleftarrow{\Pi}$. Moreover, let $(E_{t}^{x})_{t\geq 0}$ be the
$\infty$-parent ancestral process with initial condition $\\{(x,0)\\}$
associated to $\tilde{\mu}$, also constructed using $\overleftarrow{\Pi}$.
That is, in the heavier notation of Definition 2.12, $E^{x}=E^{\infty,(x,0)}$.
Then, $(E_{t}^{\epsilon,x})_{t\geq 0}$, $\epsilon>0$ and $(E_{t}^{x})_{t\geq
0}$ satisfy the following property.

######  Lemma 2.13.

For all $x>0$, if $t_{0}^{x}$ is the first time at which $(x,0)$ is affected
by a reproduction event, then a.s. there exists $\epsilon_{0}^{x}>0$ such that
for all $t\geq t_{0}^{x}$, we have

$\forall\epsilon<\epsilon_{0}^{x},\qquad E_{t}^{\epsilon,x}=E_{t}^{x}.$

Furthermore, we have a.s. for all $t\geq 0$,

$\lim\limits_{\epsilon\to
0}\mathrm{Vol}\left(E_{t}^{\epsilon,x}\cap\overline{HP}^{0}\right)=\mathrm{Vol}\left(E_{t}^{x}\cap\overline{HP}^{0}\right).$

Concretely, $\epsilon_{0}^{x}$ is the radius of a ball around $(x,0)$ small
enough to be included in the area of the first event overlapping $(x,0)$, and
not to be overlapped by any event in $\overleftarrow{\Pi}$ occurring before
$t_{0}^{x}$ (see Condition (2.3) below).

###### Proof.

Let $x>0$, and let
$(t_{0}^{x},z_{0}^{x},a_{0}^{x},b_{0}^{x},\gamma_{0}^{x})\in\overleftarrow{\Pi}$
be the first reproduction event to affect $(x,0)$. In order to show the first
part of the lemma, we want to show that a.s. there exists $\epsilon_{0}^{x}>0$
such that

$\mathcal{B}_{\epsilon_{0}^{x}}((x,0))\subseteq\mathfrak{B}_{a_{0}^{x},b_{0}^{x},\gamma_{0}^{x}}(z_{0}^{x})\text{
and }\forall(t,z,a,b,\gamma)\in\overleftarrow{\Pi}\text{ s.t.
}t<t_{0}^{x},\mathrm{Vol}\big{(}\mathcal{B}_{\epsilon_{0}^{x}}((x,0))\cap\mathfrak{B}_{a,b,\gamma}(z)\big{)}=0.$
(2.3)

Indeed, if such a $\epsilon_{0}^{x}$ exists, then for all
$0<\epsilon<\epsilon_{0}^{x}$, the first time at which
$(E_{t}^{\epsilon,x})_{t\geq 0}$ jumps is $t_{0}^{x}$. Moreover,

$\displaystyle E_{t_{0}^{x}}^{\epsilon,x}$
$\displaystyle=E_{0}^{\epsilon,x}\cup\mathfrak{B}_{a_{0}^{x},b_{0}^{x},\gamma_{0}^{x}}(z_{0}^{x})$
$\displaystyle=\mathcal{B}_{\epsilon}((x,0))\cup\mathfrak{B}_{a_{0}^{x},b_{0}^{x},\gamma_{0}^{x}}(z_{0}^{x})$
$\displaystyle=\mathfrak{B}_{a_{0}^{x},b_{0}^{x},\gamma_{0}^{x}}(z_{0}^{x}),$
$\displaystyle=E_{t_{0}^{x}}^{x},$

yielding the first part of the lemma.

As the probability that $(x,0)$ belongs to the boundary of
$\mathfrak{B}_{a_{0}^{x},b_{0}^{x},\gamma_{0}^{x}}(z_{0}^{x})$ is equal to
zero, with probability $1$ there exists $\tilde{\epsilon}>0$ such that

$\mathcal{B}_{\tilde{\epsilon}}((x,0))\subseteq\mathfrak{B}_{a_{0}^{x},b_{0}^{x},\gamma_{0}^{x}}(z_{0}^{x}).$

Let us consider the number $N_{\tilde{\epsilon}}$ of reproduction events that
affect $\mathcal{B}_{\tilde{\epsilon}}((x,0))$ during the time interval
$[0,t_{0}^{x})$, that is, all reproduction events
$(t,z,a,b,\gamma)\in\overleftarrow{\Pi}$ such that $t\in[0,t_{0}^{x})$ and

$\mathcal{B}_{\tilde{\epsilon}}((x,0))\cap\mathfrak{B}_{a,b,\gamma}(z)\neq\emptyset.$

Since the support of $\tilde{\mu}$ is bounded, $N_{\tilde{\epsilon}}$ is a.s.
finite. By construction, the point $(x,0)$ belongs to none of the finitely
many closed ellipses corresponding to these events, and so the minimal
distance $\mathfrak{e}$ between $(x,0)$ and these areas is a.s. positive. We
can then choose any $\epsilon_{0}^{x}\in(0,\mathfrak{e})$ and (2.3) holds
true, which concludes the proof of the first part of Lemma 2.13.

In order to show the second part of the lemma, let us first consider
$t\in[0,t_{0}^{x})$, and let us assume that $\epsilon_{0}^{x}$ exists (which
is almost surely satisfied). Then,

$\mathrm{Vol}\left(E_{t}^{x}\cap\overline{HP}^{0}\right)=\mathrm{Vol}\left(\\{(x,0)\\}\cap\overline{HP}^{0}\right)=0.$

Moreover, by (2.3), for all $0<\epsilon<\min(\epsilon_{0}^{x},x/2)$ we have

$\mathrm{Vol}\left(E_{t}^{\epsilon,x}\cap\overline{HP}^{0}\right)=\mathrm{Vol}\left(\mathcal{B}_{\epsilon}((x,0))\cap\overline{HP}^{0}\right)=\mathrm{Vol}(\emptyset)=0.$

Therefore,

$\lim\limits_{\epsilon\to
0}\mathrm{Vol}\left(E_{t}^{\epsilon,x}\cap\overline{HP}^{0}\right)=0=\mathrm{Vol}\left(E_{t}^{x}\cap\overline{HP}^{0}\right).$

Let us now consider $t\geq t_{0}^{x}$. For every
$\epsilon\in(0,\epsilon_{0}^{x})$, we have

$\mathrm{Vol}\left(E_{t}^{\epsilon,x}\cap\overline{HP}^{0}\right)=\mathrm{Vol}\left(E_{t}^{x}\cap\overline{HP}^{0}\right),$

and so

$\lim\limits_{\epsilon\to
0}\mathrm{Vol}\left(E_{t}^{\epsilon,x}\cap\overline{HP}^{0}\right)=\mathrm{Vol}\left(E_{t}^{x}\cap\overline{HP}^{0}\right).$

∎

We can now introduce an equivalent of $\overrightarrow{\tau}\\!_{x}$ for the
$\infty$-parent ancestral process.

######  Definition 2.14.

For all $x>0$, let
$(t_{0}^{x},z_{0}^{x},a_{0}^{x},b_{0}^{x},\gamma_{0}^{x})\in\overleftarrow{\Pi}$
be the first reproduction event to affect $(x,0)$, and let

$\tilde{\tau}_{x}:=\left\\{\begin{array}[]{ll}\min\left\\{t\geq
0:\lim\limits_{\epsilon\to
0}\mathrm{Vol}\left(E_{t}^{\epsilon,x}\cap\overline{HP}^{0}\right)>0\right\\}&\mbox{if
there exists $\epsilon_{0}^{x}>0$ satisfying (\ref{eqn:equation_epsilon})},\\\
0&\mbox{otherwise}.\end{array}\right.$

######  Lemma 2.15.

For all $x>0$, $\tilde{\tau}_{x}$ is well-defined, and is almost surely equal
to

$\displaystyle\min\left\\{t\geq
0:E_{t}^{x}\cap\overline{HP}^{0}\neq\emptyset\right\\}.$

In particular, $\tilde{\tau}_{x}\neq 0$ a.s.

###### Proof.

Let $x>0$. By Lemma 2.13, $\tilde{\tau}_{x}$ is well-defined and almost surely
equal to

$\min\Big{\\{}t\geq
0:\mathrm{Vol}\left(E_{t}^{x}\cap\overline{HP}^{0}\right)>0\Big{\\}}\geq
t_{0}^{x},$

since the support of $\tilde{\mu}$ is bounded. Moreover, since for all
$(a,b,\gamma)\in S_{\tilde{\mu}}$, the set

$\left\\{z\in\mathbb{R}^{2}:\mathfrak{B}_{a,b,\gamma}(z)\cap\overline{HP}^{0}\neq\emptyset\text{
and
}\mathrm{Vol}\left(\mathfrak{B}_{a,b,\gamma}(z)\cap\overline{HP}^{0}\right)=0\right\\}$

has zero Lebesgue measure, we have

$\min\Big{\\{}t\geq
0:\mathrm{Vol}\left(E_{t}^{x}\cap\overline{HP}^{0}\right)>0\Big{\\}}=\min\left\\{t\geq
0:E_{t}^{x}\cap\overline{HP}^{0}\neq\emptyset\right\\}\text{ a.s.,}$

which allows us to conclude. ∎

In order to use $(\tilde{\tau}_{x})_{x>0}$ to derive the desired properties of
$(\overrightarrow{\tau}\\!_{x})_{x>0}$, we shall need the following result.

######  Lemma 2.16.

For all $x>0$, $\tilde{\tau}_{x}$ and $\overrightarrow{\tau}\\!_{x}$ have the
same distribution.

###### Proof.

Recall that constructing all the $\infty$-parent ancestral process
$(E_{t}^{\epsilon,x})_{t\geq 0}$ using the same underlying Poisson point
process ensures that for all $t\geq 0$ and all $x>0$, we have

$\forall 0<\epsilon<\epsilon^{\prime},\qquad E_{t}^{\epsilon,x}\subseteq
E_{t}^{\epsilon^{\prime},x}.$ (2.4)

Moreover, for all $\epsilon>0$, we set

$\displaystyle\overrightarrow{\tau}\\!_{x}^{\epsilon}$
$\displaystyle:=\min\left\\{t>0:V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz<1\right\\}$
$\displaystyle\text{and }\quad\tilde{\tau}_{x}^{\epsilon}$
$\displaystyle:=\min\left\\{t>0:\mathrm{Vol}\left(E_{t}^{\epsilon,x}\cap\overline{HP}^{0}\right)>0\right\\}.$

Using the same argument as in the proof of Lemma 2.15, we obtain that

$\tilde{\tau}_{x}=\lim\limits_{\epsilon\to
0}\tilde{\tau}_{x}^{\epsilon}\quad\text{a.s}.$

We have a similar result for $\overrightarrow{\tau}\\!_{x}$. Indeed, for all
$0<\epsilon<\epsilon^{\prime}$, by definition of
$\overrightarrow{\tau}\\!_{x}^{\epsilon}$ we can write

$\displaystyle\int_{\mathcal{B}_{\epsilon^{\prime}}((x,0))}\omega_{\overrightarrow{\tau}\\!_{x}^{\epsilon}}^{HP}(z)dz$
$\displaystyle=\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{\overrightarrow{\tau}\\!_{x}^{\epsilon}}^{HP}(z)dz+\int_{\mathcal{B}_{\epsilon^{\prime}}((x,0))\setminus\mathcal{B}_{\epsilon}((x,0))}\omega_{\overrightarrow{\tau}\\!_{x}^{\epsilon}}^{HP}(z)dz$
$\displaystyle<V_{\epsilon}+V_{\epsilon^{\prime}}-V_{\epsilon}$
$\displaystyle<V_{\epsilon^{\prime}}.$

Therefore,
$\overrightarrow{\tau}\\!_{x}^{\epsilon^{\prime}}\leq\overrightarrow{\tau}\\!_{x}^{\epsilon}$
and $\lim\limits_{\epsilon\to 0}\overrightarrow{\tau}\\!_{x}^{\epsilon}$
exists. Moreover, for all $t\geq\overrightarrow{\tau}\\!_{x}$, by Lemma 2.10
we have

$\lim_{\epsilon\to
0}V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz<1,$

and so there exists $\epsilon_{t}>0$ such that for all
$0<\epsilon\leq\epsilon_{t}$,

$V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz<1,$

and hence for all $0<\epsilon\leq\epsilon_{t}$,
$t\geq\overrightarrow{\tau}\\!_{x}^{\epsilon}$. Similarly, if
$t>\lim\limits_{\epsilon\to 0}\overrightarrow{\tau}\\!_{x}^{\epsilon}$, then
again by Lemma 2.10, there exists $\epsilon_{t}>0$ such that for all
$0<\epsilon\leq\epsilon_{t}$,

$V_{\epsilon}^{-1}\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz<1$

and $t>\overrightarrow{\tau}\\!_{x}$. Therefore,

$\overrightarrow{\tau}\\!_{x}=\lim\limits_{\epsilon\to
0}\overrightarrow{\tau}\\!_{x}^{\epsilon}\quad\text{a.s}.$

Consequently, it is sufficient to show that
$\overrightarrow{\tau}\\!_{x}^{\epsilon}$ and $\tilde{\tau}_{x}^{\epsilon}$
have the same distribution for all $\epsilon>0$ in order to conclude the
proof.

Let $\epsilon>0$, and let $t\geq 0$. Since
$t\to\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz$ is non-
increasing and $[0,V_{\epsilon}]$-valued, we have

$\displaystyle\mathbb{P}_{M_{0}^{HP}}\left(\overrightarrow{\tau}\\!_{x}^{\epsilon}>t\right)$
$\displaystyle=\mathbb{P}_{M_{0}^{HP}}\left(\int_{\mathcal{B}_{\epsilon}((x,0))}\omega_{t}^{HP}(z)dz=V_{\epsilon}\right)$
$\displaystyle=\mathbb{P}_{M_{0}^{HP}}\left(\delta_{0}\left(\int_{\mathcal{B}_{\epsilon}((x,0))}\left(1-\omega_{t}^{HP}(z)\right)dz\right)=1\right)$
$\displaystyle=\mathbb{E}_{M_{0}^{HP}}\left[\delta_{0}\left(\int_{\mathcal{B}_{\epsilon}((x,0))}\left(1-\omega_{t}^{HP}(z)\right)dz\right)\right]$
$\displaystyle=\mathbf{E}_{\mathcal{B}_{\epsilon}((x,0))}\left[\delta_{0}\left(\int_{E_{t}^{\epsilon,x}}\left(1-\omega_{0}^{HP}(z)\right)dz\right)\right]$
$\displaystyle=\mathbf{P}_{\mathcal{B}_{\epsilon}((x,0))}\left(\delta_{0}\left(\int_{E_{t}^{\epsilon,x}}\left(1-\omega_{0}^{HP}(z)\right)dz\right)=1\right)$
$\displaystyle=\mathbf{P}_{\mathcal{B}_{\epsilon}((x,0))}\left(\mathrm{Vol}\left(E_{t}^{\epsilon,x}\cap\overline{HP^{0}}\right)=0\right)$
$\displaystyle=\mathbf{P}_{\mathcal{B}_{\epsilon}((x,0))}\left(\tilde{\tau}_{x}^{\epsilon}>t\right).$

Here we have used Proposition 2.8 to pass from the third to the fourth line. ∎

For all $x>0$, let $HP^{x}$ stand for the half-plane

$HP^{x}:=\left\\{(x^{\prime},y)\in\mathbb{R}^{2}:x^{\prime}\geq x\right\\},$

extending the notation $HP^{0}$ to the case $x\geq 0$.

In practice, the set of variables $(\tilde{\tau}_{x})_{x>0}$ is not very
convenient to use. Indeed, if $x<x^{\prime}$, even if $\tilde{\tau}_{x}$ and
$\tilde{\tau}_{x^{\prime}}$ are constructed using the same underlying Poisson
point process, this is not sufficient to have
$\tilde{\tau}_{x}\leq\tilde{\tau}_{x^{\prime}}$ a.s., as the underlying
$\infty$-parent ancestral processes have different starting positions.
Therefore, we define another sequence $(\overleftarrow{\tau}\\!_{x})_{x>0}$
such that for all $x>0$, $\overleftarrow{\tau}\\!_{x}$ and $\tilde{\tau}_{x}$
have the same distribution, but also such that all underlying $\infty$-parent
ancestral processes start from the same location $(0,0)$, thereby ensuring
that for all $0<x<x^{\prime}$, we have
$\overleftarrow{\tau}_{x}\leq\overleftarrow{\tau}_{x^{\prime}}$ a.s. Moreover,
for every $x>0$, the $\infty$-parent ancestral process associated to
$\overleftarrow{\tau}\\!_{x}$ will have the same distribution as the symmetric
of the ancestral process associated to $\tilde{\tau}_{x}$ with respect to the
axis $\\{(x/2,y):y\in\mathbb{R}\\}$. In other words, we shall be interested in
the expansion of the new process "from left to right", starting from $(0,0)$
and reaching abscissa $x$, instead of studying the expansion of $E^{x}$ "from
right to left", starting from $(x,0)$ and reaching abscissa $0$.

Let $\tilde{\mu}^{\leftarrow}$ be the finite measure on $(0,+\infty)^{2}\times
S_{\tilde{\mu}}$ defined by the property that for all measurable subsets $I,J$
of $(0,+\infty)$ and for all $-\pi/2<\gamma_{1}\leq\gamma_{2}<\pi/2$,

$\tilde{\mu}^{\leftarrow}(I\times
J\times[\gamma_{1},\gamma_{2}])=\tilde{\mu}(I\times
J\times[-\gamma_{2},-\gamma_{1}]).$

Let $(E_{t}^{\epsilon})_{t\geq 0}$, $\epsilon>0$, be a sequence of
$\infty$-parent ancestral processes associated to $\tilde{\mu}^{\leftarrow}$
with initial condition $\mathcal{B}_{\epsilon}((0,0))$, constructed using the
same underlying Poisson point process $\overline{\Pi}$, and let
$(E_{t})_{t\geq 0}$ be the $\infty$-parent ancestral process with initial
condition $\\{(0,0)\\}$ associated to $\tilde{\mu}^{\leftarrow}$, also
constructed using the Poisson point process $\overline{\Pi}$. For
definiteness, we suppose that these new objects are also constructed on the
probability space $(\mathbf{\Omega},\mathbf{\mathcal{F}},\mathbf{P})$. We
shall call Condition (2.3)’ the analogue of Condition (2.3) with $x=0$ and
$\overleftarrow{\Pi}$ replaced by $\overline{\Pi}$.

######  Definition 2.17.

Let $(t_{0},z_{0},a_{0},b_{0},\gamma_{0})\in\overleftarrow{\Pi}$ be the first
reproduction event to affect $(0,0)$, and for all $x>0$, let

$\overleftarrow{\tau}\\!_{x}:=\left\\{\begin{array}[]{ll}\min\left\\{t\geq
0:\lim\limits_{\epsilon\to 0}\mathrm{Vol}\left(E_{t}^{\epsilon}\cap
HP^{x}\right)>0\right\\}&\mbox{if there exists $\epsilon_{0}>0$ satisfying
Cond.\leavevmode\nobreak\ \eqref{eqn:equation_epsilon}',}\\\
0&\mbox{otherwise}.\end{array}\right.$

As for $(\tilde{\tau}_{x})_{x>0}$, for all $x>0$, $\overleftarrow{\tau}_{x}$
is well-defined and almost surely equal to

$\min\left\\{t\geq 0:E_{t}\cap HP^{x}\neq\emptyset\right\\}.$

Let us say that a point $z=(x,y)\in\mathbb{R}^{2}$ is at horizontal separation
$d$ of the point $z^{\prime}=(x^{\prime},y^{\prime})\in\mathbb{R}^{2}$ if, and
only if $x-x^{\prime}=d$. Then informally, $\overleftarrow{\tau}\\!_{x}$
represents the first time the $\infty$-parent ancestral process starting from
$(0,0)$ reaches points at horizontal separation of at least $x$ from the
starting location. Moreover, we have the following lemma.

######  Lemma 2.18.

The function $x\to\overleftarrow{\tau}\\!_{x}$ is nondecreasing.

###### Proof.

We distinguish two cases. If there does not exist $\epsilon_{0}>0$ satisfying
(2.3)’, then for all $x>0$, $\overleftarrow{\tau}\\!_{x}=0$, and we can
conclude.

If there exists $\epsilon_{0}>0$ satisfying (2.3), let $0<x_{1}<x_{2}$. By
similar arguments as in Lemma 2.13 and its proof, we can write that

$\displaystyle\forall
0<\epsilon\leq\min(\epsilon_{0},x_{1}/2),\quad\mathrm{Vol}\left(E_{t}^{\epsilon}\cap
HP^{x_{1}}\right)$
$\displaystyle=\mathrm{Vol}\left(E_{t}^{\min(\epsilon_{0},x_{1}/2)}\cap
HP^{x_{1}}\right)$ $\displaystyle\text{and }\quad\forall
0<\epsilon\leq\min(\epsilon_{0},x_{2}/2),\quad\mathrm{Vol}\left(E_{t}^{\epsilon}\cap
HP^{x_{2}}\right)$
$\displaystyle=\mathrm{Vol}\left(E_{t}^{\min(\epsilon_{0},x_{2}/2)})\cap
HP^{x_{2}}\right).$

Let us set $\epsilon^{\prime}=\min(\epsilon_{0},x_{1}/2,x_{2}/2)$. Then,
$HP^{x_{2}}\subset HP^{x_{1}}$ and so

$\mathrm{Vol}\left(E_{\overleftarrow{\tau}\\!_{x_{2}}}^{\epsilon^{\prime}}\cap
HP^{x_{1}}\right)\geq\mathrm{Vol}\left(E_{\overleftarrow{\tau}\\!_{x_{2}}}^{\epsilon^{\prime}}\cap
HP^{x_{2}}\right)=\lim\limits_{\epsilon\to
0}\mathrm{Vol}\left(E_{\overleftarrow{\tau}\\!_{x_{2}}}^{\epsilon}\cap
HP^{x_{2}}\right)>0$

by definition of $\overleftarrow{\tau}\\!_{x_{2}}$, and so
$\overleftarrow{\tau}\\!_{x_{1}}\leq\overleftarrow{\tau}\\!_{x_{2}}$ and we
can conclude. ∎

Notice that we are now studying the expansion of the backwards-in-time process
in the same direction as the occupied area in the forwards-in-time process.
Conversely, $(\tilde{\tau}_{x})_{x>0}$ corresponds to the expansion of the
$\infty$-parent ancestral process in the opposite direction. Moreover, we
recall that $(\overleftarrow{\tau}\\!_{x})_{x>0}$ can be seen as being
constructed using an underlying $\infty$-parent ancestral process which is the
symmetric of the one used to construct $(\tilde{\tau}_{x})_{x>0}$ with respect
to the axis $\\{(x/2,y):y\in\mathbb{R}\\}$. This observation yields the
following lemma.

######  Lemma 2.19.

For all $x>0$, $\overleftarrow{\tau}\\!_{x}$ and $\tilde{\tau}_{x}$ have the
same distribution.

###### Proof.

Let $x>0$. For all $t\geq 0$ and $\epsilon>0$, let
$\mathrm{Sym}(E_{t}^{\epsilon,x})$ be the symmetric of $E_{t}^{\epsilon,x}$
with respect to the axis $\\{(x/2,y):y\in\mathbb{R}\\}$. Then,
$(\mathrm{Sym}(E_{t}^{\epsilon,x}))_{t\geq 0}$, $\epsilon>0$ is a sequence of
$\infty$-parent ancestral processes with initial condition
$\mathcal{B}_{\epsilon}((0,0))$, all constructed using the same Poisson point
process having the same distribution as $\overline{\Pi}$ (that is, whose
intensity measure is $dt\otimes
dz\otimes\tilde{\mu}^{\leftarrow}(da,db,d\gamma)$), which can be used to
construct $\overleftarrow{\tau}\\!_{x}$. Moreover, for all $t\geq 0$ and
$\epsilon>0$,

$\mathrm{Vol}\big{(}E_{t}^{\epsilon,x}\cap\overline{HP}^{0}\big{)}>0\quad\text{if,
and only if}\quad\mathrm{Vol}\left(\mathrm{Sym}(E_{t}^{\epsilon,x})\cap
HP^{x}\right)>0,$

which allows us to conclude. ∎

In order to show Theorem 2.11, we shall use the following proposition, which
is a direct consequence of Lemmas 2.16 and 2.19.

######  Proposition 2.20.

For all $x>0$, $\overleftarrow{\tau}\\!_{x}$ and
$\overrightarrow{\tau}\\!_{x}$ have the same distribution.

Contrary to what is usually considered for stochastic growth models,
$\overleftarrow{\tau}\\!_{x}$ does not correspond to the growth of the
$\infty$-parent ancestral process in a specific direction. In particular, this
means that we cannot use the results from [10] to couple Deijfen’s model to
the $\infty$-parent ancestral process and deduce directly that its growth is
at least linear in time.

###  2.4 Numerical simulations

In order to obtain an approximation for the limiting speed of growth
$\nu^{-1}$ by means of numerical simulations, we can use the fact that

$\nu^{-1}=\left(\lim\limits_{x\to+\infty}x^{-1}\mathbf{E}[\overleftarrow{\tau}\\!_{x}]\right)^{-1},$

which is a consequence of Theorem 2.11 combined with Proposition 2.20. Indeed,
the $\infty$-parent ancestral is easier to simulate than its forwards-in-time
counterpart, since it jumps at a finite rate at any time, and since there are
no border effects to take into account while simulating the process on an
appropriately chosen compact subset of $\mathbb{R}^{2}$.

We focus on the case in which all ellipses have the same shape parameters. In
order to be able to compare the speed of growth of the occupied regions in
$\infty$-parent SLFV with different shape parameters, we assume that any given
location $z\in\mathbb{R}^{2}$ is affected by a reproduction event at rate $1$.
Therefore, we take

$\tilde{\mu}(da,db,d\gamma)=V_{a_{0},b_{0},0}^{-1}\delta_{a_{0}}(da)\otimes\delta_{b_{0}}(db)\otimes\delta_{0}(d\gamma),$

where $a_{0}\in(0,+\infty)$ and $b_{0}$ is chosen such that the volume of the
corresponding ellipse is equal to $\pi$.

For $9$ different values of $a$ ranging from $0.33$ to $3$, we simulate $30$
$\infty$-parent ancestral processes with initial condition $\\{(0,0)\\}$ and
parameters $(a,b,0)$, where $b$ is chosen as stated above. This ensures that
we compare $\infty$-parent SLFV processes for which reproduction events have
the same scale.

Figure 3: Approximate speed of growth of the occupied area in the
$\infty$-parent SLFV process, as a function of $a$. For each value of $a$,
$30$ $\infty$-parent ancestral processes with parameters $(a,b,0)$ were
simulated, in order to compute $\mathbf{E}[\overleftarrow{\tau}\\!_{x}]$ for
large values of $x$. The crosses indicate the approximate values, and the
dotted line corresponds to the linear function $\nu^{-1}(a)=2.58\,a$.

The results can be found in Figure 3. The numerical simulations show that the
speed of growth is a linear function of $a$ (over the range of $a$-values
considered). This speed is around $2.6$ times higher than the lower bound
obtained in Section 3 (equal to $a$ when $\gamma=0$), which was initially
conjectured to be the limiting speed of the process.

Figure 4: Illustration of the growth dynamics of the occupied region in the
$\infty$-parent SLFV, with $(a,b,\gamma)=(1,1,0)$. The images represent the
same $\infty$-parent SLFV at four instants $0<t_{1}<t_{2}<t_{3}<t_{4}$. The
black area represents the area occupied by real individuals, and the white
area is empty (or equivalently, filled with ghost individuals). The expansion
starts from the left-most part of the image, and goes towards the right-most
part of the image.

Numerical simulations suggest that the growth of the process seems to be
driven by "spikes" in the expansion direction, that then thicken and grow
sideways, bridging the gap with the rest of the population. See Figure 4 for
an illustration of this phenomenon. This motivates the study of a toy model of
two interacting growing piles of cubes in Section 5.

##  3 Lower bound on the speed of growth

In this section, we show that the growth of the occupied area in the
$\infty$-parent SLFV, as well as that of its dual, is at least linear in time.
More precisely, we show the following result.

######  Proposition 3.1.

There exists $\nu\geq 0$ such that

$\lim\limits_{x\to+\infty}\frac{\mathbf{E}\left[\overleftarrow{\tau}\\!_{x}\right]}{x}=\nu.$

The proof of this result can be found at the end of Section 3.3. By
Proposition 2.20, this result is then also true for
$\overrightarrow{\tau}\\!_{x}$.

The key difference with Theorem 2.11 is that Proposition 3.1 does not require
$\nu$ to be non-zero, and hence only means that the growth is at least linear
in time (and faster if $\nu=0$). In Section 4, we show that indeed we have
$\nu\neq 0$, or in other words, that the growth is exactly linear in time.

######  Remark 3.2.

Since the limiting speed of growth is given by $\nu^{-1}$, a lower bound on
the speed of growth amounts to an upper bound on
$\lim\limits_{x\to+\infty}x^{-1}\mathbf{E}\left[\overleftarrow{\tau}\\!_{x}\right]$.

In order to prove Proposition 3.1, we first establish a few auxiliary results.
In all that follows, let $\overline{\Pi}$ be a Poisson point process on
$\mathbb{R}_{+}\times\mathbb{R}^{2}\times S_{\tilde{\mu}}$ with intensity
measure $dt\otimes dz\otimes\tilde{\mu}^{\leftarrow}$, and let $(E_{t})_{t\geq
0}$ be the $\infty$-parent ancestral process with initial condition
$\\{(0,0)\\}$ and associated to $\tilde{\mu}^{\leftarrow}$, constructed using
the Poisson point process $\overline{\Pi}$. Here we use the modification of
the $\infty$-parent ancestral process introduced earlier, which allows to take
single points as initial conditions.

###  3.1 Sub-additivity

We first introduce other $\infty$-parent ancestral processes, coupled to
$(E_{t})_{t\geq 0}$ via the Poisson point process $\overline{\Pi}$. For all
$z\in\mathbb{R}^{2}$ and $s\geq 0$, let $(E_{t}^{z,s})_{t\geq 0}$ be the
$\infty$-parent ancestral process with initial condition $\\{z\\}$ associated
to $\tilde{\mu}^{\leftarrow}$, constructed using only the reproduction events
in $\Pi$ occurring strictly after time $s$. If $s=0$, then
$(E_{t}^{z,0})_{t\geq 0}$ is the regular $\infty$-parent ancestral process,
but if $s\neq 0$, then the process is constant and equal to $\\{z\\}$ during
the time interval $[0,s]$, and only after does it start following the dynamics
of an $\infty$-parent ancestral process.

Since all the processes $(E_{t}^{z,s})_{t\geq 0}$, $z\in\mathbb{R}^{2}$,
$s\geq 0$ are constructed using the same underlying Poisson point process, we
have the following result.

######  Lemma 3.3.

For all $z_{1},z_{2}\in\mathbb{R}^{2}$ and $0<s_{1}<s_{2}$, if $z_{2}\in
E_{s_{2}}^{z_{1},s_{1}}$, then a.s. for all $t\geq s_{2}$, we have

$E_{t}^{z_{2},s_{2}}\subseteq E_{t}^{z_{1},s_{1}}.$

We now introduce the following family of random variables. First, for all
$n\in\mathbb{N}$, let

$\displaystyle T_{0,n}:$ $\displaystyle=\min\left\\{t\geq
0:E_{t}^{(0,0),0}\cap
HP^{4n\mathcal{R}_{\tilde{\mu}}}\neq\emptyset\right\\}=\min\left\\{t\geq
0:E_{t}\cap
HP^{4n\mathcal{R}_{\tilde{\mu}}}\neq\emptyset\right\\}=\overleftarrow{\tau}\\!_{4n\mathcal{R}_{\tilde{\mu}}},$

and let $P_{n}$ be sampled uniformly at random among the points in
$E_{T_{0,n}}$ at horizontal separation of exactly
$4n\mathcal{R}_{\tilde{\mu}}$ from $(0,0)$. That is, $P_{n}$ is a uniform
sample from the compact set

$E_{T_{0,n}}\cap\left\\{(x,y)\in\mathbb{R}^{2}:x=4n\mathcal{R}_{\tilde{\mu}}\right\\}.$

Moreover, for all $0\leq m\leq n\in\mathbb{N}$, let

$T_{m,n}:=\min\left\\{t\geq 0:E_{t+T_{0,m}}^{P_{m},T_{0,m}}\cap
HP^{4n\mathcal{R}_{\tilde{\mu}}}\neq\emptyset\right\\},$

where $E_{t+T_{0,m}}^{P_{m},T_{0,m}}$ corresponds to the $\infty$-parent
ancestral process started from $P_{m}$ at time $T_{0,m}$.

By construction, the family $(T_{m,n})_{0\leq m\leq n}$ satisfies the
following lemma.

######  Lemma 3.4.

For all $0<m<n\in\mathbb{N}$, we have

$T_{0,n}\leq T_{0,m}+T_{m,n}.$

###### Proof.

Let $0<m<n\in\mathbb{N}$. By definition,

$P_{m}\in E_{T_{0,m}}=E_{T_{0,m}}^{(0,0),0}.$

Therefore, using Lemma 3.3, we have a.s. for all $t\geq 0$

$E_{t+T_{0,m}}^{P_{m},T_{0,m}}\subseteq E_{t+T_{0,m}}.$

In particular, this is true for $t=T_{m,n}$, hence

$E_{T_{m,n}+T_{0,m}}^{P_{m},T_{0,m}}\cap
HP^{4n\mathcal{R}_{\tilde{\mu}}}\subseteq E_{T_{m,n}+T_{0,m}}\cap
HP^{4n\mathcal{R}_{\tilde{\mu}}},$

from which we deduce (by definition of $T_{0,m}$, $P_{m}$ and $T_{m,n}$) that

$E_{T_{m,n}+T_{0,m}}\cap HP^{4n\mathcal{R}_{\tilde{\mu}}}\neq\emptyset.$

Therefore,

$\displaystyle T_{0,n}$ $\displaystyle=\min\left\\{t>0:E_{t}\cap
HP^{4n\mathcal{R}_{\tilde{\mu}}}\neq\emptyset\right\\}\leq T_{m,n}+T_{0,m}.$

∎

By invariance by translation of the underlying Poisson point processes, the
following lemmas also hold true.

######  Lemma 3.5.

For all $n\in\mathbb{N}$, the joint distribution of $(T_{n+1,n+k+1})_{k\geq
1}$ is the same as the one of $(T_{n,n+k})_{k\geq 1}$.

######  Lemma 3.6.

For all $k\in\mathbb{N}\setminus\\{0\\}$, the random variables
$(T_{nk,(n+1)k})_{n\geq 1}$ are i.i.d.

In order to use Theorem 1.10 from [31] and conclude, all we need is to show
that the following lemma is true.

######  Lemma 3.7.

For all $n\in\mathbb{N}$, we have $\mathbf{E}[T_{0,n}]<+\infty$.

Lemma 3.7 is proved at the end of Section 3.3, using the so-called express
chain that we now define. We shall then use it to complete the proof of
Proposition 3.1 (again at the end of Section 3.3).

###  3.2 Definition of the express chain

The use of what we shall call the express chain can be motivated by the
following observation. In each ellipse with centre $z=(x,y)\in\mathbb{R}^{2}$
and parameters $(a,b,\gamma)$, there exists exactly one point for which the
horizontal separation from the centre is maximal: the point with coordinates

$(x+a\cos(\theta_{max})\cos(\gamma)-b\sin(\theta_{\max})\sin(\gamma),y+a\cos(\theta_{max})\sin(\gamma)+b\sin(\theta_{max})\cos(\gamma)),$

where $\theta_{max}=\arctan(-ba^{-1}\tan(\gamma))$. This point is at
horizontal separation $\sqrt{a^{2}\cos^{2}(\gamma)+b^{2}\sin^{2}(\gamma)}$
from $z$ (see Figure 5).

Figure 5: Ellipse with parameters $(a,b,\gamma)$.

If we take this potential parent, wait until it is affected by a new
reproduction event, and repeat, we obtain a Markov jump process, jumping at
rate

$J_{\tilde{\mu}}:=\int_{S_{\tilde{\mu}}}V_{a,b,\gamma}\,\tilde{\mu}(da,db,d\gamma),$
(3.1)

and going away from $0$ at an average horizontal speed of
$J_{\tilde{\mu}}\mathbf{E}\left[\sqrt{a^{2}\cos^{2}(\gamma)+b^{2}\sin^{2}(\gamma)}\right]$
(modulo some stochasticity due to the location of the centre of the
reproduction event). See Appendix A for the proof of the geometrical
properties of ellipses used throughout this section.

Formally, the express chain is defined as follows.

######  Definition 3.8.

The express chain associated to $(E_{t})_{t\geq 0}$ (constructed using
$\overline{\Pi}$), denoted $(C_{t}^{express})_{t\geq 0}$, is the
$\mathbb{R}^{2}$-valued Markov process defined as follows.

First, we set $C_{0}^{express}=(0,0)$. Then, for all
$(t,z_{c},a,b,\gamma)\in\Pi$, if
$C_{t-}^{express}\in\mathfrak{B}_{a,b,\gamma}(z_{c})$ and
$z_{c}=(x_{c},y_{c})$, we set:

$C_{t}^{express}=\left(X_{t}^{express},Y_{t}^{express}\right)$

where

$\displaystyle X_{t}^{espress}$
$\displaystyle=x_{c}+a\cos(\theta_{max})\cos(\gamma)-b\sin(\theta_{\max})\sin(\gamma),$
$\displaystyle Y_{t}^{express}$
$\displaystyle=y_{c}+a\cos(\theta_{max})\sin(\gamma)+b\sin(\theta_{max})\cos(\gamma)$

and $\theta_{max}=\arctan(-ba^{-1}\tan(\gamma))$.

See Figure 6 for an illustration of how to construct the express chain.

Figure 6: Comparison between the express chain and the $\infty$-parent
ancestral process at time $t$, for two realisations of the $\infty$-parent
ancestral process. The ellipses indicate the reproduction events which
affected the $\infty$-parent ancestral process until time $t$. The crosses
represent the successive positions of the express chain, and the black point
indicates the location reached by the $\infty$-parent ancestral process at
time $t$ which maximises the horizontal separation from the starting location.
This location coincides with the one reached by the express chain in the first
case, but not in the second case.

The interest of the express chain lies in the following observation, whose
proof is a direct consequence of the definition of $T_{0,n}$.

######  Lemma 3.9.

Let $n\in\mathbb{N}^{*}$. For all $t\geq 0$, we have $\\{C_{t}^{express}\in
HP^{4n\mathcal{R}_{\tilde{\mu}}}\\}\subset\\{T_{0,n}\leq t\\}$.

###### Proof.

Let $t\geq 0$. Since $C_{t}^{express}\in E_{t}$ by construction, the fact the
$C_{t}^{express}\in HP^{4n\mathcal{R}_{\tilde{\mu}}}$ implies that $E_{t}\cap
HP^{4n\mathcal{R}_{\tilde{\mu}}}\neq\emptyset$, and thus that $T_{0,n}\leq t$.
∎

Therefore, if for all $n\in\mathbb{N}$, we set

$T_{0,n}^{express}:=\min\left\\{t\geq 0:C_{t}^{express}\in
HP^{4n\mathcal{R}_{\tilde{\mu}}}\right\\},$

then for all $n\in\mathbb{N}$ we have

$T_{0,n}\leq T_{0,n}^{express}\quad\text{a.s.},\qquad\text{and
so}\quad\mathbf{E}[T_{0,n}]\leq\mathbf{E}\big{[}T_{0,n}^{express}\big{]}.$
(3.2)

In other words, in order to show Lemma 3.7, it is sufficient to obtain a
similar result on $\mathbf{E}[T_{0,n}^{express}]$.

Before studying the properties of the express chain in the next section, we
introduce the following notation. For all $t\geq 0$, let $N_{t}^{express}$ be
the number of jumps of the express chain in the time interval $[0,t]$. For all
$i\in\mathbb{N}\setminus\\{0\\}$, let $t_{i}$ be the instant of the $i$-th
jump of the express chain, let $R_{i}=(R_{i}^{X},R_{i}^{Y})$ be the
coordinates of the centre of the reproduction event triggering this jump, and
let $(a_{i},b_{i},\gamma_{i})$ be the parameters of the ellipse affected by
the reproduction event. We then set for all $i\in\mathbb{N}\setminus\\{0\\}$

$D_{i}=\sqrt{a_{i}^{2}\cos^{2}(\gamma_{i})+b_{i}^{2}\sin^{2}(\gamma_{i})}.$

The random variable $D_{i}$ thus encodes the distance between the centre of
the reproduction event and the right-most point in the corresponding ellipse.
Note that the random variables $(D_{i})_{i\geq 1}$ are i.i.d.

###  3.3 Properties of the express chain

We now study some properties of the express chain, in order to obtain an upper
bound on $\mathbf{E}[T_{0,n}^{express}]$.

Let $t\geq 0$. By construction,

$\displaystyle X_{t}^{express}$
$\displaystyle=\sum_{i=1}^{N_{t}^{express}}\left(R_{i}^{X}-X_{t_{i}-}^{express}+D_{i}\right)$
(3.3)
$\displaystyle=\sum_{i=1}^{N_{t}^{express}}\Big{(}R_{i}^{X}-X_{t_{i}-}^{express}+D_{i}-\mathbf{E}[D_{1}]\Big{)}+\mathbf{E}[D_{1}]N_{t}^{express}.$

The random variables
$(R_{i}^{X}-X_{t_{i}-}^{express}+D_{i}-\mathbf{E}[D_{i}])_{i\geq 1}$ are
i.i.d, bounded and with mean $0$ (see Appendix A). We can then apply
Hoeffding’s inequality [25] and obtain the following lemma.

######  Lemma 3.10.

There exists $C_{1}>0$ such that for all $t\geq 0$ and
$n\in\mathbb{N}\setminus\\{0\\}$, for all
$k>4n\mathcal{R}_{\tilde{\mu}}\mathbf{E}[D_{1}]^{-1}$, we have

$\displaystyle\mathbf{P}\left(X_{t}^{express}<4n\mathcal{R}_{\tilde{\mu}}\Big{|}N_{t}^{express}=k\right)\leq\exp\left(-\frac{\left(\mathbf{E}[D_{1}]k-4n\mathcal{R}_{\tilde{\mu}}\right)^{2}}{C_{1}k}\right).$

Consequently, for all
$C^{\prime}>4n\mathcal{R}_{\tilde{\mu}}\mathbf{E}[D_{1}]^{-1}$,

$\mathbf{P}\left(X_{t}^{express}<4n\mathcal{R}_{\tilde{\mu}}\Big{|}N_{t}^{express}>C^{\prime}\right)\leq\exp\left(-\frac{\left(C^{\prime}\mathbf{E}[D_{1}]-4n\mathcal{R}_{\tilde{\mu}}\right)^{2}}{C_{1}C^{\prime}}\right).$

###### Proof.

First, notice that the second part of the lemma is a direct consequence of the
first part, along with the variations of the function
$x\to\exp(-(xc-d)^{2}x^{-1})$, $c,d>0$. As concerns the first part of the
lemma, let $t\geq 0$, $n\in\mathbb{N}\setminus\\{0\\}$ and
$k>4n\mathcal{R}_{\tilde{\mu}}\mathbf{E}[D_{1}]^{-1}$. Using (3.3), we have

$\displaystyle\mathbf{P}\left(X_{t}^{express}<4n\mathcal{R}_{\tilde{\mu}}\Big{|}N_{t}^{express}=k\right)$
$\displaystyle\leq\mathbf{P}\left(\sum_{i=1}^{k}(R_{i}^{X}-X_{t_{i}-}^{express}+D_{i}-\mathbf{E}[D_{1}])<4n\mathcal{R}_{\tilde{\mu}}-\mathbf{E}[D_{1}]k\Big{|}N_{t}^{express}=k\right)$
$\displaystyle=\mathbf{P}\left(\sum_{i=1}^{k}(-R_{i}^{X}+X_{t_{i}-}^{express}-D_{i}+\mathbf{E}[D_{1}])>\mathbf{E}[D_{1}]k-4n\mathcal{R}_{\tilde{\mu}}\Big{|}N_{t}^{express}=k\right).$

We conclude by using Hoeffding’s inequality together with the fact that for
all $i\geq 1$,

$\displaystyle\left|R_{i}^{X}-X_{t_{i}-}^{express}\right|$
$\displaystyle<4\mathcal{R}_{\tilde{\mu}}\qquad\text{and}\qquad\left|D_{i}-\mathbf{E}[D_{1}]\right|\leq
2\mathcal{R}_{\tilde{\mu}}.$

∎

In order to bound $\mathbf{E}[T_{0,n}^{express}]$, we also need to control the
number of jumps made by the express chain over the time interval $[0,t]$.
Recall the definition of $J_{\tilde{\mu}}$ given in (3.1).

######  Lemma 3.11.

There exists $C_{\otimes}>0$ such that for all $t>0$,

$\mathbf{P}\left(N_{t}^{express}\leq
0.1J_{\tilde{\mu}}t\right)\leq\exp(-C_{\otimes}J_{\tilde{\mu}}t).$

###### Proof.

The proof relies on the fact that the express chain jumps at rate
$J_{\tilde{\mu}}$. Hence, $N_{t}^{express}$ is Poisson distributed with
parameter $J_{\tilde{\mu}}t$.

Let $t>0$. Using a standard Chernoff bound, we obtain

$\displaystyle\mathbf{P}\left(N_{t}^{express}\leq 0.1J_{\tilde{\mu}}t\right)$
$\displaystyle\leq\frac{(eJ_{\tilde{\mu}}t)^{0.1J_{\tilde{\mu}}t}e^{-J_{\tilde{\mu}}t}}{(0.1J_{\tilde{\mu}}t)^{0.1J_{\tilde{\mu}}t}}$
$\displaystyle=\frac{\exp(0.1J_{\tilde{\mu}}t)\exp(-J_{\tilde{\mu}}t)}{\exp(0.1J_{\tilde{\mu}}t\,\ln(0.1))}$
$\displaystyle=\exp(0.1J_{\tilde{\mu}}t-J_{\tilde{\mu}}t-0.1J_{\tilde{\mu}}\ln(0.1)),$

and so taking $C_{\otimes}=J_{\tilde{\mu}}(0.9-0.1\ln(10))$ allows us to
conclude. ∎

Combining Lemmas 3.10 and 3.11, we obtain an upper bound for
$\mathbf{E}[T_{0,n}^{express}]$.

######  Lemma 3.12.

There exists $C_{2}>0$ such that for all $n\in\mathbb{N}$,

$\mathbf{E}\big{[}T_{0,n}^{express}\big{]}\leq C_{2}n.$

###### Proof.

Let $n\in\mathbb{N}$. Then,

$\displaystyle\mathbf{E}\big{[}T_{0,n}^{express}\big{]}=$
$\displaystyle\int_{0}^{\infty}\mathbf{P}\left(T_{0,n}^{express}>t\right)dt$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\mathbf{P}\left(X_{t}^{express}<4n\mathcal{R}_{\tilde{\mu}}\right)dt$
$\displaystyle\leq$
$\displaystyle\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}+\int_{\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}}^{\infty}\mathbf{P}\left(X_{t}^{express}<4n\mathcal{R}_{\tilde{\mu}}\right)dt$
$\displaystyle\leq$
$\displaystyle\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}+\int_{\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}}^{\infty}\mathbf{P}\left(N_{t}^{express}\leq\frac{J_{\tilde{\mu}}t}{10}\right)dt$
$\displaystyle+\int_{\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}}^{\infty}\mathbf{P}\left(\left\\{X_{t}^{express}<4n\mathcal{R}_{\tilde{\mu}}\right\\}\cap\left\\{N_{t}^{express}>\frac{J_{\tilde{\mu}}t}{10}\right\\}\right)$
$\displaystyle\leq$
$\displaystyle\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}+\int_{\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}}^{\infty}\exp(-C_{\otimes}J_{\tilde{\mu}}t)dt$
(3.4)
$\displaystyle+\int_{\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}}^{\infty}\exp\left(-\frac{10}{C_{1}J_{\tilde{\mu}}t}\left(\frac{J_{\tilde{\mu}}t\mathbf{E}[D_{1}]}{10}-4n\mathcal{R}_{\tilde{\mu}}\right)^{2}\right)dt.$

Here we have used Lemmas 3.10 and 3.11 to pass from the fourth to the fifth
line. Moreover, we have

$\displaystyle\int_{\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}}^{\infty}\exp$
$\displaystyle\bigg{(}-\frac{10}{C_{1}J_{\tilde{\mu}}t}\left(\frac{J_{\tilde{\mu}}t\mathbf{E}[D_{1}]}{10}-4n\mathcal{R}_{\tilde{\mu}}\right)^{2}\bigg{)}dt$
$\displaystyle=\exp\left(\frac{2\mathbf{E}[D_{1}]}{C_{1}}4n\mathcal{R}_{\tilde{\mu}}\right)\int_{\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}}^{\infty}\exp\left(-\frac{10}{C_{1}J_{\tilde{\mu}}t}\frac{t^{2}J_{\tilde{\mu}}^{2}\mathbf{E}[D_{1}]^{2}}{100}\right)\exp\left(-\frac{10}{C_{1}J_{\tilde{\mu}}t}16n^{2}\mathcal{R}_{\tilde{\mu}}^{2}\right)dt$
$\displaystyle\leq\exp\left(\frac{8n}{C_{1}}\mathcal{R}_{\tilde{\mu}}\mathbf{E}[D_{1}]\right)\int_{\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}}^{\infty}\exp\left(-\frac{J_{\tilde{\mu}}t}{10C_{1}}\mathbf{E}[D_{1}]^{2}\right)dt$
$\displaystyle\leq\exp\left(\frac{8n}{C_{1}}\mathcal{R}_{\tilde{\mu}}\mathbf{E}[D_{1}]\right)\frac{10C_{1}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]^{2}}\exp\left(-\frac{J_{\tilde{\mu}}\mathbf{E}[D_{1}]^{2}}{10C_{1}}\frac{100n\mathcal{R}_{\tilde{\mu}}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]}\right)$
$\displaystyle\leq\frac{10C_{1}}{J_{\tilde{\mu}}\mathbf{E}[D_{1}]^{2}}\exp\left(-\frac{2n}{C_{1}}\mathcal{R}_{\tilde{\mu}}\mathbf{E}[D_{1}]\right).$

Since the first term on the r.h.s of (3.4) is proportional to $n$, and the
second and third terms decrease exponentially fast in $n$, we can conclude. ∎

We can now conclude the proof of Lemma 3.7.

###### Proof.

(Lemma 3.7) Let $n\in\mathbb{N}$. By Lemma 3.12,

$\mathbf{E}[T_{0,n}^{express}]<+\infty,$

and so by (3.2),

$\mathbf{E}[T_{0,n}]<+\infty.$

∎

We conclude this section with the proof of Proposition 3.1.

###### Proof.

(Proposition 3.1) Combining Lemmas 3.12, 3.4, 3.5 and 3.6 with the fact that
$T_{0,n}\geq 0$ for all $n$, we obtain that the family $(T_{m,n})_{0\leq m\leq
n}$ satisfies all the assumptions of Theorem 1.10 from [31]. Therefore, there
exists $\nu^{\prime}\geq 0$ such that

$\lim\limits_{n\to+\infty}\frac{\mathbf{E}[T_{0,n}]}{n}=\frac{\mathbf{E}[\overleftarrow{\tau}\\!_{4n\mathcal{R}_{\tilde{\mu}}}]}{n}=\nu^{\prime}.$

We conclude by standard upper and lower bounding arguments, using the fact
that $x\to\mathbf{E}[\overleftarrow{\tau}\\!_{x}]$ is a nondecreasing function
(which is a consequence of Lemma 2.18). ∎

######  Remark 3.13.

If we see $(X_{t}^{express})_{t\geq 0}$ as a cumulative process (in the sense
of Chapter IV.3 in [3]), it is possible to use Theorem 3.1 from this Chapter
and show that the limiting horizontal speed of advance of the express chain is
equal to $J_{\tilde{\mu}}\mathbf{E}[D_{1}]$, yielding an explicit lower bound
on the speed of growth of the occupied area in the $\infty$-parent SLFV. In
particular, if all reproduction ellipses have the same shape parameters
$(a,b,\gamma)$ and if the total mass of $\tilde{\mu}$ is
$V_{a,b,\gamma}^{-1}$, which corresponds to the case investigated using
numerical simulations, then the lower bound on the speed of growth is given by
$\sqrt{a^{2}\cos^{2}(\gamma)+b^{2}\sin^{2}(\gamma)}$.

##  4 Upper bound on the speed of growth

Recall that $(E_{t})_{t\geq 0}$ is the $\infty$-parent ancestral process with
initial condition $\\{(0,0)\\}$ associated to $\tilde{\mu}^{\leftarrow}$,
constructed using $\overline{\Pi}$.

In this section, we complete the result proven in Section 3 by showing that
the growth of the $\infty$-parent SLFV and of its dual counterpart are at most
linear in time. This can be rewritten as a limiting property of
$\mathbf{E}[\overleftarrow{\tau}\\!_{x}]$ as follows.

######  Proposition 4.1.

$\lim\limits_{x\to+\infty}\frac{\mathbf{E}[\overleftarrow{\tau}\\!_{x}]}{x}>0.$

The proof can be found at the end of Section 4.3. Combining this result with
Proposition 3.1 gives us the result of Theorem 2.11.

To show the result of Proposition 4.1, we first observe that it is sufficient
to focus on the case in which all reproduction events are balls of fixed
radius. Indeed, we have the following comparison result.

######  Lemma 4.2.

Let $\overline{\Pi}^{\mathcal{R}_{\tilde{\mu}}}$ be a Poisson point process on
$\mathbb{R}_{+}\times\mathbb{R}^{2}\times S_{\tilde{\mu}}$ with intensity
measure

$\tilde{\mu}^{\leftarrow}(S_{\tilde{\mu}})dt\otimes
dz\otimes\delta_{\mathcal{R}_{\tilde{\mu}}}(da)\otimes\delta_{\mathcal{R}_{\tilde{\mu}}}(db)\otimes\delta_{0}(\gamma).$

Let $(E_{t}^{\mathcal{R}_{\tilde{\mu}}})_{t\geq 0}$ be the $\infty$-parent
ancestral process with initial condition $\\{(0,0)\\}$ constructed using
$\overline{\Pi}^{\mathcal{R}_{\tilde{\mu}}}$, and for all $x>0$, let
$\overleftarrow{\tau}_{x}^{\mathcal{R}_{\tilde{\mu}}}$ be the first time
$(E_{t}^{\mathcal{R}_{\tilde{\mu}}})_{t\geq 0}$ reaches $HP^{x}$, defined as
in Definition 2.17. Then, for all $x>0$,

$\mathbf{E}[\overleftarrow{\tau}\\!_{x}]\geq\mathbf{E}\left[\overleftarrow{\tau}\\!_{x}^{\mathcal{R}_{\tilde{\mu}}}\right].$

###### Proof.

The proof relies on the following coupling between $(E_{t})_{t\geq 0}$ and
$(E_{t}^{\mathcal{R}_{\tilde{\mu}}})_{t\geq 0}$. Instead of being independent
from $\overline{\Pi}$, the Poisson point process
$\overline{\Pi}^{\mathcal{R}_{\tilde{\mu}}}$ is constructed using the points
from $\overline{\Pi}$, as follows:

$\overline{\Pi}^{\mathcal{R}_{\tilde{\mu}}}:=\big{\\{}(t,z,\mathcal{R}_{\tilde{\mu}},\mathcal{R}_{\tilde{\mu}},0):\
(t,z,a,b,\gamma)\in\overline{\Pi}\big{\\}}.$

That is, we replace the elliptical area of each reproduction event in
$\overline{\Pi}$ by a ball of maximal radius $\mathcal{R}_{\tilde{\mu}}$.
Since
$\mathfrak{B}_{a,b,\gamma}(z)\subseteq\mathcal{B}_{\mathcal{R}_{\tilde{\mu}}}(z)$,
this coupling ensures that

$\forall t\geq 0,E_{t}\subseteq E_{t}^{\mathcal{R}_{\tilde{\mu}}}\text{ a.s.}$

Therefore, for all $x>0$,

$\min\left\\{t\geq 0:E_{t}\cap
HP^{x}\neq\emptyset\right\\}\geq\min\left\\{t\geq
0:E_{t}^{\mathcal{R}_{\tilde{\mu}}}\cap HP^{x}\right\\}\text{ a.s.},$

which allows us to conclude. ∎

To alleviate the notation, we only provide the proof of Proposition 4.1 when
$\tilde{\mu}^{\leftarrow}=\pi^{-1}\delta_{1}\otimes\delta_{1}\otimes\delta_{0}$,
so that all reproduction events happen in balls of radius $1$ and the rate at
which a given point in space is overlapped by an event is equal to $1$. The
proof can be easily generalised to balls of arbitrary fixed radius and any
intensity of events by a simple scaling of time and space. We can then use
Lemma 4.2 to obtain the corresponding result for ellipses with bounded
parameters.

###  4.1 A first-passage percolation problem

We consider the graph $\mathcal{G}$ on the vertex set $\mathbb{Z}^{2}$, in
which $(i,j)$ and $(i^{\prime},j^{\prime})$ are connected by an edge if, and
only if

$\displaystyle(i^{\prime},j^{\prime})\in\big{\\{}(i+1,j),(i-1,j),(i,j+1),(i,j-1),(i-1,j-1),(i-1,j+1)$
$\displaystyle,(i+1,j-1),$ $\displaystyle\qquad(i+1,j+1)\big{\\}}.$

To each edge $e$ of $\mathcal{G}$, we associate an independent exponential
random variable

$\mathcal{E}_{e}\sim Exp(16\times\pi^{-1}).$

$\mathcal{E}_{e}$ corresponds to the time needed to pass through the
corresponding edge. Following standard terminology in first-passage
percolation, we call it the passage time of the edge. The choice of the rate
of the exponential distribution ensures we can later compare the growth of the
$\infty$-parent ancestral process to the first-passage percolation problem we
now introduce.

If $\Gamma$ is a (potentially infinite) path formed by the edges
$e_{1},...,e_{n},...$, then the passage time of the path $\Gamma$ is defined
as

$\mathcal{E}_{\Gamma}=\sum_{e\in\Gamma}\mathcal{E}_{e}.$

If $z_{1},z_{2}\in\mathbb{Z}^{2}$, we define the first-passage time
$\mathcal{E}_{z_{1},z_{2}}$ from $z_{1}$ to $z_{2}$ (or equivalently, from
$z_{2}$ to $z_{1}$ since $\mathcal{G}$ is not oriented) as the minimum over
the passage times of all the (finite) paths going from $z_{1}$ to $z_{2}$. We
then define

$\overleftarrow{\tau}\\!_{n}^{fpp}:=\min\left\\{t\geq 0:\exists
m\in\mathbb{Z},\mathcal{E}_{(0,0),(n,m)}\leq t\right\\}.$

In other words, $\overleftarrow{\tau}\\!_{n}^{fpp}$ is the time needed to
reach a point at horizontal separation $n$ from the origin, starting from the
origin.

The interest of this first-passage percolation problem lies in the fact that
since the passage time of any given edge is a.s. stricty positive,
generalising Theorem 6.7 from [37] to our lattice yields the following result.

######  Lemma 4.3.

We have

$\lim\limits_{n\to+\infty}\frac{\mathbf{E}[\overleftarrow{\tau}\\!_{n}^{fpp}]}{n}>0.$

In order to use this lemma and show Proposition 4.1, we need to be able to
compare $\overleftarrow{\tau}\\!_{x}$ and $\overleftarrow{\tau}\\!_{n}^{fpp}$.
The main obstacle to this comparison lies in the fact that the $\infty$-parent
ancestral process is continuous in space, while the first-passage percolation
problem is defined on a graph. Therefore, we now introduce a way to
"discretise" the $\infty$-parent ancestral process.

###  4.2 Discretisation of the $\infty$-parent ancestral process

In order to discretise the $\infty$-parent ancestral process, we first place a
grid on $\mathbb{R}^{2}$, and associate a cell to each site of the lattice.
Let

$\mathcal{V}:=\left\\{(4i,4j):(i,j)\in\mathbb{Z}^{2}\right\\}$

be the underlying grid, and for all $(i,j)\in\mathbb{Z}^{2}$, let
$\mathcal{C}_{i,j}$ be the square with centre $(4i,4j)$ and side length $2$.
Each $\mathcal{C}_{i,j}$, $i,j\in\mathbb{Z}$ corresponds to the cell
associated to the site $(4i,4j)$ of the grid $\mathcal{V}$. See Figure 7 for
an illustration.

Figure 7: Grid used to discretise the $\infty$-parent ancestral process when
reproduction events are balls. Each hatched square corresponds to a cell. The
grey area around the site $(4i,4j)$, indicated by a black cross, corresponds
to the area in which centres of reproduction events have to fall to intersect
the cell $\mathcal{C}_{i,j}$. The white areas contain all the centres of
reproduction events which do not intersect any cell.

This construction satisfies the two following key properties, that are
consequences of the fact that all reproduction events are balls of radius $1$.

  1. 1.

For all $z=(x,y)\in\mathbb{R}^{2}$, unless $x=4i$, $i\in\mathbb{Z}$ or
$y=4j+2$, $j\in\mathbb{Z}$, the ball $\mathcal{B}_{1}(z)$ intersects at most
one cell.

In other words, each reproduction events intersects at most one cell a.s.
Moreover, if $\mathcal{B}_{1}(z)$ does not intersect any cell, then it means
that $z$ fell into one of the areas in white on Figure 7.

  2. 2.

If we refer to a sequence of reproduction events occurring in chronological
order and such that each reproduction event intersects the previous one as a
path of reproduction events, then any path of reproduction events for which
none of the corresponding balls intersect a cell is almost surely confined in
one of the white areas on Figure 7, in the sense that all the reproduction
event centres fall into the same white area almost surely.

The discretised version of the $\infty$-parent ancestral process, denoted by
$(D_{t})_{t\geq 0}$, is then defined as follows. For all $t\geq 0$, let

$D_{t}:=\left\\{(i,j)\in\mathbb{Z}^{2}:\mathcal{C}_{i,j}\cap
E_{t}\neq\emptyset\right\\}$

be the set of all cells which intersect the $\infty$-parent ancestral process
associated to
$\tilde{\mu}^{\leftarrow}=\pi^{-1}\delta_{1}\otimes\delta_{1}\otimes\delta_{0}$
(and with initial condition $(0,0)$) at time $t$. Let
$(\overleftarrow{\tau}\\!_{n}^{discr})_{n\geq 0}$ be the random variables
defined for all $n\geq 0$ by

$\overleftarrow{\tau}\\!_{n}^{discr}:=\min\left\\{t\geq 0:\exists
m\in\mathbb{Z},(n,m)\in D_{t}\right\\}.$

Due to the structure of the grid and the size of the cells, we have the
following result.

######  Lemma 4.4.

For all $n\in\mathbb{N}\setminus\\{0\\}$, we have

$\overleftarrow{\tau}\\!_{n}^{discr}\leq\overleftarrow{\tau}\\!_{4n}\quad\text{a.s}.$

###### Proof.

Let $n\in\mathbb{N}\setminus\\{0\\}$. Let $z\in\mathbb{R}^{2}$ be the centre
of the reproduction event which occurs at time $\overleftarrow{\tau}\\!_{4n}$
and makes the $\infty$-parent ancestral process reach for the first time a
point at horizontal separation $4n$ from the origin. Then, a.s. there exists
$j\in\mathbb{Z}$ such that

$\mathcal{B}_{1}(z)\cap\mathcal{C}_{n,j}\neq\emptyset,$

and hence
$\overleftarrow{\tau}\\!_{n}^{discr}\leq\overleftarrow{\tau}\\!_{4n}$. ∎

In all that follows, for all $(i,j),(i^{\prime},j^{\prime})\in\mathbb{Z}^{2}$,
we say that

  * •

the cell $\mathcal{C}_{i,j}$ is active at time $t$ if $(i,j)\in D_{t}$;

  * •

the cell $\mathcal{C}_{i,j}$ is activated at time $t$ if $(i,j)\in D_{t}$ and
$(i,j)\notin D_{t-}$;

  * •

the cell $\mathcal{C}_{i^{\prime},j^{\prime}}$ activates $\mathcal{C}_{i,j}$
at time $t$ if there exists $s<t$ such that
$\mathcal{C}_{i^{\prime},j^{\prime}}$ is active at time $s$, and if there
exists a path of reproduction events starting from
$\mathcal{C}_{i^{\prime},j^{\prime}}$ at time $s$, initially overlapping an
area of $\mathcal{C}_{i^{\prime},j^{\prime}}$ containing type $1$ individuals,
and reaching $\mathcal{C}_{i,j}$ for the first time at time $t$ while not
intersecting any other cell in the time interval $[s,t]$.

Notice that under this terminology, a cell can activate another one which is
already active. Moreover, with probability one the only cells that the cell
$\mathcal{C}_{i^{\prime},j^{\prime}}$ can activate are its nearest neighbours
in the graph $\mathcal{G}$, that is, the cells $\mathcal{C}_{i,j}$ such that

$\sqrt{(i-i^{\prime})^{2}+(j-j^{\prime})^{2}}\leq\sqrt{2}.$

###  4.3 Comparison to the first-passage percolation problem

Let us now compare the growth of the discretised $\infty$-parent ancestral
process $(D_{t})_{t\geq 0}$ to that of the first-passage percolation problem
introduced earlier. Recall that

$\overleftarrow{\tau}\\!_{n}^{fpp}:=\min\left\\{t\geq 0:\exists
m\in\mathbb{Z},\mathcal{E}_{(0,0),(n,m)}\leq t\right\\}$

is the time needed by the process associated to the first-passage percolation
problem to reach a point at horizontal separation $n$ from the origin,
starting from the origin.

######  Proposition 4.5.

The random variable $\overleftarrow{\tau}\\!_{n}^{discr}$ is stochastically
bounded from below by $\overleftarrow{\tau}\\!_{n}^{fpp}$. That is, for all
$t\geq 0$,

$\mathbf{P}\left(\overleftarrow{\tau}\\!_{n}^{discr}\geq
t\right)\geq\mathbf{P}\left(\overleftarrow{\tau}\\!_{n}^{fpp}\geq t\right).$

###### Proof.

First, notice that in the discretised $\infty$-parent ancestral process, a
cell can be activated by several neighbouring cells at the same time, but a.s.
an active cell can only activate neighbouring cells one after another.
Conversely, in the first-passage percolation problem, the edges leading to a
same cell are almost surely crossed at different times, but only the first
time one of these edges is crossed matters. Therefore, it is sufficient to
show that the time needed to cross an edge in the process associated to the
first-passage percolation problem is stochastically bounded from above by the
time needed for a cell to be activated once at least one of its neighbours is
active.

Then, assume that at some time $t$, the cell $\mathcal{C}_{i,j}$ was just
activated, while one of its neighbours is still not active. Let $e$ denote the
edge between $(i,j)$ and $(i^{\prime},j^{\prime})$. In order for
$\mathcal{C}_{i^{\prime},j^{\prime}}$ to be activated (not necessarily by
$\mathcal{C}_{i,j}$), there needs to be a reproduction events occurring after
time $t$ and intersecting $\mathcal{C}_{i^{\prime},j^{\prime}}$. Such
reproduction events occur at a rate bounded from above by $\pi^{-1}$ times the
volume of the area of potential centres $z$ such that
$\mathcal{B}_{1}(z)\cap\mathcal{C}_{i^{\prime},j^{\prime}}\neq 0$, which is
itself bounded by $16\pi^{-1}$. Therefore, the time needed for
$\mathcal{C}_{i^{\prime},j^{\prime}}$ to be activated is stochastically
bounded from below by $\mathcal{E}_{e}$, and we can conclude. ∎

######  Remark 4.6.

Due to correlations between activations by neighbouring cells, it woud be more
difficult to construct a coupling with the first-passage percolation problem.
However, the stochastic comparison we obtained is sufficient.

We can now show Proposition 4.1.

###### Proof.

(Proposition 4.1) By Proposition 4.5 and Lemma 4.4, for all
$n\in\mathbb{N}\setminus\\{0\\}$,

$\mathbf{E}[\overleftarrow{\tau}\\!_{4n}]\geq\mathbf{E}[\overleftarrow{\tau}\\!_{n}^{discr}]\geq\mathbf{E}[\overleftarrow{\tau}\\!_{n}^{fpp}].$

By Lemma 4.3, we know that

$\lim\limits_{n\to+\infty}\frac{\mathbf{E}[\overleftarrow{\tau}\\!_{n}^{fpp}]}{4n}>0.$

Since we know that
$\lim\limits_{n\to+\infty}\mathbf{E}[\overleftarrow{\tau}\\!_{n}]\times
n^{-1}$ exists by Proposition 3.1, we obtain that

$\lim\limits_{n\to+\infty}\frac{\mathbf{E}[\overleftarrow{\tau}\\!_{4n}]}{4n}>0.$

We conclude by using the fact that $x\to\overleftarrow{\tau}\\!_{x}$ is
nondecreasing by Lemma 2.18. ∎

We conclude this section with the proof of Theorem 2.11.

###### Proof.

(Theorem 2.11) By Proposition 3.1, we know that there exists $\nu\geq 0$ such
that

$\lim\limits_{x\to+\infty}\frac{\mathbf{E}[\overleftarrow{\tau}_{x}]}{x}=\nu.$

Moreover, by Proposition 4.1, we know that

$\lim\limits_{x\to+\infty}\frac{\mathbf{E}[\overleftarrow{\tau}_{x}]}{x}>0.$

Therefore, $\nu>0$ and we conclude by using Proposition 2.20. ∎

##  5 The two column growth process

Figure 8: Growth dynamics of the two columns growth process. The red arrows
indicate possible growth events occurring at rate 1.

The numerical simulations performed in Section 2.4 show that the speed of
growth of the occupied region in the $\infty$-parent SLFV is higher than the
one conjectured using the express chain. Moreover, the results suggest that
the growth of the process is driven by relatively unfrequent "spikes" than
then thicken and grow sideways. In order to investigate how these spikes can
significantly increase the growth speed, we introduce the following toy model.
We consider two adjacent piles of cubes, thereafter referred to as the left
pile and the right pile. Each cube has height $1$. A cube is added on top of
the left (_resp._ , right) pile at rate $1$, independently from the other
pile. Moreover, the piles can also grow sideways: if there is a cube at height
$h$ in the left (_resp._ , right) pile, and no cube at such height in the
other pile, then a cube is added to the right (_resp._ , left) pile at height
$h$ at rate $1$. In particular, if the left pile is one cube higher than the
right (_resp._ , left) pile, then the total rate at which the height of the
right (_resp._ , left) pile increases of $1$ is equal to $2$, as the growth
can be due to a new cube falling on top of the right (_resp._ , left) pile as
well as sideways growth of the left (_resp._ , right) pile. See Figure 8 for
an illustration of the dynamics of the process.

######  Remark 5.1.

The term "pile" can be slightly misleading, since it is possible to have holes
in it, due to the other pile growing sideways. See Figure 8 for an
illustration.

The interest of the process lies in the following observation. If $l_{t}$
(_resp._ , $r_{t}$) represents the maximal height reached by the left (_resp._
, right) pile, then the cubes at height $h<\min(l_{t},r_{t})$ no longer
contribute to the growth of the process. Therefore, we can "reset" the process
whenever $l_{t}=r_{t}$.

We now define this process, called the two columns growth process, rigorously.

###  5.1 The two column growth process: Definition and properties

The state space over which the process is defined is the set
$\widetilde{\mathcal{S}}$ defined by

$\widetilde{\mathcal{S}}:=\left\\{(i,j)\in\mathbb{N}\times\mathbb{N}:i\leq
j\right\\}.$

If $(i,j)\in\widetilde{\mathcal{S}}$, then $i$ represents the height reached
by the lowest pile, and $j$ the height reached by the highest pile.

######  Definition 5.2.

Let $(i,j)\in\widetilde{\mathcal{S}}$. The two columns growth process
$(G_{t})_{t\geq 0}=(m_{t},M_{t})_{t\geq 0}$ with initial condition $(i,j)$,
thereafter referred to as the 2-CGP, is the continuous-time Markov chain on
the state space $\widetilde{\mathcal{S}}$ whose transition rates are as
follows. For all $(i,j)\in\widetilde{\mathcal{S}}$,

  1. 1.

If $i=j$, then $(i,j)\to(i,i+1)$ at rate $2$, and no other transitions are
possible.

  2. 2.

If $j\geq i+1$, then

$(i,j)\to\left\\{\begin{array}[]{ll}(i,j+1)&\mbox{at rate }1,\\\
(i+1,j)&\mbox{at rate }2,\\\ (i+k,j),k\in\llbracket 2,j-i\rrbracket&\mbox{at
rate }1\mbox{ if }j>i+1,\end{array}\right.$

and no other transitions are possible.

These transition rates exactly encode the dynamics described earlier. Indeed,
if $i=j$, then both piles have the same height, and cubes fall onto each one
of the piles at rate $1$, yielding a total transition rate of $2$ as we do not
record which pile is the highest. If $i\neq j$, then each pile can grow
upwards at rate $1$, but the highest pile can also grow sideways. Depending on
where the highest pile grows sideways, this results in a more or less sharp
increase in the height of the lowest pile.

Earlier, we saw that it was possible to "reset" the process whenever the two
piles had the same height. In order to do so, let
$\mathcal{S}_{\square\\!\square}$ be the set

$\mathcal{S}_{\square\\!\square}:=\left\\{(i,i):i\in\mathbb{N}\right\\}$

of all configurations such that both piles have the same height, and let
$T_{\square}$ be the time of first return of $(G_{t})_{t\geq 0}$ to a state
belonging to $\mathcal{S}_{\square\\!\square}$. Our first result is an upper
bound on $\mathbf{E}_{(0,0)}\left[T_{\square}\right]$.

######  Lemma 5.3.

We have

$\mathbf{E}_{(0,0)}\left[T_{\square}\right]\leq 3/2.$

###### Proof.

In order to do so, we first notice that $T_{\square}$ is equal to the sum of
the time needed to exit $(0,0)$, which follows an exponential distribution
with parameter $2$, and of the time needed to reach a state in
$\mathcal{S}_{\square\\!\square}$ starting from $(0,1)$.

The first step yields an expected contribution of $1/2$ to
$\mathbf{E}_{(0,0)}[T_{\square}]$. Once the process is in the state $(0,1)$,
assuming without loss of generality that the cube fell on the right pile
(which is now the highest pile), we assign an exponential clock with parameter
$1$ to the highest cube of the right pile, which rings whenever the cube
attempts to grow sideways. Whenever the highest cube changes, the exponential
clock is assigned to the new highest cube. When the clock rings for the first
time, we distinguish two cases:

  * •

Either there is no cube at the same height in the left pile, and the cube can
grow sideways. Then, we are back to a state in
$\mathcal{S}_{\square\\!\square}$, though perhaps not for the first time.

  * •

Either there is already a cube at the same height in the left pile. Then, we
know we already went back to a state in $\mathcal{S}_{\square\\!\square}$.

Therefore,

$\mathbf{E}_{(0,0)}\left[T_{\square}\right]\leq 1/2+1,$

allowing us to conclude. ∎

We can use this lemma to obtain an upper bound on
$\mathbf{E}_{(0,0)}[M_{T_{\square}}]$ and
$\mathbf{E}_{(0,0)}[m_{T_{\square}}]$.

######  Lemma 5.4.

We have

$\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\right]\leq
2\quad\text{and}\quad\mathbf{E}_{(0,0)}\left[m_{T_{\square}}\right]\leq 2.$

###### Proof.

First, as $G_{T_{\square}}\in\mathcal{S}_{\square\\!\square}$, we have
$M_{T_{\square}}=m_{T_{\square}}$, and it is sufficient to provide an upper
bound on $\mathbf{E}_{(0,0)}[M_{T_{\square}}]$. Let us set

$\widetilde{T}:=\min\left\\{t\geq 0:M_{t}\neq m_{t}\right\\},$

which corresponds to the time needed for the process to leave the state
$(0,0)$. For all $t\in[\widetilde{T},T_{\square})$, we have $M_{t}>m_{t}$, and
the lowest pile does not contribute to the growth of the highest pile.
Therefore, over the time interval $[\widetilde{T},T_{\square})$, the only way
the highest pile can grow is by new cubes falling on top of it, whose total
number is given by $M_{T_{\square}}-M_{\widetilde{T}}=M_{T_{\square}}-1$.
Moreover, since the clocks corresponding to the upward growth of the highest
pile are independent of the clocks responsible for the upward growth of the
smallest pile and for the sidewise growth at the top of the highest pile,
conditionally on $T_{\square}-\widetilde{T}$, the variable
$M_{T_{\square}}-M_{\widetilde{T}}$ follows a Poisson distribution with
parameter $T_{\square}-\widetilde{T}$. Therefore,

$\displaystyle\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\right]$
$\displaystyle=\mathbf{E}_{(0,0)}\big{[}M_{\widetilde{T}}\big{]}+\mathbf{E}_{(0,0)}\big{[}M_{T_{\square}}-M_{\widetilde{T}}\big{]}$
$\displaystyle=1+\mathbf{E}_{(0,0)}\Big{[}\mathbf{E}_{(0,0)}\big{[}M_{T_{\square}}-M_{\widetilde{T}}\big{|}T_{\square}-\widetilde{T}\big{]}\Big{]}$
$\displaystyle=1+\mathbf{E}_{(0,0)}\big{[}T_{\square}-\widetilde{T}\big{]}$
$\displaystyle=1-1/2+\mathbf{E}_{(0,0)}\big{[}T_{\square}\big{]}$
$\displaystyle\leq 1/2+3/2$

by Lemma 5.3, which allows us to conclude. ∎

In order to study the speed of growth of the process, we see the 2-CGP as a
cumulative process, in the sense of Chapter VI in [3]. We obtain the following
theoretical result.

######  Theorem 5.5.

We have

$\lim\limits_{t\to+\infty}\frac{M_{t}}{t}=\lim\limits_{t\to+\infty}\frac{m_{t}}{t}=\frac{\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\right]}{\mathbf{E}_{(0,0)}\left[T_{\square}\right]}\text{
a.s.}$

###### Proof.

Let $T_{\square}^{0}=0$, and for all $n\geq 1$, let $T_{\square}^{n}$ be the
time of n-th return of $(G_{t})_{t\geq 0}$ to a state in
$\mathcal{S}_{\square\\!\square}$. For all $t>0$, let
$N(t)=\max\\{n\in\mathbb{N}:T_{\square}^{n}\leq t\\}$. We can rewrite $m_{t}$
and $M_{t}$ as

$\displaystyle m_{t}$
$\displaystyle=\sum_{n=1}^{N(t)}\big{(}M_{T_{\square}^{n}}-M_{T_{\square}^{n}-1}\big{)}+m_{t}-M_{T_{\square}^{N(t)}}$
$\displaystyle\text{and }M_{t}$
$\displaystyle=\sum_{n=1}^{N(t)}\big{(}M_{T_{\square}^{n}}-M_{T_{\square}^{n}-1}\big{)}+M_{t}-M_{T_{\square}^{N(t)}}.$

Here we have used the fact that for all $n\in\mathbb{N}$,
$M_{T_{\square}^{n}}=m_{T_{\square}^{n}}$.

Observe that the random variables
$(M_{T_{\square}^{n}}-M_{T_{\square}^{n-1}})_{n\geq 1}$ are i.i.d. The same is
true for the random variables $(T_{\square}^{n}-T_{\square}^{n-1})_{n\geq 1}$.
If we can show that

$\mathbf{E}_{(0,0)}\left[\max_{0\leq t\leq
T_{\square}}M_{t}\right]<+\infty\quad\text{and}\quad\mathbf{E}_{(0,0)}\left[\max_{0\leq
t\leq T_{\square}}m_{t}\right]<+\infty,$ (5.1)

then we can use Theorem 3.1 from Chapter VI in [3] and conclude. As $t\to
M_{t}$ and $t\to m_{t}$ are non-decreasing, this amounts to showing that
$\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\right]<+\infty$ and
$\mathbf{E}_{(0,0)}[m_{T_{\square}}]<+\infty$, which is a direct consequence
of Lemma 5.4. ∎

In order to use this result and obtain the limiting speed of growth of the
2-CGP, we need to be able to compute $\mathbf{E}_{(0,0)}[M_{T_{\square}}]$ and
$\mathbf{E}_{(0,0)}[T_{\square}]$. As a first step, we introduce the
martingale problem satisfied by the 2-CGP, which we shall use to obtain a
relation between $\mathbf{E}_{(0,0)}[M_{T_{\square}}]$ and
$\mathbf{E}_{(0,0)}[T_{\square}]$.

Let $C_{b}(\widetilde{\mathcal{S}})$ be the space of bounded functions
$f:\widetilde{\mathcal{S}}\to\mathbb{R}$. The generator $\mathcal{G}$ of the
2-CGP acting on functions $f\in C_{b}(\widetilde{\mathcal{S}})$ is defined as
follows. For all $f\in C_{b}(\widetilde{\mathcal{S}})$ and for all
$(i,j)\in\widetilde{\mathcal{S}})$,

$\displaystyle\mathcal{G}f(i,j)=$ $\displaystyle\mathds{1}_{\\{i=j\\}}\times
2\left(f(i,i+1)-f(i,j)\right)$
$\displaystyle+\mathds{1}_{\\{i+1=j\\}}\times\left[2(f(i+1,j)-f(i,j))+f(i,j+1)-f(i,j)\right]$
$\displaystyle+\mathds{1}_{\\{j\geq
i+2\\}}\times\left[2(f(i+1,j)-f(i,j))+f(i,j+1)-f(i,j)+\sum_{k=2}^{j-i}(f(i+k,j)-f(i,j))\right].$

The 2-CGP is then a solution to the following martingale problem.

######  Lemma 5.6.

Let $(i,j)\in\widetilde{\mathcal{S}}$, and let $(G_{t})_{t\geq
0}=(m_{t},M_{t})_{t\geq 0}$ be the 2-CGP with initial condition $(i,j)$. Then,
for all $f\in C_{b}(\widetilde{\mathcal{S}})$,

$\left(f(G_{t})-f((i,j))-\int_{0}^{t}\mathcal{G}f(G_{s})ds\right)_{t\geq 0}$

is a martingale.

We use the martingale problem with functions of the form $f_{d}:(i,j)\to
j\mathds{1}_{\\{j<d\\}}$, $d\in\mathbb{N}\setminus\\{0,1\\}$. Indeed, for all
$d\in\mathbb{N}\setminus\\{0,1\\}$ and $s\geq 0$, if $G_{s}=(m_{s},M_{s})$ is
such that $M_{s}-m_{s}\geq 1$, then

$\mathcal{G}f_{d}(G_{s})=\mathds{1}_{\\{M_{s}<d-1\\}}-\mathds{1}_{\\{M_{s}=d-1\\}}(d-1)$
(5.2)

and if $m_{s}=M_{s}$, then

$\mathcal{G}f_{d}(G_{s})=2\mathds{1}_{\\{M_{s}<d-1\\}}-2(d-1)\mathds{1}_{\\{M_{s}=d-1\\}}.$
(5.3)

We obtain the following result.

######  Lemma 5.7.

We have

$\mathbf{E}_{(0,0)}[M_{T_{\square}}]=\frac{1}{2}+\mathbf{E}_{(0,0)}[T_{\square}].$

###### Proof.

For all $d\in\mathbb{N}\setminus\\{0,1\\}$, let $T_{d}:=\inf\\{t\geq
0:M_{t}\geq d\\}$. We use the martingale problem stated in Lemma 5.6 with the
function $f_{d}:(i,j)\to j\mathds{1}_{\\{j<d\\}}$,
$d\in\mathbb{N}\setminus\\{0,1\\}$, and the stopping time $T_{\square}\wedge
T_{d}$. We obtain

$\displaystyle\mathbf{E}_{(0,0)}\left[M_{T_{\square}\wedge T_{d}}\right]$
$\displaystyle=\mathbf{E}_{(0,0)}\left[M_{T_{\square}\wedge
T_{d}}\mathds{1}_{\\{M_{T_{\square}\wedge T_{d}<d+2}\\}}\right]$
$\displaystyle=0+\mathbf{E}_{(0,0)}\left[\int_{0}^{T_{d}\wedge
T_{d}}\mathcal{G}f_{d+2}(G_{s})ds\right]$
$\displaystyle=\mathbf{E}_{(0,0)}\left[\int_{0}^{T_{1}}\mathcal{G}f_{d+2}(G_{s})ds\right]+\mathbf{E}_{(0,0)}\left[\int_{T_{1}}^{T_{\square}\wedge
T_{d}}\mathcal{G}f_{d+2}(G_{s})ds\right]$
$\displaystyle=\mathbf{E}_{(0,0)}\left[2T_{1}\right]+\mathbf{E}_{(0,0)}\left[T_{\square}\wedge
T_{d}-T_{1}\right]$
$\displaystyle=\mathbf{E}_{(0,0)}\left[T_{1}\right]+\mathbf{E}_{(0,0)}\left[T_{\square}\mathds{1}_{\\{T_{\square}\leq
T_{d}\\}}\right]+\mathbf{E}_{(0,0)}\left[T_{d}\mathds{1}_{\\{T_{d}<T_{\square}\\}}\right]$
$\displaystyle=\frac{1}{2}+\mathbf{E}_{(0,0)}\left[T_{\square}\mathds{1}_{\\{T_{\square}\leq
T_{d}\\}}\right]+\mathbf{E}_{(0,0)}\left[T_{d}\mathds{1}_{\\{T_{d}<T_{\square}\\}}\right].$

Here we have used (5.2) and (5.3) to pass from the second to the third line.
Moreover, we also have

$\mathbf{E}_{(0,0)}\left[M_{T_{\square}\wedge
T_{d}}\right]=\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\mathds{1}_{\\{T_{\square}\leq
T_{d}\\}}\right]+\mathbf{E}_{(0,0)}\left[d\mathds{1}_{\\{T_{\square}>T_{d}\\}}\right].$

Therefore, if we show that

$\displaystyle\lim\limits_{d\to+\infty}\mathbf{E}_{(0,0)}\left[T_{\square}\mathds{1}_{\\{T_{\square}\leq
T_{d}\\}}\right]$ $\displaystyle=\mathbf{E}_{(0,0)}\left[T_{\square}\right],$
(5.4)
$\displaystyle\lim\limits_{d\to+\infty}\mathbf{E}_{(0,0)}\left[T_{d}\mathds{1}_{\\{T_{d}<T_{\square}\\}}\right]$
$\displaystyle=0,$ (5.5)
$\displaystyle\lim\limits_{d\to+\infty}\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\mathds{1}_{\\{T_{\square}\leq
T_{d}\\}}\right]$
$\displaystyle=\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\right],$ (5.6)
$\displaystyle\text{and}\quad\lim\limits_{d\to+\infty}\mathbf{E}_{(0,0)}\left[M_{T_{d}}\mathds{1}_{\\{T_{d}<T_{\square}\\}}\right]$
$\displaystyle=0,$ (5.7)

then we shall be able to conclude.

In order to do so, recall that in order to come back to the set
$\mathcal{S}_{\square\\!\square}$, the process first needs to leave the state
$(0,0)$, which occurs at time $T_{1}\sim Exp(2)$. Without loss of generality,
we assume that the first cube falls on the right pile. As in the proof of
Lemma 5.3, we assign an exponential clock $T_{\to}\sim Exp(1)$ to the highest
cube of the right pile, and move this exponential clock to the new highest
cube in this pile whenever it grows. Then, reasoning as in the proof of Lemma
5.3, we have $T_{\square}\leq T_{\to}+T_{1}$. Moreover, $M_{T_{\square}}\leq
d$ if, and only if at most $d-1$ cubes fall on the right pile during the time
interval $[T_{1},T_{\square}]$. Therefore, if $\mathcal{P}(\lambda)$,
$\lambda>0$ stands for a Poisson random variable with parameter $\lambda$,

$\displaystyle\mathbf{P}_{(0,0)}(T_{\square}>T_{d})$
$\displaystyle\leq\mathbf{P}(\mathcal{P}(T_{\to})\geq d-1)$
$\displaystyle\leq\int_{0}^{\infty}e^{-t}\mathbf{P}(\mathcal{P}(t)\geq d-1)dt$
$\displaystyle\leq\int_{0}^{\infty}e^{-t}\frac{\mathbf{E}[\mathcal{P}(t)]}{d-1}dt$
$\displaystyle\leq\frac{1}{d-1}\int_{0}^{\infty}te^{-t}dt$
$\displaystyle\xrightarrow[d\to+\infty]{}0.$

Therefore, by the dominated convergence theorem and Lemmas 5.3 and 5.4, we can
write

$\lim\limits_{d\to+\infty}\mathbf{E}_{(0,0)}\left[T_{\square}\mathds{1}_{\\{T_{d}<T_{\square}\\}}\right]=0\quad\text{and}\quad\lim\limits_{d\to+\infty}\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\mathds{1}_{\\{T_{d}<T_{\square}\\}}\right]=0,$

from which we deduce (5.4) and (5.6). Moreover,

$\mathbf{E}_{(0,0)}\left[T_{d}\mathds{1}_{\\{T_{d}<T_{\square}\\}}\right]\leq\mathbf{E}_{(0,0)}\left[T_{\square}\mathds{1}_{\\{T_{d}<T_{\square}\\}}\right],$

giving (5.5), and as $t\to M_{t}$ is non-decreasing,

$\mathbf{E}_{(0,0)}\left[M_{T_{d}}\mathds{1}_{\\{T_{d}<T_{\square}\\}}\right]\leq\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\mathds{1}_{\\{T_{d}<T_{\square}\\}}\right],$

allowing us to conclude, again by a dominated convergence argument. ∎

Using this result together with Theorem 5.5, we obtain a new expression for
the speed of growth of the process, along with explicit lower and upper
bounds.

######  Proposition 5.8.

We have

$\lim\limits_{t\to+\infty}\frac{M_{t}}{t}=1+\frac{1}{2\mathbf{E}_{(0,0)}[T_{\square}]}\in[4/3,2]\qquad\text{
a.s.}$

###### Proof.

By Theorem 5.5,

$\lim\limits_{t\to+\infty}\frac{M_{t}}{t}=\frac{\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\right]}{\mathbf{E}_{(0,0)}\left[T_{\square}\right]}\text{
a.s.}$

Moreover, by Lemma 5.7,

$\displaystyle\frac{\mathbf{E}_{(0,0)}\left[M_{T_{\square}}\right]}{\mathbf{E}_{(0,0)}\left[T_{\square}\right]}$
$\displaystyle=\frac{1/2+\mathbf{E}_{(0,0)}\left[T_{\square}\right]}{\mathbf{E}_{(0,0)}\left[T_{\square}\right]}=1+\frac{1}{2\mathbf{E}_{(0,0)}\left[T_{\square}\right]}.$

By Lemma 5.3, $\mathbf{E}_{(0,0)}\left[T_{\square}\right]\leq 3/2$ and so

$1+\frac{1}{2\mathbf{E}_{(0,0)}\left[T_{\square}\right]}\geq 1+1/3=4/3.$

Moreover, as $T_{\square}\geq T_{1}$ and as $T_{1}\sim Exp(2)$, we have
$\mathbf{E}_{(0,0)}\left[T_{\square}\right]\geq 1/2$. Therefore,

$1+\frac{1}{2\mathbf{E}_{(0,0)}\left[T_{\square}\right]}\leq 1+1=2,$

allowing us to conclude. ∎

######  Remark 5.9.

Without the interaction with the other pile, the speed of growth of an
isolated pile of cubes would be of $1$. Therefore, this first result means
that the interaction between the two piles increases the growth speed by a
factor of at least $1.33$, and at most $2$. Considering more than two
interacting piles would increase the growth speed even more, and yield a
factor closer to the one obtained for the SLFV (whose value of $2.58$ was
obtained through numerical simulations in Section 2.4).

###  5.2 The discretised two columns growth process

The main obstacle to the study of the speed of growth of the 2-CGP lies in the
fact that the process does not jump at a constant rate: the bigger the height
difference between the two columns, the faster the process jumps. In order to
circumvent this problem, we now introduce a discretised version of the two
columns growth process. Then, we explain how to couple it to the 2-CGP, and
use its invariant distribution to obtain an approximation for
$\mathbf{E}_{(0,0)}[T_{\square}]$ and for the speed of growth. In all that
follows, let $(G_{t})_{t\geq 0}=(m_{t},M_{t})_{t\geq 0}$ be the 2-CGP with
initial condition $(0,0)$. Recall that $T_{\square}$ is the time of first
return of $(G_{t})_{t\geq 0}$ to a state in $\mathcal{S}_{\square\\!\square}$,
and for all $d\in\mathbb{N}\setminus\\{0,1\\}$, $T_{d}=\inf\\{t\geq
0:M_{t}\geq d\\}$.

Moreover, let $N\in\mathbb{N}\setminus\\{0,1\\}$ and $\epsilon>0$. In order to
construct the discretised 2-CGP, we make the following observation. If $t\geq
0$, then the probability that the process $(G_{t})_{t\geq 0}$ jumps at least
once in the time interval $[t,t+\epsilon)$ is equal to

$1-\exp\left(-\epsilon(M_{t}-m_{t}+2)\right)\approx\epsilon(M_{t}-m_{t}+2),$

and the probability that it jumps at least twice is bounded from above by

$\left(1-\exp\left(-\epsilon(M_{t}-m_{t}+3)\right)\right)^{2}\approx\epsilon^{2}(M_{t}-m_{t}+3)^{2}.$

Therefore, if $\epsilon$ is small enough and if we are able to control
$M_{t}-m_{t}$, then we can consider that at most one growth event occurs
during a time interval of length $\epsilon$. We use this idea to construct the
discretised 2-CGP. In order to ease the notation, we only describe the
dynamics of the height difference between the two piles, and define the
discretised 2-CGP on the state space $\mathbb{N}$. Note that it is possible to
recover the complete process from the evolution of the height difference.
Moreover, we shall often abuse notation and say that the discretised 2-CGP
starts from the state $(0,0)$, or that it comes back to a state in
$\mathcal{S}_{\square\\!\square}$ when it comes back to the state $0$.

######  Definition 5.10.

For all $\epsilon>0$ and $N\in\mathbb{N}\setminus\\{0,1\\}$, the discretised
2-CGP $(\hat{G}_{n}^{(N,\epsilon)})_{n\in\mathbb{N}}$ with time step
$\epsilon$ and maximal height difference $N$ is the $\llbracket
0,N\rrbracket$-valued discrete-time Markov chain with initial condition
$\hat{G}_{0}^{(N,\epsilon)}=0$ and whose transition probabilities
$(\hat{p}_{i,j}^{(N,\epsilon)})_{(i,j)\in\llbracket 0,N\rrbracket^{2}}$ are
defined as follows.

  1. 1.

If $i=0$, $\hat{p}_{0,0}^{(N,\epsilon)}=\exp(-2\epsilon)$,
$\hat{p}_{0,1}^{(N,\epsilon)}=1-\exp(-2\epsilon)$, and for all $j\in\llbracket
2,N\rrbracket$, $\hat{p}_{0,j}^{(N,\epsilon)}=0$.

  2. 2.

For all $i\in\llbracket 1,N-1\rrbracket$,

$\hat{p}_{i,j}^{(N,\epsilon)}:=\left\\{\begin{array}[]{ll}0&\mbox{if
}j>i+1,\\\ \frac{2}{i+2}(1-\exp(-(i+2)\epsilon))&\mbox{if }j=i-1,\\\
\exp(-(i+2)\epsilon)&\mbox{if }j=i,\\\
\frac{1}{i+2}(1-\exp(-(i+2)\epsilon))&\mbox{if }j=i+1\mbox{ or (if $i\neq 1$)
}0\leq j\leq i-2,\end{array}\right.$

  3. 3.

If $i=N$,

$\hat{p}_{N,j}^{(N,\epsilon)}:=\left\\{\begin{array}[]{ll}0&\mbox{if
}j>i+1,\\\ \frac{2}{N+2}(1-\exp(-(N+2)\epsilon))&\mbox{if }j=i-1,\\\
1-\frac{N+1}{N+2}(1-\exp(-(N+2)\epsilon))&\mbox{if }j=i,\\\
\frac{1}{N+2}(1-\exp(-(N+2)\epsilon))&\mbox{if }0\leq j\leq
i-2,\end{array}\right.$

This process has dynamics similar to those of the 2-CGP, except when the
height difference between the two piles is equal to $N$: the growth of the
highest pile in then blocked until the lower pile grows. This bound ensures we
can compute the invariant distribution of the process.

Before studying the properties of the discretised 2-CGP, we explain how to
couple it to the (continuous-time) 2-CGP. In order to do so, for all
$n\in\mathbb{N}$, let $t_{n}:=n\epsilon$. Moreover, let
$\hat{T}_{p}^{(\epsilon)}$ be the smallest positive integer such that
$(G_{t})_{t\geq 0}$ jumps at least twice in the time interval
$[t_{\hat{T}_{p}^{(\epsilon)}},t_{\hat{T}_{p}^{(\epsilon)}+1})$, and let
$\hat{T}_{N}^{(\epsilon)}$ be the smallest positive integer such that there
exists $t\in[t_{\hat{T}_{N}^{(\epsilon)}},t_{\hat{T}_{N}^{(\epsilon)}+1})$
such that $M_{t}\geq N$. We then construct the coupled discretised 2-CGP
$(\hat{G}_{n}^{N,\epsilon})_{n\in\mathbb{N}}$ as follows.

  1. 1.

First, we set $\hat{G}_{0}^{(N,\epsilon)}=0$.

  2. 2.

For all $n\in\llbracket
0,\min(\hat{T}_{N}^{(\epsilon)},\hat{T}_{p}^{(\epsilon)})-1\rrbracket$, we set

$\hat{G}_{n+1}^{(N,\epsilon)}=M_{t_{n+1}}-m_{t_{n+1}}.$

  3. 3.

If $\hat{T}_{p}^{(\epsilon)}\leq\hat{T}_{N}^{(\epsilon)}$,
$\hat{G}_{\hat{T}_{p}^{(\epsilon)}+1}^{(N,\epsilon)}$ is taken equal to the
value of $M_{t}-m_{t}$ after the first jump of $(G_{t})_{t\geq 0}$ over the
time interval $[t_{\hat{T}_{p}^{(\epsilon)}},t_{\hat{T}_{p}^{(\epsilon)}+1})$.
Otherwise, we set
$\hat{G}_{\hat{T}_{N}^{(\epsilon)}+1}^{(N,\epsilon)}=M_{t_{\hat{T}_{N}^{(\epsilon)}+1}}-m_{t_{\hat{T}_{N}^{(\epsilon)}+1}}$.

  4. 4.

For
$n>\min\left(\hat{T}_{p}^{(\epsilon)}+1,\hat{T}_{N}^{(\epsilon)}+1\right)$,
the coupling no longer holds, and
$(\hat{G}_{n}^{(N,\epsilon)})_{n\in\mathbb{N}}$ evolves according to the
dynamics described in Definition 5.10.

This coupling satisfies the following property.

######  Lemma 5.11.

Let $\epsilon>0$ and $N\in\mathbb{N}\setminus\\{0,1\\}$. The coupling of the
discretised 2-CGP $(\hat{G}_{n}^{(N,\epsilon)})_{n\in\mathbb{N}}$ with
timestep $\epsilon$ and maximal height difference $N$ to the original 2-CGP
$(G_{t})_{t\geq 0}$ holds until time
$\min\left(\hat{T}_{p}^{(\epsilon)},\hat{T}_{N}^{(\epsilon)}\right)$. In other
words, for all $n\in\mathbb{N}$, if
$n\leq\min\left(\hat{T}_{p}^{(\epsilon)},\hat{T}_{N}^{(\epsilon)}\right)$,
then

$\hat{G}_{n}^{(N,\epsilon)}=M_{t_{n}}-m_{t_{n}}.$

Moreover, let $\hat{T}_{\square}^{(N,\epsilon)}$ be defined as

$\hat{T}_{\square}^{(N,\epsilon)}:=\min\left\\{n\in\mathbb{N}\setminus\\{0\\}:\hat{G}_{n}^{(N,\epsilon)}=0\text{
but }\hat{G}_{n-1}^{(N,\epsilon)}\neq 0\right\\}.$

If
$\hat{T}_{\square}^{(N,\epsilon)}\leq\min\left(\hat{T}_{N}^{(\epsilon)},\hat{T}_{p}^{(\epsilon)}\right)$,
then
$\epsilon\hat{T}_{\square}^{(N,\epsilon)}-\epsilon<T_{\square}\leq\epsilon\hat{T}_{\square}^{(N,\epsilon)}$
a.s.

Notice that contrary to $T_{\square}$, the random variable
$\hat{T}_{\square}^{(N,\epsilon)}$ does not correspond exactly to the time of
first return of $\hat{G}_{n}^{(N,\epsilon)}$ to $0$, but rather to the time
needed for the process to exit state $0$ and then return to it. For instance,
if $\hat{G}_{1}^{(N,\epsilon)}=0$, then the time of first return to state $0$
of $(\hat{G}_{n}^{(N,\epsilon)})_{n\in\mathbb{N}}$ is equal to $1$, while
$\hat{T}_{\square}^{(N,\epsilon)}>1$.

###### Proof.

The first part of the lemma is a direct consequence of the coupling. Next,
assume that
$\hat{T}_{\square}^{(N,\epsilon)}\leq\min\left(\hat{T}_{N}^{(\epsilon)},\hat{T}_{p}^{(\epsilon)}\right)$.
By definition of $\hat{T}_{\square}^{(N,\epsilon)}$, we know that

$M_{t_{\hat{T}_{\square}^{(N,\epsilon)}}}-m_{t_{\hat{T}_{\square}^{(N,\epsilon)}}}=0,$
(5.8)

and for all $n\in\llbracket 0,\hat{T}_{\square}^{(N,\epsilon)}\rrbracket$,

$M_{t_{n}}-m_{t_{n}}\neq 0.$

Moreover, $(G_{t})_{t\geq 0}$ jumps at most once over each time interval
$[t_{n},t_{n+1})$, $n\in\llbracket
0,\hat{T}_{\square}^{(N,\epsilon)}-1\rrbracket$. Therefore, for all
$n\in\llbracket 0,\hat{T}_{\square}^{(N,\epsilon)}-1\rrbracket$ and
$t\in[t_{n},t_{n+1})$,

$M_{t}-m_{t}\in\left\\{M_{t_{n}}-m_{t_{n}},M_{t_{n+1}}-m_{t_{n+1}}\right\\},$

from which we deduce that

$\forall t\in\Big{[}0,t_{\hat{T}_{\square}^{(N,\epsilon)}-1}\Big{]},\quad
M_{t}-m_{t}\neq 0.$

Therefore,

$T_{\square}>t_{\hat{T}_{\square}^{(N,\epsilon)}-1}=\epsilon\left(\hat{T}_{\square}^{(N,\epsilon)}-1\right).$

Moreover, by (5.8), we have

$T_{\square}\leq
t_{\hat{T}_{\square}^{(N,\epsilon)}}=\epsilon\hat{T}_{\square}^{(N,\epsilon)},$

and we can conclude. ∎

The interest of the discretised 2-CGP lies in the fact that it is possible to
compute $\mathbf{E}_{(0,0)}[\hat{T}_{\square}^{(N,\epsilon)}]$ explicitly,
using the invariant distribution of the process. The quantity
$\epsilon\mathbf{E}_{(0,0)}[\hat{T}_{\square}^{(N,\epsilon)}]$ is a good
approximation for $\mathbf{E}_{(0,0)}[T_{\square}]$ when $\epsilon\to 0$ and
$N\to+\infty$, as stated in the following result.

######  Proposition 5.12.

We have

$\lim_{\begin{subarray}{c}\epsilon\to 0\\\ N\to+\infty\\\ N^{2}\epsilon\to
0\end{subarray}}\epsilon\mathbf{E}_{(0,0)}\left[\hat{T}_{\square}^{(N,\epsilon)}\right]=\mathbf{E}_{(0,0)}[T_{\square}].$

In order to show Proposition 5.12, we need two technical lemmas. The first
one, Lemma 5.13, states that for large $N$, with high probability, the 2-CGP
comes back to a state in $\mathcal{S}_{\square\\!\square}$ before reaching
height $N$. The second one, Lemma 5.14, states that before coming back to a
state in $\mathcal{S}_{\square\\!\square}$, under suitable conditions on
$\epsilon$ and $N$, with high probability, at most one growth event occurs in
each interval of length $\epsilon$. Notice that these two lemmas describe the
properties of the original (non-discretised) 2-CGP. We shall use them to show
that in the limiting regime, the coupling between the discretised 2-CGP and
the original process still holds at time $\hat{T}_{\square}^{(N,\epsilon)}$,
as whether the coupling breaks depends on the dynamics of the original 2-CGP.

######  Lemma 5.13.

For the non-discretised 2-CGP $(G_{t})_{t\geq 0}=(m_{t},M_{t})_{t\geq 0}$, for
all $\epsilon>0$ and $N\in\mathbb{N}\setminus\\{0,1\\}$, we have

$\mathbf{P}_{(0,0)}\left(T_{N}\leq
T_{\square}+\epsilon\right)\leq\frac{1}{2^{N-1}}\exp(\epsilon),$

where we recall that $T_{N}=\inf\left\\{t\geq 0:M_{t}\geq N\right\\}$.

###### Proof.

Let $\epsilon>0$ and $N\in\mathbb{N}\setminus\\{0,1\\}$. Adapting the proof of
Lemma 5.7, we obtain that

$\mathbf{P}_{(0,0)}\left(T_{N}\leq
T_{\square}+\epsilon\right)\leq\mathbf{P}_{(0,0)}\left(\mathcal{P}\left(T_{\to}+\epsilon\right)\geq
N-1\right),$

where $T_{\to}\sim Exp(1)$ and where $\mathcal{P}(\lambda)$, $\lambda>0$
stands for a Poisson random variable with parameter $\lambda$. Therefore,

$\displaystyle\mathbf{P}_{(0,0)}\left(T_{N}\leq T_{\square}+\epsilon\right)$
$\displaystyle\leq\int_{0}^{\infty}e^{-t}e^{-(t+\epsilon)}\,\left(\sum_{i=N-1}^{+\infty}\frac{(t+\epsilon)^{i}}{i!}\right)dt$
$\displaystyle=e^{-\epsilon}\left[\sum_{i=N-1}^{+\infty}\int_{0}^{\infty}e^{-2t}\,\frac{(t+\epsilon)^{i}}{i!}dt\right].$

Moreover, for all $i\geq N-1$,

$\displaystyle\int_{0}^{\infty}e^{-2t}\,\frac{(t+\epsilon)^{i}}{i!}dt$
$\displaystyle=\left[-\frac{1}{2}e^{-2t}\,\frac{(t+\epsilon)^{i}}{i!}\right]_{0}^{\infty}+\int_{0}^{\infty}\frac{1}{2}e^{-2t}\,\frac{(t+\epsilon)^{i-1}}{(i-1)!}dt$
$\displaystyle=\frac{1}{2}\frac{\epsilon^{i}}{i!}+\frac{1}{2}\int_{0}^{\infty}e^{-2t}\,\frac{(t+\epsilon)^{i-1}}{(i-1)!}dt$
$\displaystyle=\sum_{j=0}^{i}\frac{1}{2^{j+1}}\frac{\epsilon^{i-j}}{(i-j)!}$
$\displaystyle=\frac{1}{2^{i+1}}\left(\sum_{j=0}^{i}\frac{(2\epsilon)^{i-j}}{(i-j)!}\right)$
$\displaystyle\leq\frac{1}{2^{i+1}}\exp(2\epsilon).$

Therefore,

$\displaystyle\mathbf{P}_{(0,0)}(T_{N}\leq T_{\square}+\epsilon)$
$\displaystyle\leq\sum_{i=N-1}^{+\infty}\frac{1}{2^{i+1}}e^{2\epsilon}e^{-\epsilon}\leq
e^{\epsilon}\frac{1}{2^{N-1}},$

which concludes the proof. ∎

######  Lemma 5.14.

In the notation of Lemma 5.13, we have

$\mathbf{P}_{(0,0)}\left(\left.\hat{T}_{p}^{(\epsilon)}<\frac{T_{\square}}{\epsilon}\right|T_{N}>T_{\square}+\epsilon\right)\leq
2\left(1-\exp(-(N+2)\epsilon))\right)+\frac{1}{1-\exp(-\epsilon)}\left(1-\exp(-(N+2)\epsilon))\right)^{2}.$

###### Proof.

Consider the time intervals $[t_{0},t_{1})$, $[t_{1},t_{2})$, …, $[t_{\lfloor
T_{1}\epsilon^{-1}\rfloor},t_{\lfloor T_{1}\epsilon^{-1}\rfloor+1})$, …,
$[t_{\lfloor T_{\square}\epsilon^{-1}\rfloor},t_{\lfloor
T_{\square}\epsilon^{-1}\rfloor+1})$. We work conditionally on
$T_{N}>T_{\square}+\epsilon$. In order to have
$\hat{T}_{p}^{(\epsilon)}<T_{\square}\epsilon^{-1}$, one of the following
events needs to occur.

  1. 1.

Another growth event occurs in the time interval $[t_{\lfloor
T_{1}\epsilon^{-1}\rfloor},t_{\lfloor T_{1}\epsilon^{-1}\rfloor+1})$.

This occurs with probability bounded from above by $1-\exp(-(N+2)\epsilon)$.

  2. 2.

At least two growth events occur in at least one of the time intervals
$[t_{\lfloor T_{1}\epsilon^{-1}\rfloor+1},t_{\lfloor
T_{1}\epsilon^{-1}\rfloor+2})$, …, $[t_{\lfloor
T_{\square}\epsilon^{-1}\rfloor-1},t_{\lfloor
T_{\square}\epsilon^{-1}\rfloor})$.

For each time interval, this occurs with probability bounded from above by
$(1-\exp(-(N+2)\epsilon))^{2}$.

  3. 3.

Another growth event occurs in the time interval $[t_{\lfloor
T_{\square}\epsilon^{-1}\rfloor},t_{\lfloor
T_{\square}\epsilon^{-1}\rfloor+1})$.

Again, this occurs with probability bounded from above by
$1-\exp(-(N+2)\epsilon)$.

Moreover, if only one growth event occurs during the time interval
$[t_{\lfloor T_{1}\epsilon^{-1}\rfloor},t_{\lfloor
T_{1}\epsilon^{-1}\rfloor+1})$, then the random variable $\lfloor
T_{\square}\epsilon^{-1}\rfloor-\lfloor T_{1}\epsilon^{-1}\rfloor$ is bounded
from above by a geometric law with success probability $1-\exp(-\epsilon)$.
Therefore,

$\displaystyle\mathbf{P}_{(0,0)}\left(\left.\hat{T}_{p}^{(\epsilon)}<\frac{T_{\square}}{\epsilon}\right|T_{N}>T_{\square}+\epsilon\right)$
$\displaystyle\leq
2(1-\exp(-(N+2)\epsilon))+\sum_{k=2}^{+\infty}k\exp(-\epsilon)^{k-1}(1-\exp(-\epsilon))(1-\exp(-(N+2)\epsilon))^{2}$
$\displaystyle\leq
2\left(1-\exp(-(N+2)\epsilon))\right)+\frac{1}{1-\exp(-\epsilon)}\left(1-\exp(-(N+2)dt))\right)^{2},$

and we can conclude. ∎

We can now show Proposition 5.12.

###### Proof.

(Proposition 5.12) By Lemma 5.11, if
$T_{\square}\leq\min(\hat{T}_{N}^{(\epsilon)},\epsilon\hat{T}_{p}^{\epsilon})$,
then

$\epsilon\hat{T}_{\square}^{(N,\epsilon)}-\epsilon<T_{\square}\leq\epsilon\hat{T}_{\square}^{(N,\epsilon)}.$

In particular, this is true if $T_{N}-\epsilon>T_{\square}$ and
$\hat{T}_{p}^{(\epsilon)}\geq T_{\square}\epsilon^{-1}$. Then, by case
disjunction,

$\displaystyle\mathbf{E}_{(0,0)}[T_{\square}]=$
$\displaystyle\mathbf{E}_{(0,0)}\left[T_{\square}\mathds{1}_{\\{T_{N}<T_{\square}+\epsilon\\}}\right]+\mathbf{E}_{(0,0)}\left[T_{\square}\mathds{1}_{\\{T_{N}>T_{\square}+\epsilon\\}}\mathds{1}_{\\{\hat{T}_{p}{(\epsilon)}<T_{\square}\epsilon^{-1}\\}}\right]$
$\displaystyle+\mathbf{E}_{(0,0)}\left[T_{\square}\mathds{1}_{\\{T_{N}>T_{\square}+\epsilon\\}}\mathds{1}_{\\{\hat{T}_{p}{(\epsilon)}\geq
T_{\square}\epsilon^{-1}\\}}\right],$

and similarly,

$\displaystyle\mathbf{E}_{(0,0)}\left[\hat{T}_{\square}^{(N,\epsilon)}\right]=$
$\displaystyle\mathbf{E}_{(0,0)}\left[\hat{T}_{\square}^{(N,\epsilon)}\mathds{1}_{\\{T_{N}<T_{\square}+\epsilon\\}}\right]+\mathbf{E}_{(0,0)}\left[\hat{T}_{\square}^{(N,\epsilon)}\mathds{1}_{\\{T_{N}>T_{\square}+\epsilon\\}}\mathds{1}_{\\{\hat{T}_{p}{(\epsilon)}<T_{\square}\epsilon^{-1}\\}}\right]$
$\displaystyle+\mathbf{E}_{(0,0)}\left[\hat{T}_{\square}^{(N,\epsilon)}\mathds{1}_{\\{T_{N}>T_{\square}+\epsilon\\}}\mathds{1}_{\\{\hat{T}_{p}{(\epsilon)}\geq
T_{\square}\epsilon^{-1}\\}}\right].$

Therefore,

$\displaystyle\epsilon\mathbf{E}_{(0,0)}\Big{[}$