theorem]Proposition

# The scaling limit of the root component in the Wired Minimal Spanning Forest
of the Poisson Weighted Infinite Tree

Omer Angel and Delphin Sénizergues Department of Mathematics, University of
British Columbia<EMAIL_ADDRESS>UPL, Univ. Paris Nanterre, CNRS,
F92000 Nanterre France<EMAIL_ADDRESS>
###### Abstract

In this paper we prove a scaling limit result for the component of the root in
the Wired Minimal Spanning Forest (WMSF) of the Poisson-Weighted Infinite Tree
(PWIT), where the latter tree arises as the local weak limit of the Minimal
Spanning Tree (MST) on the complete graph endowed with i.i.d. weights on its
edges. The limiting object $\mathcal{M}^{\infty}$ can be obtained by
aggregating independent Brownian trees using two types of gluing procedures:
one that we call the _Brownian tree aggregation process_ and resembles the so-
called _stick-breaking construction of the Brownian tree_ ; and another one
that we call the _chain_ construction, which simply corresponds to gluing a
sequence of metric spaces along a line.

##  1 Introduction

Let $(V,E,w)$ be a finite, connected, weighted graph, where $(V,E)$ is the
underlying graph and
$w:E\rightarrow\mathopen{[}0\mathclose{}\mathpunct{},\infty\mathclose{)}$ is
the weight function. A spanning tree of $(V,E)$ is a tree that is a subgraph
of $(V,E)$ with vertex set $V$. A minimal spanning tree (MST) $T$ of $(V,E,w)$
satisfies

$\displaystyle\sum_{e\in T}w(e)=\min\left\\{\sum_{e\in
T^{\prime}}w(e)\mathrel{}\middle|\mathrel{}T^{\prime}\text{ is a spanning tree
of }(V,E)\right\\}.$

This combinatorial optimization problem has been extensively studied in
computer science, combinatorics and probability. We refer to [3, Section 1.1]
for a overview of the probabilistic results related to the MST problem.

In this paper we place ourselves in the following random setting: we consider
$K_{n}$ the complete graph on $n$ vertices and assign i.i.d. weights to the
edges distributed as uniform random variables in
$\mathopen{[}0\mathclose{}\mathpunct{},1\mathclose{]}$. In that case the edge-
weights are almost surely distinct so there almost surely exist a unique MST,
which we call $\mathrm{M}_{n}$. We are interested in some asymptotic
properties of $\mathrm{M}_{n}$ as $n\rightarrow\infty$ and, in particular, in
different ways of defining a notion of limit for this object.

##### The scaling limit of $\mathrm{M}_{n}$.

For a measured metric space $(X,d,\mu)$, we denote by
$\operatorname{Scale}(a,b;X)$ the space $X$ with distances scaled by $a$ and
measure scaled by $b$. In [3], the authors consider $\mathrm{M}_{n}$ as random
measured metric space, by endowing its set of vertices with the graph distance
and the counting measure. They obtain the following scaling limit convergence.
We shall recall the definition of the Gromov–Hausdorff–Prokhorov topology
later in Section 2.1. [Theorem 1.1 of [3]] Seen as a measured metric space, we
have the following convergence in the Gromov–Hausdorff–Prokhorov topology

$\displaystyle\mathrm{Scale}(n^{-\frac{1}{3}},n^{-1};\mathrm{M}_{n})\underset{n\rightarrow\infty}{\rightarrow}\mathcal{M}^{\mathrm{comp}},$

where the limiting object $\mathcal{M}^{\mathrm{comp}}$ is a compact random
tree that has Minkowski dimension equal to $3$ almost surely. The limiting
tree appearing in the statement of the theorem is not constructed in an
explicit way in [3] and remains quite mysterious. This object is believed to
be universal in the sense that it is expected to arise as the scaling limit of
the MST of random graphs that exhibit mean field behavior when their edges are
endowed with i.i.d. weights: random regular graphs, random graphs with given
degree sequence, inhomogeneous graphs, high dimensional hypercubes and more.
One step in this direction is the work [6] of Addario-Berry and Sen where they
rigorously prove such a result for the uniform $3$-regular graph. In the
recent preprint [12], Broutin and Marckert present an explicit construction of
$\mathcal{M}^{\mathrm{comp}}$ as the _convex minorant tree_ obtained from a
trajectory of Brownian motion with parabolic drift, together with a sequence
of independent uniform random variable $(U_{i})_{i\geq 1}$ on
$\mathopen{(}0\mathclose{}\mathpunct{},1\mathclose{)}$, and refer to the
object $\mathcal{M}^{\mathrm{comp}}$ as the _parabolic Brownian tree_. This
new construction allows them to get more insight on the structure of the
object, and in particular, it allows them to prove that the Hausdorff
dimension of the object is $3$.

##### The local weak limit.

Another way of considering a limit of the object $M_{n}$ is to look at what
happens locally around a random vertex. Some general theorem of Aldous and
Steele [9] ensures that for the appropriate _weighted_ local topology (one
that accounts for the weights of the edges as well as the structure of the
graph), if the underlying weighted graph converges locally in that topology
towards a limit, then the corresponding MST converges as well towards the
Wired Minimal Spanning Forest (WMSF) of the limiting graph. In the case of
$M_{n}$, the underlying graph $K_{n}$ endowed with appropriate i.i.d. weights
on the edges is known to converge to the Poisson-Weighted Infinite Tree (PWIT)
and so the weighted local limit of $M_{n}$ is the WMSF of the PWIT.

In [2], the same convergence is proved in the specific case of the object
$M_{n}$ and the structure of the limit is investigated in more depth. Last, in
the recent paper [25], Nachmias and Pang prove that under some mild
conditions, which apply in our case, this convergence in the weighted local
topology transfers to a convergence in the usual local topology when one
forgets the weights. Note that in this case the limiting object is only a tree
instead of being a forest: any vertex that is not in the root component of the
WMSF is at distance infinity from the root when using the usual graph
distance. [[9, 2, 25]] We have the convergence

$\displaystyle\mathrm{M}_{n}\underset{n\rightarrow\infty}{\rightarrow}\mathrm{M}^{\infty},$

in the local weak topology, where $\mathrm{M}^{\infty}$ is the root component
of the WMSF of the PWIT. The limiting tree is shown in [2] to be one-ended and
having roughly cubic volume growth. These results are refined in [25], where
the authors also prove results about the random walk on such a graph; in
particular they prove that the spectral dimension and typical displacement
exponent are almost surely respectively $\frac{3}{2}$ and $\frac{1}{4}$.

##### Towards a non-compact scaling limit.

The goal of this paper is to introduce $\mathcal{M}^{\infty}$, a continuous
non-compact version of the object $\mathrm{M}_{n}$ the MST of $K_{n}$. This
should be reminiscent of the case of the uniform spanning tree $T_{n}$ of
$K_{n}$, which is simply a uniform labeled tree on $n$ vertices. It is well-
known in that case that three limiting objects can be associated to $T_{n}$:
its scaling limit called the _Brownian CRT_ (after rescaling the distances by
$n^{1/2}$), its local weak limit which is an infinite discrete tree called the
_Kesten tree_ ; and its non-compact scaling limit called the _infinite
Brownian CRT_. This non-compact limit arises as several different limits: the
scaling limit of the Kesten tree, the local limit of the Brownian CRT (i.e.
what we see in the CRT when we zoom in around a random point), and as the
scaling limit of $T_{n}$ when the distances are rescaled by a quantity
$o(n^{1/2})$.

In our case, the goal is to introduce the corresponding continuous non-compact
object $\mathcal{M}^{\infty}$ for $M_{n}$ and prove one of these convergences,
namely the analogue of the first one, i.e. that $\mathcal{M}^{\infty}$ is the
scaling limit of $\mathrm{M}^{\infty}$, see Figure 1. The two other
convergences will be studied in a separate paper.

Figure 1: The objects $M_{n}$, $\mathcal{M}^{\mathrm{comp}}$,
$\mathrm{M}^{\infty}$ and $\mathcal{M}^{\infty}$ and their relationships. This
paper focuses on the red arrow. The solid black arrows are already proved in
the literature, and the dashed ones are conjectured.

###  1.1 Structure of the root component of the WMSF of the PWIT

(a) Decomposition of the invasion percolation tree $\mathrm{T}^{\infty}$ into
ponds that are linked along an infinite spine by forward maximal edges, drawn
in dotted line. The edges in the dark red pond
$\mathrm{T}_{\alpha_{2}}^{\bullet}$ all have weight smaller than $\alpha_{2}$,
the forward maximal edge that exits it. (b) Decomposition of the invasion
percolation tree $T(u)$ into ponds. The invasion percolation starting from $u$
explores the orange pond first, then the purple one and the dark red one. The
rest of the invasion coincides with that of the root from there.

Figure 2: Structure of invasion percolation clusters on the PWIT

First we describe the objects of study. Let $\mathbb{U}$ be the Ulam-Harris
tree; this is the tree with vertex set $V(\mathbb{U})=\bigcup_{k\geq
0}\mathbb{N}^{k}$, and for each $k\geq 1$ and each vertex
$v=(n_{1},\dots,n_{k})\in\mathbb{N}^{k}$ , an edge between $v$ and its parent
$(n_{1},\dots,n_{k-1})$. For each $v=(n_{1},\dots,n_{k})\in\mathbb{U}$ , let
$(x_{i},i\geq 1)$ be the atoms of a homogeneous rate one Poisson process on
$\mathopen{[}0\mathclose{}\mathpunct{},\infty\mathclose{)}$, and for each
$i\geq 1$ give the edge from $v$ to its child $v_{i}=(n_{1},\dots,n_{k},i)$
the weight $W({v,v_{i}})=x$. Writing $W=\\{W(e),e\in E(\mathbb{U})\\}$, where
$E(\mathbb{U})$ is the edge set of $\mathbb{U}$ , the Poisson-weighted
infinite tree is the tree $\mathbb{U}$ rooted at $\emptyset$, endowed with the
weight function $W$, which we write as a triple $(\mathbb{U},\emptyset,W)$.

##### Invasion percolation, forward maximal edges and ponds.

Invasion percolation (also called Prim’s algorithm) on a rooted locally finite
weighted graph with distinct weights is defined as follows: the algorithm
grows a component from the root of the graph, adding at each step the lowest
weight edge leaving the current component. When performed on a finite
connected weighted graph, this procedure yields the minimal spanning tree of
that graph. Performed on an infinite graph however, it yields an infinite tree
whose set of vertices is a proper subset of the original graph, which we call
the invasion percolation cluster of the root.

We consider $\mathrm{T}^{\infty}$ the invasion cluster of the root of the
weighted rooted tree $(\mathbb{U},\emptyset,W)$. It is known that
$\mathrm{T}^{\infty}$ is almost surely one-ended and that it admits a
decomposition into ponds, by cutting at so-called _forward maximal edges_. We
say that an edge $e\in E(\mathrm{T}^{\infty})$ is forward maximal if it
disconnects the root $\emptyset$ from infinity and is such that if there
exists $e^{\prime}\in E(\mathrm{T}^{\infty})$ such that $e$ is in the path
joining $\emptyset$ and $e^{\prime}$ then $W(e^{\prime})<W(e)$. In terms of
the invasion percolation process described above, this means that any edge
explored after $e$ will have a smaller weight than $e$.

Denote by $\alpha_{1}>\alpha_{2}>\dots$ the respective weights of the
successive forward maximal edges $e_{1},e_{2},\dots$ in the order in which
they are discovered by the invasion percolation process, and denote by
$\mathscr{A}=\\{\alpha_{1},\alpha_{2},\dots\\}$ the set of all their values.
We can remove those edges in order to disconnect $\mathrm{T}^{\infty}$ into
finite connected components that we call _ponds_. We denote those ponds by
$(\mathrm{T}^{\bullet}_{\alpha})_{\alpha\in\mathscr{A}}$, by indexing them by
the weight of the corresponding forward maximal edge exiting the component,
which we call the _type_ (called _activation time_ in [25]) of the vertices
belonging to the corresponding pond. Here, each
$\mathrm{T}^{\bullet}_{\alpha}$ is then a finite rooted tree, rooted at its
vertex closest to the root and pointed at the vertex from which the next
forward maximal edge exits. See Figure 2 for an illustration.

##### Invasion percolation from every vertex.

For any vertex $u\in V(\mathbb{U})$, we can repeat the procedure above _from
vertex $u$_ by just considering $T(u)$ the invasion percolation cluster of the
rooted weighted tree $\mathbb{U}(u)=(U,u,W)$, the PWIT re-rooted at vertex
$u$. Let $\widetilde{\mathrm{M}}$ be the union of all the trees $(T(u),u\in
V(\mathbb{U}))$. The forest $\widetilde{\mathrm{M}}$ is the so-called Wired
Minimal Spanning Forest of the PWIT. Now the tree $\mathrm{M}^{\infty}$, which
we study in this paper, is defined as the connected component of the forest
$\widetilde{\mathrm{M}}$ containing the root $\emptyset$ of $\mathbb{U}$. It
is almost surely one-ended, see [2, Corollary 7.2].

The tree $\mathrm{M}^{\infty}$ of course contains
$\mathrm{T}^{\infty}=T(\emptyset)$, the invasion percolation cluster of the
root. Consider a vertex $u\in V(\mathbb{U})$ that is such that
$u\notin\mathrm{T}^{\infty}$ and its invasion percolation cluster $T(u)$. The
tree $T(u)$ can be disconnected along its own forward maximal edges in the
same way as $T(\emptyset)$. Now if we suppose that $u\in\mathrm{M}^{\infty}$
then the facts that $T(\emptyset)$ and $T(u)$ are one-ended, as well as
$\mathrm{M}^{\infty}$, imply that the ponds and forward maximal edges of
$T(\emptyset)$ and $T(u)$ coincide except for a finite number of them: this
ensures that the invasion percolation process from vertex $u$ visits a finite
number of ponds before starting to explore one of the ponds
$(\mathrm{T}^{\bullet}_{\alpha})_{\alpha\in\mathscr{A}}$. In order to avoid
confusion, for $u\in\mathrm{M}^{\infty}$, we call any of the forward maximal
edges of $T(u)$ that is not a forward maximal edge of $T(\emptyset)$ a _back-
edge_. Any pond of $T(u)$ that does not overlap with $\mathrm{T}^{\infty}$ is
denoted by $T_{\beta}$, where $\beta$ is the weight of the first back-edge
visited by invasion percolation started from that vertex. Again we say that
all the vertices in $T_{\beta}$ have type $\beta$.

Now, consider $\mathrm{M}^{\infty}$ and as before we disconnect it along the
forward maximal edges $e_{1},e_{2}\dots$ of $\mathrm{T}^{\infty}$. We denote
by $(\mathrm{M}^{\bullet}_{\alpha})_{\alpha\in\mathscr{A}}$ the obtained
sequence of connected components. Each of those component
$\mathrm{M}^{\bullet}_{\alpha}$ can actually be decomposed again into the
union of $\mathrm{T}^{\bullet}_{\alpha}$ and a set of trees $T_{\beta}$
connected to each other along a tree structure by back-edges. In fact,
$\mathrm{M}^{\bullet}_{\alpha}$ can be obtained from
$\mathrm{T}^{\bullet}_{\alpha}$ by running the so-called Poisson–Galton–Watson
Aggregation Process which we recall in Section 3. The idea of this paper is to
define a similar structure in the continuum and prove that it is indeed the
scaling limit of $\mathrm{M}^{\infty}$, resp. $\mathrm{T}^{\infty}$.

###  1.2 Summary of the results

Figure 3: Construction of the non-compact continuous object
$\mathcal{M}^{\infty}$ (on the bottom) from a Poisson point process
$\mathscr{P}$ (on top). The color of the trees below correspond to the atoms
of the PPP above. Note that in $\mathscr{P}$ there is an accumulation of atoms
near $\infty$ (not represented here).

We define an infinite one-ended continuum random tree $\mathcal{M}^{\infty}$
jointly with $\mathcal{T}^{\infty}\subset\mathcal{M}^{\infty}$, which are the
continuous analogues of respectively $\mathrm{M}^{\infty}$ and
$\mathrm{T}^{\infty}$.

##### Joint construction of $\mathcal{T}^{\infty}$ and $\mathcal{M}^{\infty}$.

We first explain how we can describe the object $\mathcal{M}^{\infty}$ from
another object that we denote by $\mathcal{M}^{\bullet}$, which we then
discuss below. This is slightly analogous to the construction of Brownian
motion by chaining of excursions sampled from the Ito measure, though in our
construction each component has another one immediately following it (among
other differences). Let

  * •

$\mathscr{P}$ be a Poisson Point Process on
$\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$ with intensity
$\frac{1}{t}\mathrm{d}t$,

  * •

$\left((\mathcal{T}^{\bullet}_{x},\mathcal{M}^{\bullet}_{x}),\
x\in\mathscr{P}\right)$ a collection of random variables, where conditionally
on $\mathscr{P}$ they are independent and we have

$\displaystyle(\mathcal{T}^{\bullet}_{x},\mathcal{M}^{\bullet}_{x})\overset{(d)}{=}\left(\mathrm{Scale}(x^{-1},x^{-2};\mathcal{T}^{\bullet}),\
\mathrm{Scale}(x^{-1},x^{-3};\mathcal{M}^{\bullet})\right),$

where $(\mathcal{T}^{\bullet},\mathcal{M}^{\bullet})$ is a couple of random
pointed rooted measured trees such that
$\mathcal{T}^{\bullet}\subset\mathcal{M}^{\bullet}$ that we will describe in
more details later on.

Then $\mathcal{M}^{\infty}$ is constructed from the sequence
$(\mathcal{M}_{x}^{\bullet})_{x\in\mathscr{P}}$ as

$\displaystyle\mathrm{Chain}(\mathcal{M}_{x}^{\bullet},\ x\in\mathscr{P}).$

The tree $\mathcal{T}^{\infty}$ is constructed in a similar manner from
$(\mathcal{T}_{x}^{\bullet})_{x\in\mathscr{P}}$ and can be seen as a subset of
$\mathcal{M}^{\infty}$. The $\mathrm{Chain}$ construction will be defined in
detail in Section 2, but essentially consists in taking all the sequence of
rooted pointed trees $(\mathcal{M}_{x}^{\bullet})_{x\in\mathscr{P}}$ and
gluing them head to tail in decreasing order of $x$-value. The resulting space
is completed by adding an extra point $\rho_{\infty}$ which is characterized
by $\rho_{\infty}:=\lim_{x\rightarrow\infty,x\in\mathscr{P}}\rho_{x}$, where
for any $x\in\mathscr{P}$, the point $\rho_{x}$ denotes the common root of
$\mathcal{T}^{\bullet}_{x}$ and $\mathcal{M}^{\bullet}_{x}$. The obtained
space is an unbounded complete measured tree, rooted at $\rho_{\infty}$.

Notice the similar structure to that of
$(\mathrm{T}^{\infty},\mathrm{M}^{\infty})$: the point process $\mathscr{P}$
corresponds to the set $\mathscr{A}$ of weights of forward maximal edges, the
trees $(\mathcal{T}^{\bullet}_{x},\mathcal{M}^{\bullet}_{x})$ for
$x\in\mathscr{P}$ take the place of the
$(\mathrm{T}_{\alpha}^{\bullet},\mathrm{M}_{\alpha}^{\bullet})$ for
$\alpha\in\mathscr{A}$, and the Chain construction mimics the way that we can
get $\mathrm{M}^{\infty}$ and $\mathrm{T}^{\infty}$ from linking all the
$(\mathrm{M}_{\alpha}^{\bullet})_{\alpha\in\mathscr{A}}$ (resp. the
$(\mathrm{T}_{\alpha}^{\bullet})_{\alpha\in\mathscr{A}}$) using the forward
maximal edges.

##### Convergence results.

Before discussing the construction of
$(\mathcal{T}^{\bullet},\mathcal{M}^{\bullet})$, which are the building blocks
of $\mathcal{T}^{\infty}$ and $\mathcal{M}^{\infty}$, we state our scaling
limit convergence result. The result below is expressed in the local
Gromov–Hausdorff–Prokhorov topology, the definition of which is recalled in
Section 2. The tree $\mathcal{M}^{\infty}$ is the scaling limit of
$\mathrm{M}^{\infty}$ for the local Gromov-Hausdorff-Prokhorov topology. More
precisely

$\displaystyle\mathrm{Scale}\left(r,\frac{r^{3}}{C_{0}};\mathrm{M}^{\infty}\right)\underset{r\rightarrow
0}{\rightarrow}\mathcal{M}^{\infty},$

where the constant $C_{0}$ is defined later in (47). The above convergence
happens jointly with

$\displaystyle\mathrm{Scale}\left(r,r^{2};\mathrm{T}^{\infty}\right)\underset{r\rightarrow
0}{\rightarrow}\mathcal{T}^{\infty}.$

Note that invasion percolation on a $d$-regular tree and its scaling limit
were investigated in [11]. While the behavior does not depend significantly on
$d$, the extension to the PWIT was only investigated later in [5]. In [11],
the tree generated by the invasion percolation process started at the root is
studied using its encoding via functions such as the contour function, height
function or Lukasiewicz path. The scaling limit is then described as well by
its encoding by a height function that is described as the solution to some
stochastic differential equation. Heuristically, the obtained tree should have
the same distribution as $\mathcal{T}^{\infty}$, up to some deterministic
scaling, even though our description can seem quite different as first glance.

##### Construction of $\mathcal{M}^{\bullet}$ from $\mathcal{T}^{\bullet}$:
the Brownian tree aggregation process.

We now describe in more details the distribution of the pair
$(\mathcal{T}^{\bullet},\mathcal{M}^{\bullet})$ used to construct
$\mathcal{T}^{\infty}$ and $\mathcal{M}^{\infty}$. They are seen as (nested)
random metric spaces, endowed with a common root $\rho$ and distinguished
point $v$, and measures (one for each) that are not necessarily normalized to
be probability measures.

First, $\mathcal{T}^{\bullet}$ has the distribution of a Brownian tree with
random mass $X$, where $X\overset{(d)}{=}\sqrt{2}\cdot\Gamma$ where $\Gamma$
has distribution $\mathrm{Gamma}(\frac{1}{2})$, endowed with a randomly chosen
point $v$ (on top of being rooted at some point $\rho$). The construction of
$\mathcal{M}^{\bullet}$ from $\mathcal{T}^{\bullet}$ then uses a process that
we call the _Brownian tree aggregation process_ (which is analogous to the
Poisson-Galton-Watson aggregation process used in the discrete setting to
describe the construction of $\mathrm{M}_{\alpha}^{\bullet}$ for any
$\alpha>1$). A useful analogy for the reader is the stick-breaking
construction of the CRT, where line segments of random lengths are glued at
uniform points on the union of all past segments. However, unlike that
process, there are infinitely many attached trees being added in any time
interval.

Here is a first intuitive way of understanding the construction of
$\mathcal{M}^{\bullet}$. This is a growth process for a tree, started at time
$t=1$ (which turns out to be more convenient) with the _seed_
$\mathcal{T}^{\bullet}$. Immediately after time $t=1$, independent Brownian
trees arrive and are attached to the existing tree by having their root
identified to a point already present. At time $t$, attachments of trees of
mass $y$ to an element of mass $\mathrm{d}m$ of the current structure arrive
at rate

$\frac{1}{\sqrt{2\pi}y^{3/2}}\exp\left(-\frac{t^{2}}{2}y\right)\mathrm{d}y\mathrm{d}m,$
(1)

where each corresponding tree of mass $y$ is a Brownian tree with the given
mass (conditionally on all the rest). The object $\mathcal{M}^{\bullet}$ is
the limit as $t\to\infty$ of the process defined above, still rooted and
pointed at respectively $\rho$ and $v$.

The way that we define such a process formally deviates slightly from this
intuition, and start with the process giving the total mass of the tree at
time $t$. We would like to define a pure jump Markov process
$(\mathcal{W}_{t})_{t\geq 1}$ in terms of its generator. Since this Markov
process is inhomogeneous in time, we actually give the generator of the
process $((t,\mathcal{W}_{t}))_{t\geq 1}$, which is time-homogeneous. The
generator $\mathbf{A}$ of the process $(t,\mathcal{W}_{t})$ is so that for any
smooth and compactly supported function
$f:(0,\infty)\times\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}\rightarrow\mathbb{R}$
we have

$\displaystyle\mathbf{A}f(t,x)=\partial_{t}f(t,x)+x\cdot\int_{0}^{\infty}\left(f(t,x+y)-f(t,x)\right)\frac{\exp\left(-\frac{t^{2}}{2}y\right)}{y^{\frac{3}{2}}\sqrt{2\pi}}\mathrm{d}y.$
(2)

This process represent the evolution of the total mass that is present at time
$t$. The process $(\mathcal{W}_{t})_{t\geq 1}$ is a subordinator and a pure
jump process, and so has countably many jumps. For every jump time $t$ of the
process, take an independent Brownian tree

$\displaystyle\mathcal{T}_{t}$

with mass given by the size of the jump
$w_{t}:=\mathcal{W}_{t}-\mathcal{W}_{t-}$. For every such times $t$, a point
$X_{t}$ is sampled proportionally to the mass measure on

$\displaystyle\mathcal{T}^{\bullet}\sqcup\bigsqcup_{1<s<t}\mathcal{T}_{t}.$
(3)

The tree at time $t$ is obtained by gluing the root of every
$\mathcal{T}_{s}$, for all jump times $s\leq t$ of the process $\mathcal{W}$,
to the corresponding point $X_{s}$. For $t=\infty$, we take the completion of
the obtained space. The fact that this construction makes sense and almost
surely yields a compact metric space is proved in a more general setting in
the corresponding section.

We should also mention how the measure on $\mathcal{M}^{\bullet}$ is obtained:
The total mass of the tree constructed up to time $t$ is $\mathcal{W}_{t}$ and
tends to infinity. The measure on $\mathcal{M}^{\bullet}$ arises as the
scaling limit as $t\to\infty$ of the mass-measure on the trees that have index
less than $t$. As it turns out, the correct normalization is $\frac{1}{t}$.
The measure $\mu$ on $\mathcal{M}^{\bullet}$ is defined as
$\mu:=\lim_{t\rightarrow\infty}\frac{1}{t}\mu_{t}$, where $\mu_{t}$ is the
measure on the tree at time $t$, and the limit is a weak limit of measures.
The limiting measure on $\mathcal{M}^{\bullet}$ has total mass given by

$\displaystyle Y:=\lim_{t\to\infty}\frac{\mathcal{W}_{t}}{t}.$

More about that in Section 3.

Thanks to the construction of $\mathcal{M}^{\bullet}$ as an aggregation
process, we can determine some of its geometric properties. The random metric
space $\mathcal{M}^{\bullet}$ is well-defined as a random variable in
$\mathbb{L}^{\bullet,c}$. Furthermore

  1. (i)

Its mass and diameter have moments of all orders.

  2. (ii)

It almost surely has Minkowski and Hausdorff dimension $3$.

  3. (iii)

Its measure is almost surely non-zero, diffuse and has full support.

Note that, as a result, the non-compact random metric space
$\mathcal{M}^{\infty}$ which is a countable union of copies of
$\mathcal{M}^{\bullet}$ also has Hausdorff dimension $3$ and diffuse mass
measure with full support.

###  1.3 Organization of this paper

We start in Section 2 by recalling some facts about the
Gromov–Hausdorff–Prokhorov topology and introducing the two constructions
called $\mathrm{Chain}$ and aggregation process ($\mathrm{Aggreg}$)
respectively. These constructions are particularly important because we
interpret all our objects in this framework. In particular, we state some
sufficient conditions for the convergence of such structures. In Section 3 we
then use these constructions to rigorously construct the trees
$\mathcal{M}^{\bullet}$ and $\mathcal{M}^{\infty}$. In particular, we prove
Theorem 1.2 in Section 3.1. Then in Section 4, we study the tree
$\mathrm{M}^{\infty}$ and decompose it in a way that use again the
$\mathrm{Chain}$ and $\mathrm{Aggreg}$ constructions in an analogous way. We
then prove Theorem 1.2 in Section 4.8 by using results from Section 2 that
ensure that the limit of objects constructed as a $\mathrm{Chain}$ (resp.
$\mathrm{Aggreg}$) of discrete objects can be described as a $\mathrm{Chain}$
(resp. $\mathrm{Aggreg}$) of continuous ones.

Table 1: Table of notation of the main objects of the paper Discrete objects
---
$\mathscr{A}$ | : |  set of weights of forward maximal edges of $\mathrm{T}^{\infty}$
$\mathrm{T}^{\infty}$ | : |  invasion percolation cluster of the root of the PWIT
$\mathrm{T}_{\alpha}^{\bullet}$ | : |  defined for $\alpha\in\mathscr{A}$; it is the pond located just before the forward maximal edge of weight $\alpha$
$M_{n}$ | : |  minimal spanning tree of $K_{n}$ endowed with independent random weights
$\mathrm{M}^{\infty}$ | : |  the root component of the WMSF of the PWIT
$\mathrm{M}_{\alpha}^{\bullet}$ | : |  the connected component of $\mathrm{M}^{\infty}$ after removing the forward maximal edges of $\mathrm{T}^{\infty}$ that contains $\mathrm{T}_{\alpha}^{\bullet}$
Continuous objects
$\mathscr{P}$ | : |  Poisson point process on $\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$ with intensity measure $\frac{\differential t}{t}$
$\mathcal{T}^{\infty}$ | : |  scaling limit of $\mathrm{T}^{\infty}$
$\mathcal{T}^{\bullet}$ | : |  scaling limit of $\mathrm{T}_{\alpha}^{\bullet}$
$\mathcal{M}^{\infty}$ | : |  scaling limit of $\mathrm{M}^{\infty}$
$\mathcal{M}^{\bullet}$ | : |  scaling limit of $\mathrm{M}_{\alpha}^{\bullet}$
$(\mathcal{W}_{t})_{t\geq 1}$ | : |  weight process used in the construction of $\mathcal{M}^{\bullet}$
$X$ | : |  total mass of $\mathcal{T}^{\bullet}$ and starting value of $(\mathcal{W}_{t})_{t\geq 1}$, i.e. $X=\mathcal{W}_{1}$
$Y$ | : |  total mass of $\mathcal{M}^{\bullet}$, obtained as $Y=\lim_{t\rightarrow\infty}\frac{\mathcal{W}_{t}}{t}$
$\mathscr{J}$ | : |  set of jumps of the weight process $(\mathcal{W}_{t})_{t\geq 1}$
$(w_{t})_{t\in\mathscr{J}}$ | : |  jumps $w_{t}=\mathcal{W}_{t}-\mathcal{W}_{t-}$ of the weight process
$\mathcal{T}_{t}$ | : |  Brownian CRT with total mass $w_{t}$
Constructions
$\mathrm{mass}(\mathsf{X})$ | : |  total mass of the measure carried by the metric measured space $\mathsf{X}$
$\mathrm{diam}(\mathsf{X})$ | : |  diameter of the metric (possibly measured) space $\mathsf{X}$
$\mathrm{Scale}(a,b;\mathsf{X})$ | : |  version of the metric measured space $\mathsf{X}$ where the distances are scaled by $a$ and the mass is scaled by $b$
$\mathrm{Chain}$ | : |  construction taking a sequence of length spaces and linking them, each one to the next
$\mathrm{Aggreg}$ | : |  construction that yields a metric space from the result of an aggregation process
$\mathbb{L}$ | : |  set of equivalence classes of rooted, measured, locally compact length spaces
$\mathbb{L}^{c}$ | : |  set of equivalence classes of rooted, measured, compact length spaces
$\mathbb{L}^{\bullet,c}$ | : |  set of equivalence classes of rooted, pointed, measured, compact length spaces
Other symbols
$C_{0}$ | : |  some positive constant defined in (47)
$r$ | : |  positive parameter destined to tend to $0$
$\operatorname{cst}$ | : |  a positive constant whose value may change along the lines

##  2 Constructions and convergences

In this section, we introduce two constructions from sequences of measured
metric space. One of them is fairly simple and consists in linking a sequence
of metric spaces in a linear manner: we call it the _chain_ construction. The
other one is slightly more involved and consists of gluing some ordered family
of metric spaces on top of each other in a random way: this is called an
_aggregation process_. It is a natural extension of the so called "line-
breaking" construction of random trees [7, 8, 16], already extended to more
general metric spaces [17, 27], where we do not constrain the blocks that we
glue together to arrive in a discrete order anymore.

We state some properties of these two constructions (in particular convergence
results) and we explain how to interpret our objects of interest in this
setting. These results will be the base of the rest of our arguments.

In this section we only expose results and give an idea of how they are used
later in the paper. The proofs of the results stated here will be postponed to
the Appendix.

###  2.1 Reminder about the pointed GH and GHP topologies and local versions

In this section, we give a brief reminder about the definition of the
Hausdorff, Gromov–Hausdorff and Gromov–Hausdorff–Prokhorov topologies on
compact metric spaces, and their local equivalent on boundedly finite rooted
metric spaces. We will mainly follow the exposition of [1]. This section can
be skipped at first reading.

Let $(X,d^{X})$ be a Polish metric space and ${\mathcal{B}}(X)$ be its set of
Borel measurable subsets. The diameter of $A\in{\mathcal{B}}(X)$ is given by:

$\operatorname{diam}(A)=\sup\left\\{d^{X}(x,y)\mathrel{}\middle|\mathrel{}x,y\in
A\right\\}.$

For $A,B\in{\mathcal{B}}(X)$, we set:

$d_{\text{H}}^{X}(A,B)=\inf\\{\epsilon>0;\ A\subset B^{\epsilon}\
\mathrm{and}\ B\subset A^{\epsilon}\\},$

the Hausdorff distance between $A$ and $B$, where

$A^{\epsilon}=\left\\{x\in X\mathrel{}\middle|\mathrel{}\inf_{y\in
A}d^{X}(x,y)<\epsilon\right\\}$ (4)

is the $\epsilon$\- fattening of $A$. If $(X,d^{X})$ is compact, then the
space of compact subsets of $X$, endowed with the Hausdorff distance, is
compact.

Let $\mathrm{Mes}_{f}(X)$ denote the set of all finite Borel measures on $X$.
If $\mu,\nu\in\mathrm{Mes}_{f}(X)$, we set:

$d_{\text{P}}^{X}(\mu,\nu)=\inf\left\\{\epsilon>0\mathrel{}\middle|\mathrel{}\mu(A)\leq\nu(A^{\epsilon})+\epsilon\text{
and }\nu(A)\leq\mu(A^{\epsilon})+\epsilon\ \text{ for any closed set
}A\right\\},$

the Prokhorov distance between $\mu$ and $\nu$. It is well known, that
$(\mathrm{Mes}_{f}(X),d_{\text{P}}^{X})$ is a Polish metric space, and that
the topology generated by $d_{\text{P}}^{X}$ is exactly the topology of weak
convergence.

The notion of Prokhorov distance can be extended in the following way. A Borel
measure is said to be locally finite if the measure of any bounded Borel set
is finite. Let $\mathrm{Mes}(X)$ denote the set of all locally finite Borel
measures on $X$. Let $\rho$ be a distinguished element of $X$, which we call
the root. We consider the closed ball of radius $R$ centered at $\rho$:

$X^{(R)}=\left\\{x\in X\mathrel{}\middle|\mathrel{}d^{X}(\rho,x)\leq
R\right\\},$ (5)

and for $\mu\in\mathrm{Mes}(X)$ its restriction $\mu^{(R)}$ to $X^{(R)}$:

$\mu^{(R)}(dx)=\mathbf{1}_{X^{(R)}}(x)\;\mu(dx).$ (6)

If $\mu,\nu\in\mathrm{Mes}(X)$, the generalized Prokhorov distance between
$\mu$ and $\nu$ is defined as:

$d_{\text{gP}}^{X}(\mu,\nu)=\int_{0}^{\infty}\exp{-R}\left(1\wedge
d^{X}_{\text{P}}\left(\mu^{(R)},\nu^{(R)}\right)\right)\ \differential R.$ (7)

The function $d_{\text{gP}}^{X}$ is well defined and is a distance.
Furthermore $(\mathrm{Mes}(X),d_{\text{gP}}^{X})$ is a Polish metric space,
and the topology generated by $d_{\text{gP}}^{X}$ is exactly the topology of
vague convergence. When there is no ambiguity on the metric space $(X,d^{X})$,
we may write $d$, $d_{\text{H}}$, and $d_{\text{P}}$ instead of $d^{X}$,
$d^{X}_{\text{H}}$ and $d^{X}_{\text{P}}$.

If $\Phi:X\rightarrow X^{\prime}$ is a Borel-measurable map between two Polish
metric spaces and if $\mu$ is a Borel measure on $X$, we will note
$\Phi_{*}\mu$ the image measure on $X^{\prime}$ defined by
$\Phi_{*}\mu(A)=\mu(\Phi^{-1}(A))$, for any Borel set $A\subset X$.

######  Definition .

  * •

A rooted measured metric space $\mathsf{X}=(X,d,\rho,\mu)$ is a metric space
$(X,d)$ with a distinguished element $\rho\in X$, called the root, and a
locally finite Borel measure $\mu$.

  * •

Two rooted measured metric spaces $\mathsf{X}=(X,d,\rho,\mu)$ and
$\mathsf{X}^{\prime}=(X^{\prime},d^{\prime},\rho^{\prime},\mu^{\prime})$ are
said to be GHP-isometric if there exists an isometric one-to-one map
$\Phi:X\rightarrow X^{\prime}$ such that $\Phi(\rho)=\rho^{\prime}$ and
$\Phi_{*}\mu=\mu^{\prime}$. In that case, $\Phi$ is called a GHP-isometry.

Notice that if $(X,d)$ is compact, then a locally finite measure on $X$ is
finite and belongs to $\mathrm{Mes}_{f}(X)$. We will now use a procedure due
to Edwards [15] and developed by Gromov [18] to compare any two compact rooted
measured metric spaces, even if they are not subspaces of the same Polish
metric space.

####  2.1.1 Gromov-Hausdorff-Prokhorov metric for compact spaces

Let $(X,d)$ and $(X^{\prime},d^{\prime})$ be two compact metric spaces. The
Gromov-Hausdorff metric between $(X,d)$ and $(X^{\prime},d^{\prime})$ is given
by:

$d_{\text{GH}}^{c}((X,d),(X^{\prime},d^{\prime}))=\inf_{\varphi,\varphi^{\prime},Z}d_{\text{H}}^{Z}(\varphi(X),\varphi^{\prime}(X^{\prime})),$
(8)

where the infimum is taken over all isometric embeddings
$\varphi:X\hookrightarrow Z$ and $\varphi^{\prime}:X^{\prime}\hookrightarrow
Z$ into some common Polish metric space $(Z,d^{Z})$. The last display actually
defines a metric on the set of isometry classes of compact metric spaces.

Now, we introduce the Gromov–Hausdorff–Prokhorov distance for compact spaces.
Let $\mathsf{X}=(X,d,\rho,\mu)$ and
$\mathsf{X}^{\prime}=(X^{\prime},d^{\prime},\rho^{\prime},\mu^{\prime})$ be
two compact rooted measured metric spaces, and define:

$d_{\mathrm{GHP}}^{c}(\mathsf{X},\mathsf{X}^{\prime})=\inf_{\Phi,\Phi^{\prime},Z}\left(d^{Z}(\Phi(\rho),\Phi^{\prime}(\rho^{\prime}))+d_{\text{H}}^{Z}(\Phi(X),\Phi^{\prime}(X^{\prime}))+d_{\text{P}}^{Z}(\Phi_{*}\mu,\Phi_{*}^{\prime}\mu^{\prime})\right),$
(9)

where the infimum is taken over all isometric embeddings
$\Phi:X\hookrightarrow Z$ and $\Phi^{\prime}:X^{\prime}\hookrightarrow Z$ into
some common Polish metric space $(Z,d^{Z})$.

Note that equation (9) does not actually define a distance, as
$d_{\mathrm{GHP}}^{c}(\mathsf{X},\mathsf{X}^{\prime})=0$ if $\mathsf{X}$ and
$\mathsf{X}^{\prime}$ are GHP-isometric. Therefore, we shall consider
$\mathbb{K}$, the set of GHP-isometry classes of compact rooted measured
metric space and identify a compact rooted measured metric space with its
class in $\mathbb{K}$. Then the function $d_{\mathrm{GHP}}^{c}$ is finite on
$\mathbb{K}^{2}$.

[Theorem 2.5 of [1]]

  * (i)

The function $d_{\mathrm{GHP}}^{c}$ defines a distance on $\mathbb{K}$.

  * (ii)

The space $(\mathbb{K},d_{\mathrm{GHP}}^{c})$ is a Polish metric space.

The function $d_{\mathrm{GHP}}^{c}$ is called the Gromov–Hausdorff–Prokhorov
distance. The Gromov–Hausdorff–Prokhorov distance could be defined without
reference to any root. However, the introduction of the root is necessary to
extend the definition of the Gromov–Hausdorff–Prokhorov distance to spaces
that are not necessarily compact.

We sometimes need to endow our spaces with extra structure, like an additional
distinguished point for example. In that case, it is possible to modify the
definition of the Gromov–Hausdorff–Prokhorov distance to account for this
extra piece of data and the corresponding space $\mathbb{K}^{\bullet}$ of
equivalence classes of root _pointed_ compact metric spaces is also Polish. We
denote by $d_{\mathrm{GHP}}^{\bullet,c}$ the corresponding distance.

####  2.1.2 Gromov–Hausdorff–Prokhorov distance for locally compact length
spaces

A metric space $(X,d)$ is a length space if for every $x,y\in X$ , we have
$d(x,y)=\inf\\{\mathrm{Len}(\gamma)\\}$, \- where the infimum is taken over
all rectifiable curves
$\gamma:\mathopen{[}0\mathclose{}\mathpunct{},1\mathclose{]}\rightarrow X$
such that $\gamma(0)=x$ and $\gamma(1)=y$ , and where $\mathrm{Len}(\gamma)$
is the length of the rectifiable curve $\gamma$.

Let $\mathbb{L}$ be the set of GHP-isometry classes of rooted, measured,
complete and locally compact length spaces and identify a rooted, measured,
complete and locally compact length space with its class in $\mathbb{L}$. We
will also denote by $\mathbb{L}^{c}=\mathbb{K}\cap\mathbb{L}$, and
$\mathbb{L}^{\bullet,c}$ for the space of their pointed counterparts.

If $\mathsf{X}=(X,d,\rho,\mu)\in\mathbb{L}$, then for $R\geq 0$ we will
consider its restriction to the closed ball of radius $R$ centered at $\rho$,
denoted by $\mathsf{X}^{(R)}=(X^{(R)},d^{(R)},\rho^{(R)},\mu^{(R)})$, where
$X^{(R)}$ is defined by (5), the distance $d^{(R)}$ is the restriction of $d$
to $X^{(R)}$, and the measure $\mu^{(R)}$ is defined by (6). If $\mathsf{X}$
belongs to $\mathbb{L}$ , then $\mathsf{X}^{(R)}$ belongs to $\mathbb{K}$ for
all $R\geq 0$.

The following distance is well defined on $\mathbb{L}\times\mathbb{L}$:

$d_{\mathrm{GHP}}(\mathsf{X},\mathsf{Y})=\int_{0}^{\infty}e^{-R}\cdot\left(1\wedge
d^{c}_{\mathrm{GHP}}\left(\mathsf{X}^{(R)},\mathsf{Y}^{(R)}\right)\right)\differential
R.$

The weight $e^{-R}$ and truncation are arbitrary and their choice does not
affect the resulting topology.

[Theorem 2.9 and Proposition 2.10 of [1]] We have the following:

  * (i)

The function $d_{\mathrm{GHP}}$ defines a distance on $\mathbb{L}$.

  * (ii)

The space $(\mathbb{L},d_{\mathrm{GHP}})$ is a Polish metric space.

  * (iii)

The two distances $d_{\mathrm{GHP}}$ and $d_{\mathrm{GHP}}^{c}$ define the
same topology on $\mathbb{K}\cap\mathbb{L}$.

All the objects that we study in this paper can be understood in that context
of length spaces. The discrete objects that we are considering (discrete
trees) can be seen as length spaces by considering their _cable graph_ ,
meaning that we consider every edge in the tree as a segment of length $1$.
The continuous objects that we are considering are all weak limits of such
discrete objects that are indeed length spaces. This makes them length spaces
as well by the fact that taking Gromov–Hausdorff limits preserves the property
of being a length space, see [13, Theorem 7.5.1].

###  2.2 The chain construction

In this section, we present a construction that takes a family of compact
measured metric spaces and glues them sequentially along a path.

####  2.2.1 Construction

Suppose that we are given a discrete set
$\mathscr{N}\subset\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$
and a family $(\mathsf{L}_{t}^{\bullet},t\in\mathscr{N})$ of rooted pointed
measured compact length spaces indexed by the set $\mathscr{N}$, where

$\displaystyle\mathsf{L}_{t}^{\bullet}=(L_{t},d_{t},\rho_{t},v_{t},\nu_{t}).$

We would like to make sense of the object obtained by linking every
$\mathsf{L}_{t}^{\bullet}$ by identifying the distinguished point $v_{t}$ to
the root $\rho_{t^{\prime}}$ of the next block in ascending order. For the
result to be well defined we make the following assumptions.

Assume that

  1. (Ch1)

For any $A>0$ we have

$\displaystyle\sum_{t\in\mathscr{N}\cap\mathopen{(}A\mathclose{}\mathpunct{},\infty\mathclose{)}}d_{t}(\rho_{t},v_{t})<\infty\quad\text{
and
}\quad\sum_{t\in\mathscr{N}\cap\mathopen{(}A\mathclose{}\mathpunct{},\infty\mathclose{)}}\nu_{t}(L_{t})<\infty$

  2. (Ch2)

We have

$\displaystyle\sup_{t\in\mathscr{N}\cap\mathopen{(}A\mathclose{}\mathpunct{},\infty\mathclose{)}}\operatorname{diam}(\mathsf{L}_{t}^{\bullet})\underset{A\rightarrow\infty}{\rightarrow}0.$

  3. (Ch3)

For any $\epsilon>0$, one of the two following conditions is satisfied:

    1. (Ch3a)

$\sum_{t\in\mathscr{N}\cap\mathopen{(}0\mathclose{}\mathpunct{},\epsilon\mathclose{)}}d_{t}(\rho_{t},v_{t})=\infty$,

    2. (Ch3b)

$\mathscr{N}\cap\mathopen{(}0\mathclose{}\mathpunct{},\epsilon\mathclose{)}$
is finite.

We define $L_{\infty}=\\{\rho_{\infty}\\}$ and consider

$\displaystyle\bigsqcup_{t\in\mathscr{N}\cup\\{\rho_{\infty}\\}}L_{t}$

which we endow with the pseudo-distance $d$ defined as

$d(x,y)=d(y,x)=\begin{cases}d_{t}(x,y)&\text{if }x,y\in L_{t},\\\
d_{t}(x,v_{t})+\displaystyle\sum_{r\in\mathscr{N}\cap(s,t)}d_{r}(\rho_{r},v_{r})+d_{s}(\rho_{s},y)&\text{if
}(x,y)\in L_{s}\times L_{t},\ t<s.\end{cases}$ (10)

Then $\mathrm{Chain}(\mathsf{L}_{t}^{\bullet},t\in\mathscr{N})$ is defined as

$\displaystyle\left(\bigsqcup_{t\in\mathscr{N}\cup\\{\rho_{\infty}\\}}L_{t}\right)/\sim$

where $\sim$ is the equivalence relation generated by $x\sim y$ if $d(x,y)=0$.
The space $\mathrm{Chain}(\mathsf{L}_{t}^{\bullet},t\in\mathscr{N})$ is seen
as a rooted measured metric space after endowing the space with the projection
of the distance $d$ and the measure $\sum_{t\in\mathscr{N}}\nu_{t}$ and the
root $\rho_{\infty}$.

######  Lemma .

If assumptions (Ch1), (Ch2), (Ch3) hold then
$\mathrm{Chain}(\mathsf{L}_{t}^{\bullet},t\in\mathscr{N})$ is an element of
$\mathbb{L}$.

The proof of this lemma can be found in Appendix A.

######  Remark .

If only (Ch3) fails, then a length space in $\mathbb{L}$ can still be
constructed from $\mathrm{Chain}(\mathsf{L}_{t}^{\bullet},t\in\mathscr{N})$,
except that it needs to be completed with a second point $\rho_{0}$ on the
other side. Also, note that in our definition, we exclude the possibility that
the set $\mathscr{N}$ could be dense. The construction could be made more
general to accommodate for such situations, but this would require the
addition of more (possibly an uncountable infinity) completion points in
between the links of the chain. Such a construction appears for example in
[30, Section 6.1].

####  2.2.2 Convergence of chains

In order to prove scaling limit convergence results for the discrete objects
mentioned above to the continuous ones, we provide here sufficient conditions
for convergence. Since we will consider convergence of random such objects, we
state the condition for a random object constructed as
$\mathrm{Chain}(\mathsf{L}_{t}^{\bullet,r},t\in\mathscr{N}^{r})$ to converge
to a limit expressed as
$\mathrm{Chain}(\mathsf{L}_{t}^{\bullet,0},t\in\mathscr{N}^{0})$ as
$r\rightarrow 0$. We assume that for any $r\geq 0$, so in particular for $r=0$
as well, the point process $\mathscr{N}^{r}$ and the family
$(\mathsf{L}_{t}^{\bullet,r},t\in\mathscr{N}^{r})$ satisfy (Ch1), (Ch2), (Ch3)
almost surely. Additionally we assume the following:

  1. (ChConv1)

The process $((t,\mathsf{L}_{t}^{\bullet,r}),\ t\in\mathscr{N}^{r})$ converges
in distribution as $r\rightarrow 0$ to $((t,\mathsf{L}_{t}^{\bullet,0}),\
t\in\mathscr{N}^{0})$ as point processes on
$\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$ with marks in
$\mathbb{L}^{\bullet,c}$. We mean by that that for any compact set
$K\subset\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$, the
random measure

$\displaystyle\sum_{t\in\mathscr{N}^{r}\cap
K}\delta_{(x,\mathsf{L}_{t}^{\bullet,r})}\overset{(d)}{\rightarrow}\sum_{t\in\mathscr{N}^{0}\cap
K}\delta_{(x,\mathsf{L}_{t}^{\bullet,0})}$

as $r\rightarrow 0$ for the topology of the weak convergence of measures on
$K\times\mathbb{L}^{\bullet,c}$.

  2. (ChConv2)

For any $\epsilon>0$

$\displaystyle\limsup_{A\rightarrow\infty}\limsup_{r\rightarrow
0}\mathbb{P}\left(\mathrm{mass}\left(\mathrm{Chain}(\mathsf{L}_{t}^{\bullet,r},t\in\mathscr{N}^{r}\cap\mathopen{(}A\mathclose{}\mathpunct{},\infty\mathclose{)})\right)>\epsilon\right)=0.$

  3. (ChConv3)

For any $\epsilon>0$

$\displaystyle\limsup_{A\rightarrow\infty}\limsup_{r\rightarrow
0}\mathbb{P}\left(\mathrm{diam}\left(\mathrm{Chain}(\mathsf{L}_{t}^{\bullet,r},t\in\mathscr{N}^{r}\cap\mathopen{(}A\mathclose{}\mathpunct{},\infty\mathclose{)})\right)>\epsilon\right)=0.$

Then we have the following result. Suppose that for all $r>0$, the family
$(\mathsf{L}_{t}^{\bullet,r},t\in\mathscr{N}^{r})$ satisfies (Ch1), (Ch2) and
(Ch3) almost surely. Additionally, assume that
$(\mathsf{L}_{t}^{\bullet,0},t\in\mathscr{N}^{0})$ satisfies (Ch1), (Ch2) and
(Ch3)(Ch3a) almost surely. Then, under (ChConv1),(ChConv2), (ChConv3), we have
the following convergence in distribution in the space $\mathbb{L}$,

$\displaystyle\mathrm{Chain}(\mathsf{L}_{t}^{\bullet,r},t\in\mathscr{N}^{r})\underset{r\rightarrow
0}{\rightarrow}\mathrm{Chain}(\mathsf{L}_{t}^{\bullet,0},t\in\mathscr{N}^{0}).$

The proof of this proposition can be found in Appendix A.

###  2.3 Aggregation processes

The second construction that we will use is that of _aggregation processes_.
In this case, we create a metric space by gluing together a (potentially
infinite) collection of measured metric spaces $\left(\mathsf{B}_{t},\
t\in\mathscr{J}\right)$ in a _random_ manner. The reader familiar with the
stick-breaking construction of the CRT may wish to keep that in mind as a
useful analogy. Indeed, the stick breaking construction can be defined as a
special case of an aggregation process as defined here. However, unlike the
stick breaking construction, here it is possible that infinitely many blocks
are being glued within a finite time interval. (For any $t<\infty$ he blocks
used up to time $t$ still have finite total mass.)

####  2.3.1 Construction

Since the definition of this process is a bit more involved, we express it for
a particular case where the blocks
$\left(\mathsf{B}_{t},t\in\mathscr{J}\right)$ that we glue together are random
and satisfy some _ad hoc_ assumptions that will be satisfied by the objects
studied in this paper. We start with

  * •

a _weight process_ $(W_{t})_{t\geq t_{0}}$, which is a càdlàg increasing pure-
jump process, such that $W_{t_{0}}>0$, (we denote by $\mathscr{J}$ its set of
jump times),

  * •

a _seed_ $\mathsf{S}^{\bullet}=(S,\rho,v,d_{S},\nu_{S})$ which is a compact,
rooted, pointed, measured length space (_i.e._ an element of
$\mathbb{L}^{\bullet,c}$) and has total mass equal to $W_{t_{0}}$.

  * •

a _block-distribution family_ $\eta=\left(\eta(t,w):\ t>t_{0},w>0\right)$ of
distributions on $\mathbb{L}^{c}$ which is such that for any $t>t_{0},w>0$, an
object $\mathsf{B}(t,w)$ sampled under the measure $\eta(t,w)$ almost surely
has mass $w$.

From these, we construct below some rooted (pointed) random metric space

$\displaystyle\mathrm{Aggreg}((W_{t})_{t\geq
t_{0}},\mathsf{S}^{\bullet},\eta).$

For each jump time $t\in\mathscr{J}$ we let $w_{t}=W_{t}-W_{t-}$ and we let
$w_{t_{0}}=W_{t_{0}}$. Then we sample a random _block_
$\mathsf{B}_{t}=(B_{t},d_{t},\rho_{t},\nu_{t})$, whose distribution is
$\eta(t,w_{t})$, independently for all $t\in\mathscr{J}$. Then the object of
interest in constructed from the collection
$\left(\mathsf{B}_{t},t\in\mathscr{J}\right)$, by quotienting the set

$\displaystyle S\sqcup\bigsqcup_{t>t_{0}}B_{t},$

by the appropriate identification of pairs of points: For every jump time
$t\in\mathscr{J}$, we pick a random point $X_{t}$ on the set
$\bigsqcup_{t_{0}\leq s<t}B_{s}$ using a normalized version of the measure
$\sum_{t_{0}\leq s<t}\nu_{s}$. Define the equivalence relation $\sim$ as
generated by the relations $X_{t}\sim\rho_{t}$ for each $t\in\mathscr{J}$. The
object of interest is then the _metric gluing_

$\displaystyle(A_{\infty},\rho,v,d):=\overline{\left(\mathsf{S}^{\bullet}\sqcup\bigsqcup_{t>t_{0}}\mathsf{B}_{t}\right)/\sim}$

in the sense of [13], where the root $\rho$ and distinguished point $v$ are
inherited from the seed $\mathsf{S}^{\bullet}$. The overline in the last
display represent the operation of completion with respect to the distance
defined by the metric gluing. Note that the completion and the gluing
operation both preserve the property of being a length space. We will see
below that under some reasonable assumptions, this space can be shown to be
almost surely compact.

For any $t\geq t_{0}$, we can consider the probability measure
$\mu_{t}:=W_{t}^{-1}\sum_{t_{0}\leq s\leq t}\nu_{s}$. Again, under some
reasonable assumptions, this sequence of measures almost surely converge for
the weak topology as $t\rightarrow\infty$ so that $A_{\infty}$ can be endowed
with a probability measure $\mu_{\infty}:=\lim_{t\rightarrow\infty}\mu_{t}$.
This makes

$\displaystyle\mathrm{Aggreg}((W_{t})_{t\geq
t_{0}},\mathsf{S}^{\bullet},\eta):=\mathsf{A}_{\infty}^{\bullet}=(A_{\infty},\rho,v,d,\mu_{\infty})$

an element of $\mathbb{L}^{\bullet,c}$ almost surely. In what comes next we
will work under the following two assumptions, which are satisfied in all the
cases studied in this paper.

  1. (AG1)

For some deterministic non-increasing function
$\delta:\mathopen{[}t_{0}\mathclose{}\mathpunct{},\infty\mathclose{)}\rightarrow\mathopen{[}0\mathclose{}\mathpunct{},1\mathclose{]}$
that tends to $0$ as $t\rightarrow\infty$, for some random variables $Z_{1}$
and $Z_{2}$ we have for all $t\geq t_{0}$,

$\displaystyle W_{t}\leq Z_{1}\cdot t\qquad\text{ and }\qquad w_{t}\leq
Z_{2}\cdot t^{-2+\delta(t)}.$ (11)

  2. (AG2)

We have

$\displaystyle\sup_{t\geq
t_{0},w>0}\mathbb{P}\left(\frac{\operatorname{diam}\mathsf{B}(t,w)}{\sqrt{w}}\geq
x\right)<C_{1}\cdot\exp\left(-C_{2}x\right)$

The first assumption ensures that the jumps of the process get indeed smaller
and smaller in a quantifiable way. The second one ensures that the scaling of
the distances in the blocks are of the order of the square root of their mass.
Those assumptions above have been taken so that they apply to our setting in a
way that give us some quantitative results that depend on the specific
exponents $2$ and $\frac{1}{2}$ that appear in (AG2) and (AG1). The arguments
that we use in the proofs are quite robust and can be modified to account for
other assumptions of the same type.

Under assumptions (AG1) and (AG2), the object

$\displaystyle\mathsf{A}_{\infty}^{\bullet}=\mathrm{Aggreg}((W_{t})_{t\geq
t_{0}},\mathsf{S}^{\bullet},\eta)$

is almost surely well-defined as an $\mathbb{L}^{\bullet,c}$-valued random
variable. Furthermore, if the random variables $Z_{1}$ and $Z_{2}$ have
moments of all orders, then the Hausdorff distance
$d_{\mathrm{H}}(\mathsf{A}_{\infty}^{\bullet},\mathsf{S}^{\bullet})$ also has
moments of all order. The proof of this proposition can be found in Section
B.4 of the Appendix. We remark that without some assumptions similar to the
above it is possible to give examples of similar aggregation processes where
the distances between some points in $\mathsf{A}_{t}$ are infinite, even for
finite $t$.

####  2.3.2 Extra conditions on the aggregation process to get Hausdorff and
Minkowski dimension

Finally, we introduce more precise conditions on the blocks and weight process
(which will be satisfied in our applications) that allow us to compute the
almost sure Hausdorff and Minkowski dimension of our object
$\mathsf{A}_{\infty}^{\bullet}$. We use $N_{\epsilon}(\mathsf{X})$ to denote
the minimal number of balls of radius $\epsilon$ needed to cover a metric
space $\mathsf{X}$.

  1. (AGMink)

There exists some constant $C$ such that for all $t\geq t_{0}$ and $w>0$ we
have for $\mathsf{B}(t,w)\sim\eta(t,w)$,

$\displaystyle\mathbb{E}\left[N_{\epsilon}(\mathsf{B}(t,w))\right]\leq
C\cdot(1\vee w\epsilon^{-2+o\mathopen{}\left(1\right)}).$ (12)

  1. (AGHaus1)

The weight process $(W_{t})_{t\geq t_{0}}$ satisfies

$\displaystyle W_{t}\sim Z_{3}\cdot t,$

as $t\rightarrow\infty$ for some random variable $Z_{3}$.

  2. (AGHaus2)

The weight process $(W_{t})_{t\geq t_{0}}$ is such that for all $\delta>0$,

$\displaystyle\lim_{t\rightarrow\infty}\frac{\sum_{s\leq
t}w_{s}\mathbf{1}_{\\{w_{s}\geq s^{-2-\delta}\\}}}{W_{t}}=1.$

  3. (AGHaus3)

The measure $\nu_{t,w}$ carried on
$\mathsf{B}(t,w)=(B_{t,w},\rho_{t,w},d_{t,w},\nu_{t,w})$ has almost surely
full support and there exists positive $a$ and $b$ such that uniformly in $t$
and $w$, for $U(t,w)$ a random point in $\mathsf{B}(t,w)$ sampled under a
normalized version of the measure $\nu_{t,w}$ we have

$\displaystyle\mathbb{P}\left(d(\rho_{t,w},U(t,w))\geq
aw^{\frac{1}{2}}\right)\geq b.$

Assume (AG1) and (AG2) are satisfied so that $\mathsf{A}_{\infty}^{\bullet}$
is almost surely well-defined as a random element of $\mathbb{L}^{\bullet,c}$.
Then

  1. (i)

If we further assume that $\mathsf{S}^{\bullet}$ has upper Minkowski dimension
less than $3$ almost surely and that (AGMink) holds, then a.s. the upper
Minkowski dimension of $\mathsf{A}_{\infty}^{\bullet}$ satisfies

$\displaystyle\overline{\dim}_{\mathrm{Mink}}(\mathsf{A}_{\infty}^{\bullet})\leq
3.$

  2. (ii)

If we further assume that (AGHaus1),(AGHaus2) and (AGHaus3) hold, then almost
surely the measure $\mu_{\infty}$ is diffuse and has full support and

$\displaystyle\dim_{\mathrm{Haus}}(\mathsf{A}_{\infty}^{\bullet})\geq 3.$

In particular if all those conditions are satisfied, then the Minkowski
dimension of $\mathsf{A}_{\infty}^{\bullet}$ almost surely exists and
$\dim_{\mathrm{Mink}}(\mathsf{A}_{\infty}^{\bullet})=\dim_{\mathrm{Haus}}(\mathsf{A}_{\infty}^{\bullet})=3$
a.s.. The proof of this proposition is divided in two parts, the proofs of (i)
and (ii) can respectively be found in Section B.5 and Section B.6 of the
Appendix.

####  2.3.3 Convergence of aggregation processes

Now, assume that for any $r>0$ we have a weight process $(W_{t}^{r})_{t\geq
t_{0}}$, a seed $\mathsf{S}^{\bullet,r}$ and a block-distribution family
$\eta^{r}=\left(\eta^{r}(t,w):\ t>t_{0},w>0\right)$, for which we assume that
(AG1) and (AG2) are satisfied. We make the following assumptions to ensure a
convergence result on aggregation processes.

  1. (AGConv1)

On every compact interval we have the following convergence in distribution
for the Skorokhod topology:

$\displaystyle(W_{t}^{r})_{t\geq t_{0}}\rightarrow(W_{t})_{t\geq
t_{0}}\quad\text{as}\quad r\rightarrow 0.$

  2. (AGConv2)

We have $\mathsf{\mathsf{missing}}S^{\bullet,r}\underset{r\rightarrow
0}{\rightarrow}\mathsf{\mathsf{missing}}S^{\bullet}$ in distribution in
$\mathbb{L}^{\bullet,c}$.

  3. (AGConv3)

For any $t>t_{0}$ and $w>0$ we have $\eta^{r}(t,w)\rightarrow\eta(t,w)$ for
the Prokhorov distance (on the set of probability measures over
$\mathbb{L}^{c}$) as $r\rightarrow 0$, and the mapping $(t,w)\mapsto\eta(t,w)$
is continuous (for the Prokhorov distance as well).

  4. (AGConv4)

For some non-increasing function
$\delta:\mathopen{[}t_{0}\mathclose{}\mathpunct{},\infty\mathclose{)}\rightarrow\mathopen{[}0\mathclose{}\mathpunct{},1\mathclose{]}$
that tends to $0$ as $t\rightarrow\infty$, for some random variables
$Z_{i}^{r}$ with $i\in\\{1,2\\}$ where the family $(Z_{i}^{r})_{0<r\leq 1}$ is
tight, we have for all $t\geq t_{0}$,

$\displaystyle W_{t}^{r}\leq Z_{1}^{r}\cdot t\quad\text{ and }\quad
w_{t}^{r}\leq Z_{2}^{r}\cdot t^{-2+\delta(t)}.$ (13)

  5. (AGConv5)

There exists constant $c,C>0$ such that we have

$\displaystyle\sup_{0<r\leq
1}\sup_{t>t_{0},w>0}\mathbb{P}\left(\frac{\operatorname{diam}\mathsf{B}^{r}(t,w)}{\sqrt{w}}\geq
x\right)\leq C\exp(-cx).$

Under the assumptions above, we have the convergence in
$\mathbb{L}^{\bullet,c}$ in distribution

$\displaystyle\mathrm{Aggreg}((W_{t}^{r})_{t\geq
t_{0}},\mathsf{S}^{\bullet,r},\eta^{r})\underset{r\rightarrow
0}{\rightarrow}\mathrm{Aggreg}((W_{t})_{t\geq
t_{0}},\mathsf{S}^{\bullet},\eta),$

The proof of this proposition can be found in Section B.7 of the Appendix.

##  3 The tree $\mathcal{M}^{\infty}$

In this section, we use the results presented in the previous section to
rigorously construct our object $\mathcal{M}^{\infty}$. First, we need to
justify that the tree $\mathcal{M}^{\bullet}$ is well-defined as a random
variable in $\mathbb{L}^{\bullet,c}$. Then, the tree $\mathcal{M}^{\infty}$ is
constructed as

$\displaystyle\mathcal{M}^{\infty}:=\mathrm{Chain}\left(\mathcal{M}_{x}^{\bullet},\
x\in\mathscr{P}\right),$ (14)

where:

  * •

the point process $\mathscr{P}$ is a Poisson point process with intensity
$\frac{\differential t}{t}$ on
$\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$,

  * •

conditionally on $\mathscr{P}$ the sequence
$\left(\mathcal{M}_{x}^{\bullet},x\in\mathscr{P}\right)$ is independent and

$\displaystyle\mathcal{M}^{\bullet}_{x}\overset{(d)}{=}\mathrm{Scale}(x^{-1},x^{-3};\mathcal{M}^{\bullet}),$

for the random measured tree $\mathcal{M}^{\bullet}$.

We need to check that conditions (Ch1), (Ch2), (Ch3) hold almost surely in
order for $\mathcal{M}^{\infty}$ to be a well-defined. In fact, checking that
$\mathbb{E}\left[\operatorname{diam}(\mathcal{M}^{\bullet})\right]$ and
$\mathbb{E}\left[\mathrm{mass}(\mathcal{M}^{\bullet})\right]$ are finite will
be enough to ensure that those assumptions hold.

On our way to define $\mathcal{M}^{\bullet}$, we first introduce
$\widetilde{\mathcal{M}}^{\bullet}$ as the result of an aggregation process

$\displaystyle\widetilde{\mathcal{M}}^{\bullet}:=\mathrm{Aggreg}((\mathcal{W}_{t})_{t\geq
1},\mathcal{T}^{\bullet},\eta^{\mathrm{Br}})$ (15)

where

  * •

the seed $\mathcal{T}^{\bullet}$ is given by a Brownian tree of random mass
$X$ with density $\frac{\exp\left(-\frac{x}{2}\right)}{\sqrt{2\pi
x}}\mathrm{d}x$, pointed at a uniform random point,

  * •

the weight process $(\mathcal{W}_{t})_{t\geq 1}$, started at time $1$ with
value $X$, is an increasing pure-jump càdlàg process and the process
$((t,\mathcal{W}_{t}))_{t\geq 1}$ is a Markov Feller process with generator
$\mathbf{A}$ so that for any smooth and compactly supported function
$f:(0,\infty)\times\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}\rightarrow\mathbb{R}$
we have

$\displaystyle\mathbf{A}f(t,x)=\partial_{t}f(t,x)+\int_{0}^{\infty}\left(f(t,x+y)-f(t,x)\right)\cdot
x\cdot\frac{\exp\left(-\frac{t^{2}}{2}y\right)}{y^{\frac{3}{2}}\sqrt{2\pi}}\differential
y,$ (16)

  * •

the block-distribution family $\eta^{\mathrm{Br}}$ is so that for any
$t>1,w>0$, the distribution $\eta^{\mathrm{Br}}(t,w)$ is that of a Brownian
tree with mass $w$. This means that a tree under $\eta^{\mathrm{Br}}(t,w)$ is
distributed as $\mathrm{Scale}(w^{\frac{1}{2}},w;\mathcal{T})$, where
$\mathcal{T}$ is the Brownian tree of mass $1$111Here we use Aldous’
normalization: the tree $\mathcal{T}$ can be defined by its contour function,
which is given by twice a Brownian excursion of duration $1$, see [23, Section
2.3]..

Then, denoting $Y:=\lim\frac{\mathcal{W}_{t}}{t}$ our random tree
$\mathcal{M}^{\bullet}$ is obtained as

$\displaystyle\mathcal{M}^{\bullet}:=\mathrm{Scale}(1,Y;\widetilde{\mathcal{M}}^{\bullet}).$
(17)

In this section, we want to prove that (AG1) and (AG2) hold for the
construction above, in order to apply Proposition 2.3.1. We also need to check
that the random variable $Y$ indeed exists and is non-zero almost surely.

All of that, along with the proof of other properties of the random metric
space $\mathcal{M}^{\bullet}$, will be the content of the proof of Theorem
1.2, which is the goal of this section.

###  3.1 Properties of the weight process and proof of Theorem 1.2

We start by stating the properties of the weight process
$(\mathcal{W}_{t})_{t\geq 1}$ that we need in the proof of Theorem 1.2 and
that we prove in the rest of the section. Recall that we write
$w_{t}=\mathcal{W}_{t}-\mathcal{W}_{t-}$ for any $t>1$. The process
$(\mathcal{W}_{t})_{t\geq 1}$ introduced above satisfies the following
properties

  1. (i)

Almost surely,

$\displaystyle\frac{\mathcal{W}_{t}}{t}\underset{t\rightarrow\infty}{\rightarrow}Y,$

where the limiting random variable is almost surely positive.

  2. (ii)

The random variable

$\displaystyle Z_{1}:=\sup_{t\geq 1}\left(\frac{\mathcal{W}_{t}}{t}\right)$

is almost surely finite and has an exponentially small tail.

  3. (iii)

The random variable

$\displaystyle Z_{2}:=\sup_{t\geq
1}\left(\frac{w_{t}}{t^{-2}(1\vee\log^{2}t)}\right)$

is almost surely finite and has an exponentially small tail.

  4. (iv)

For any $\delta>0$ we almost surely have

$\displaystyle\sum_{1\leq s\leq t}w_{t}\mathbf{1}_{\\{w_{s}\leq
s^{-2-\delta}\\}}=o(t).$

We now prove Theorem 1.2 from the definition of $\mathcal{M}^{\bullet}$ given
by (15) and (17), using the properties of the weight process given in
Proposition 3.1.

###### Proof of Theorem 1.2.

We first need to apply Proposition 2.3.1 first to get that
$\widetilde{\mathcal{M}}^{\bullet}$ is well-defined as a random variable in
$\mathbb{L}^{\bullet,c}$. The fact that $\mathcal{M}^{\bullet}$ is well-
defined then just follows. For that we can check conditions (AG1) and (AG2).
Assumption (AG2) is actually immediate in this case using the fact that the
diameter of the Brownian tree of mass $1$ has an exponential moment (use for
example [21] and the construction of the CRT from the Brownian excursion, see
[23, Section 2.3]). Then, assumption (AG1) is satisfied thanks to items (ii)
and (iii) of Proposition 3.1. This entails that
$\widetilde{\mathcal{M}}^{\bullet}$ is well-defined as a random variable in
$\mathbb{L}^{\bullet,c}$ and the Hausdorff distance between
$\widetilde{\mathcal{M}}^{\bullet}$ and its seed $\mathcal{T}^{\bullet}$ has
moments of all orders. Since $\operatorname{diam}(\mathcal{T}^{\bullet})$ has
moments of all orders as well, this entails that the same is true for
$\operatorname{diam}(\widetilde{\mathcal{M}}^{\bullet})$. Now, since
$\mathcal{M}^{\bullet}=\mathrm{Scale}(1,Y;\widetilde{\mathcal{M}}^{\bullet})$
the above discussion also shows that $\mathcal{M}^{\bullet}$ is well-defined
and its diameter has moments of all orders. Since Proposition 3.1.(ii) ensures
that $Y$ has moments of all order, this allows us to conclude that Theorem
1.2.(i) holds.

For Theorem 1.2.(ii), we use Proposition 2.3.2. For that it suffices to check
that the assumptions (AGMink) and (AGHaus1),(AGHaus2) and (AGHaus3) hold for
the aggregation process that defines $\widetilde{\mathcal{M}}^{\bullet}$.
Conditions (AGMink) and (AGHaus3) follow easily from the fact that
$\eta^{\mathrm{Br}}(t,w)$ is the law of a Brownian tree of mass $w$.
Conditions (AGHaus1),(AGHaus2) follow from Proposition 3.1. This concludes the
proof of Theorem 1.2. ∎

The rest of the section is devoted to proving Proposition 3.1.

###  3.2 Definition of the weight process from a Poisson Point Process

In this section, we first present an alternative construction of the process
$(\mathcal{W}_{t})_{t\geq 1}$ from a Poisson point process. This will allow
for some easier computations in the next section.

##### Some stochastic differential equation driven by a Poisson point process.

In order to match the setting of [19], we assume in this section that we are
working on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ on which we
can define the random variable $X$, the total mass of the tree
$\mathcal{T}^{\bullet}$, together with an independent Poisson point process
$\mathscr{Q}$ on
$\mathopen{(}1\mathclose{}\mathpunct{},\infty\mathclose{)}\times\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}$
with intensity

$\displaystyle u(t,y)\cdot\differential t\otimes\differential
m\otimes\differential y\qquad\text{where}\qquad
u(t,y)=\frac{\exp\left(-\frac{t^{2}}{2}y\right)}{y^{\frac{3}{2}}\sqrt{2\pi}}.$

Note already that

$\displaystyle\int_{0}^{\infty}y\cdot u(t,y)\differential
y=\frac{1}{t}\qquad\text{and}\qquad\int_{0}^{\infty}y^{2}\cdot
u(t,y)\differential y=\frac{1}{t^{3}},$ (18)

as this will be useful later on.

Denote by
$\mathcal{B}\left(\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}\right)$
the set of Borel-measurable subsets of
$\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}$. For any
$U\in\mathcal{B}\left(\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}\right)$
introduce
$N_{\mathscr{Q}}(t,U):=\\#\mathscr{Q}\cap(\mathopen{(}0\mathclose{}\mathpunct{},t\mathclose{]}\times
U)$ and consider the (right-continuous version of the) filtration
$(\mathcal{F}_{t})_{t\geq 1}$ generated by those processes together with the
random variable $X$, i.e.

$\displaystyle\mathcal{F}_{t}=\sigma\left(X,\left(N_{\mathscr{Q}}(s,U),\ 1\leq
s\leq t,\
U\in\mathcal{B}\left(\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}\right)\right)\right),$

so that $(t\mapsto N_{\mathscr{Q}}(t,U))$ is $(\mathcal{F}_{t})$-adapted, for
every
$U\in\mathcal{B}\left(\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}\right)$,
and $X$ is $\mathcal{F}_{1}$-measurable. Following [19, Definition 3.1], it is
natural to introduce the corresponding _compensator_ measure
$\hat{N}_{\mathscr{Q}}(t,\cdot)$ where for any
$U\in\mathcal{B}\left(\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}\right)$
we have
$\hat{N}_{\mathscr{Q}}(t,U)=\int_{1}^{t}\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}}\mathbf{1}_{\\{(m,y)\in
U\\}}\cdot u(s,y)\differential s\differential m\differential y$.

From there we introduce the following equation on an unknown function
$(\mathcal{X}(t))_{t\geq 1}$

$\displaystyle\mathcal{X}(t)=\mathcal{X}(1)+\sum_{(s,m,y)\in\mathscr{Q}}y\cdot\mathbf{1}_{\\{m<\mathcal{X}(s-),\
s\leq t\\}},$

which can also be expressed as the following jump-type SDE

$\displaystyle\mathcal{X}(t)=\mathcal{X}(1)+\int_{1}^{t}\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}}g(\mathcal{X}(s-),m,y)N_{\mathscr{Q}}(\differential
s\differential m\differential y)$ (19)

where $g(x,m,y)=y\mathbf{1}_{\\{m<x\\}}$. From [19, Theorem 9.1], the above
SDE admits a unique solution $(\mathcal{X}(t))_{t\geq 1}$ for any (possibly
random and in that case $\mathcal{F}_{1}$-measurable) starting value $\xi$,
and this solution is càdlàg and $(\mathcal{F}_{t})$-adapted. Moreover, if the
starting value has a second moment, it follows from the construction of the
solution that the obtained process $(\mathcal{X}(t))_{t\geq 1}$ is so that
$\mathbb{E}\left[\mathcal{X}(t)^{2}\right]<\infty$ for all $t\geq 1$.

##### Definition and properties of $(\mathcal{W}_{t})_{t\geq 1}$.

The process $(\mathcal{W}_{t})_{t\geq 1}$ is then defined as the solution of
(19), started from the value $X$ at time $1$, where $X$ is the total mass of
the tree $\mathcal{T}^{\bullet}$. Recall that $X$ admits a second moment so
that, thanks to the above paragraph we get that for all $t\geq 1$ we have
$\mathbb{E}\left[(\mathcal{W}_{t})^{2}\right]<\infty$. Note also that by
construction, $(\mathcal{W}_{t})_{t\geq 1}$ is almost surely positive and
increasing. Now, writing $\mathcal{W}_{t}$ as

$\displaystyle\mathcal{W}_{t}=X+\int_{1}^{t}\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}}g(\mathcal{W}_{s-},m,y)N_{\mathscr{Q}}(\differential
s\differential m\differential y)$ (20)

ensures that $(\mathcal{W}_{t})_{t\geq 1}$ is a semi-martingale in the sense
of [19, Definition 4.1], as the term $g(\mathcal{W}_{s-},m,y)$ appearing in
the integral is $(\mathcal{F}_{s})$-predictable.

Applying the Itô formula [19, Theorem 5.1] with any twice differentiable
function $F$ we get that, for $t\geq 1$,

$\displaystyle F(\mathcal{W}_{t})-F(\mathcal{W}_{1})$
$\displaystyle=\int_{1}^{t}\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}}\left(F(\mathcal{W}_{s-}+g(\mathcal{W}_{s-},m,y))-F(\mathcal{W}_{s-})\right)N_{\mathscr{Q}}(\differential
s\differential m\differential y).$

Now, we aim at subtracting a term from the last display in a way that we get a
martingale. The standard way to do that is to subtract a term containing the
same integral but against the compensator $\hat{N}_{\mathscr{Q}}(\differential
s\differential m\differential y)$ instead of the point process
$N_{\mathscr{Q}}(\differential s\differential m\differential y)$. For this we
first need to check some integrability conditions: We can check that when we
integrate the absolute value of the integrand against the compensator
$\hat{N}_{\mathscr{Q}}(\differential s\differential m\differential y)$ we get

$\displaystyle\int_{1}^{t}\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}}\left|F(\mathcal{W}_{s-}+g(\mathcal{W}_{s-},m,y))-F(\mathcal{W}_{s-})\right|\hat{N}_{\mathscr{Q}}(\differential
s\differential m\differential y)$
$\displaystyle=\int_{1}^{t}\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}}\left|F(\mathcal{W}_{s-}+y\cdot\mathbf{1}_{\\{m<\mathcal{W}_{s-}\\}})-F(\mathcal{W}_{s-})\right|u(s,y)\differential
s\differential m\differential y$
$\displaystyle=\int_{1}^{t}\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}}\left|F(\mathcal{W}_{s-}+y)-F(\mathcal{W}_{s-})\right|\cdot\mathcal{W}_{s-}\cdot
u(s,y)\differential s\differential y.$

Now suppose that $F$ is taken so that it additionally satisfies that for any
$x,y\geq 0$ we have $|F(x+y)-F(x)|\leq C\cdot(1+x+y)\cdot y$. Note that this
holds for any differentiable $F$ with bounded derivative for example, as well
as for the function $x\mapsto x^{2}$. For such a function $F$, the above
integral is bounded above by

$\displaystyle\int_{1}^{t}\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}}C\cdot(1+\mathcal{W}_{s-}+y)\cdot
y\cdot\mathcal{W}_{s-}\cdot u(s,y)\differential s\differential
y=C\int_{1}^{t}\left(\frac{\mathcal{W}_{s-}}{s}+\frac{\mathcal{W}_{s-}^{2}}{s}+\frac{\mathcal{W}_{s-}}{s^{3}}\right)\differential
s,$

and now the expectation of the right-hand-side is finite, for any $t\geq 1$.
This allows us to use [19, (3.8) in Chapter 2], which ensures that the
compensated process

$\displaystyle
F(\mathcal{W}_{t})-F(\mathcal{W}_{1})-\int_{1}^{t}\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}}\left(F(\mathcal{W}_{s-}+y)-F(\mathcal{W}_{s-})\right)\cdot\mathcal{W}_{s-}\cdot
u(s,y)\differential s\differential y$ (21)

is an $(\mathcal{F}_{t})$-martingale. This will be useful later on.

Note in particular that

$\displaystyle\mathbb{E}\left[F(\mathcal{W}_{t+h})-F(\mathcal{W}_{t})\mathrel{}\middle|\mathrel{}\mathcal{F}_{t}\right]$
$\displaystyle=\int_{t}^{t+h}\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}}\mathbb{E}\left[\left(F(\mathcal{W}_{s-}+y)-F(\mathcal{W}_{s-})\right)\mathcal{W}_{s-})\mathrel{}\middle|\mathrel{}\mathcal{F}_{t}\right]u(s,y)\differential
s\differential y$ $\displaystyle\underset{h\rightarrow
0+}{=}h\cdot\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}}\left(F(\mathcal{W}_{t}+y)-F(\mathcal{W}_{t})\right)\cdot\mathcal{W}_{t}\cdot
u(t,y)\differential y+o(h).$

This entails that $(\mathcal{W}_{t})_{t\geq 1}$ is a time-inhomogeneous Feller
process whose generator $\mathbf{A}_{t}$ at time $t$ acts on functions $f$ as

$\displaystyle\mathbf{A}_{t}f(x)=\int_{\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}}\left(f(x+y)-f(x)\right)\cdot
x\cdot u(t,y)\differential y.$

The reason for describing in (16) the generator of the process
$((t,\mathcal{W}_{t}))_{t\geq 1}$ instead of $(\mathcal{W}_{t})_{t\geq 1}$ is
just a matter of technicality in the proofs, the former is time-homogeneous
whereas the latter is not. From the above we get the following lemma.

######  Lemma .

The process $((t,\mathcal{W}_{t}))_{t\geq 1}$ defined above is a time-
homogeneous Markov Feller process with generator $\mathbf{A}$ characterized by
its action on, say, the smooth functions
$f:\mathopen{[}1\mathclose{}\mathpunct{},\infty\mathclose{)}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$
with compact support, with

$\displaystyle\mathbf{A}f(t,x)=\partial_{t}f(t,x)+\int_{0}^{\infty}\left(f(t,x+y)-f(t,x)\right)\cdot
x\cdot\frac{\exp\left(-\frac{t^{2}}{2}y\right)}{y^{\frac{3}{2}}\sqrt{2\pi}}\cdot\differential
y.$ (22)

##### Intuition behind this description of $(\mathcal{W}_{t})_{t\geq 1}$.

From all of the above, the process $(\mathcal{W}_{t})_{t\geq 1}$ satisfies
$\mathcal{W}_{1}=X$ and for all $t\geq 1$,

$\displaystyle\mathcal{W}_{t}=X+\sum_{(s,m,y)\in\mathscr{Q}}y\cdot\mathbf{1}_{\\{m<\mathcal{W}_{s-},\
s\leq t\\}}.$ (23)

A way to imagine this process informally is to think of the first dimension as
time, the second dimension as space, and the third one as a count of mass, so
that every point $(s,m,y)$ is seen as a point at time $s$, at position $m$,
with mass $y$. This way we can imagine letting the time run starting from $1$
and following the value $\mathcal{W}_{t}$ of the function started at height
$X$. Then at any time $t$ that an atom $(t,m,y)$ is present, if the point is
below $\mathcal{W}_{t-}$ then the function grows by an amount
$w_{t}=\mathcal{W}_{t}-\mathcal{W}_{t-}$, which corresponds to the mass of
that point.

###  3.3 A priori estimates on the weight process

For any measurable subset $E$ of
$\mathopen{(}1\mathclose{}\mathpunct{},\infty\mathclose{)}\times\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}^{2}$
we define

$\displaystyle
M^{\mathscr{Q}}(E):=\sum_{(s,m,y)\in\mathscr{Q}}y\cdot\mathbf{1}_{\\{(s,m,y)\in
E\\}}.$ (24)

In the time-space-mass interpretation of the construction appearing at the end
of Section 3.2, the quantity $M^{\mathscr{Q}}(E)$ corresponds to the total
mass of points present in the set $E$. We are interested in this quantity
because of the result stated below, which states that if the process
$\mathcal{W}$ is below some level $A$ at time $t$ and the total mass
$M^{\mathscr{Q}}(\mathopen{(}t\mathclose{}\mathpunct{},t^{\prime}\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},A+B\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)})$
of points present below level $A+B$ in the time interval
$\mathopen{(}t\mathclose{}\mathpunct{},t^{\prime}\mathclose{]}$ is smaller
than $B$, then the process cannot reach level $A+B$ by time $t^{\prime}$.

######  Lemma .

For two different times $t<t^{\prime}$ and any two constants $A,B>0$, on the
event where

$\displaystyle\mathcal{W}_{t}\leq A\qquad\text{and}\qquad
M^{\mathscr{Q}}(\mathopen{(}t\mathclose{}\mathpunct{},t^{\prime}\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},A+B\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)})<B$

we have

$\displaystyle\mathcal{W}_{t^{\prime}}<A+B.$

###### Proof.

Assume that we are on the event where the first condition is satisfied and
where $\mathcal{W}_{t^{\prime}}>A+B$. Then there exists some
$t^{\prime\prime}=\inf\left\\{t\leq s\leq
t^{\prime}\mathrel{}\middle|\mathrel{}\mathcal{W}_{s}\geq A+B\right\\}$. We
then have

$\displaystyle A+B\leq\mathcal{W}_{t^{\prime\prime}}$
$\displaystyle=\mathcal{W}_{t}+\sum_{(s,m,y)\in\mathscr{Q}}y\cdot\mathbf{1}_{\\{m<\mathcal{W}_{s-},\
t<s\leq t^{\prime\prime}\\}}$ $\displaystyle\leq
A+M^{\mathscr{Q}}(\mathopen{(}t\mathclose{}\mathpunct{},t^{\prime}\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},A+B\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)})<A+B,$

which is a contradiction, so the event that we considered in the first place
is empty. This concludes the proof by contradiction. ∎

Using Campbell’s theorem for Poisson point processes (see for example [22,
Section 3.2] for a reference), we get that for any $\lambda\in\mathbb{C}$ for
which the right-hand-side makes sense we have

$\displaystyle\mathbb{E}\left[\exp(\lambda\cdot
M^{\mathscr{Q}}(\mathopen{(}c\mathclose{}\mathpunct{},d\mathclose{]}\times\mathopen{(}a\mathclose{}\mathpunct{},b\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}))\right]$
$\displaystyle=\int_{c}^{d}\differential{t}\int_{a}^{b}\differential{m}\int_{0}^{\infty}(e^{\lambda
y}-1)\cdot\frac{e^{-\frac{t^{2}}{2}y}}{y^{\frac{3}{2}}\sqrt{2\pi}}\differential{y}$
$\displaystyle=(b-a)\int_{c}^{d}F(\lambda,t)\differential{t},$

where we write $F(\lambda,t)=\int_{0}^{\infty}(e^{\lambda
y}-1)\cdot\frac{e^{-\frac{t^{2}}{2}y}}{y^{\frac{3}{2}}\sqrt{2\pi}}\differential{y}$,
again, for values of $\lambda$ for which this integral makes sense. Now we
have the following control on the values of $F(\lambda,t)$.

######  Lemma .

For $\lambda<\frac{t^{2}}{2}$ the value $F(\lambda,t)$ is well-defined and

$\displaystyle
F(\lambda,t)\underset{\begin{subarray}{c}\lambda,t\rightarrow\infty\\\
\lambda\ll
t^{2}\end{subarray}}{=}\frac{\lambda}{t}+O\left(\frac{\lambda^{2}}{t^{3}}\right)$

###### Proof.

We just replace $e^{\lambda y}-1$ with its Taylor series expansion

$\displaystyle F(\lambda,t)=\int_{0}^{\infty}(e^{\lambda
y}-1)\cdot\frac{e^{-\frac{t^{2}}{2}y}}{y^{\frac{3}{2}}\sqrt{2\pi}}\differential{y}$
$\displaystyle=\int_{0}^{\infty}\left(\sum_{k=1}^{\infty}\frac{(\lambda
y)^{k}}{k!}\right)\cdot\frac{e^{-\frac{t^{2}}{2}y}}{y^{\frac{3}{2}}\sqrt{2\pi}}\differential{y}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{\lambda^{k}}{\sqrt{2\pi}k!}\cdot\int_{0}^{\infty}y^{k-\frac{3}{2}}\cdot
e^{-\frac{t^{2}}{2}y}\differential{y}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{\lambda^{k}}{\sqrt{2\pi}k!}\cdot\left(\frac{t^{2}}{2}\right)^{-k+\frac{1}{2}}\cdot\Gamma\left(k-\frac{1}{2}\right)$
$\displaystyle=\frac{t}{\sqrt{2}}\cdot\sum_{k=1}^{\infty}\cdot\left(\frac{2\lambda}{t^{2}}\right)^{k}\cdot\frac{\Gamma\left(k-\frac{1}{2}\right)}{\sqrt{2\pi}k!}$
$\displaystyle=\frac{\lambda}{t}+\frac{\lambda^{2}}{t^{3}}\cdot
G\left(\frac{2\lambda}{t^{2}}\right),$

where $G(x)$ is a power series with radius of convergence $1$ and $G(0)\neq
0$. This concludes the proof. ∎

Using those two results (Lemma 3.3 and Lemma 3.3) repeatedly, together with
some Chernoff bounds, we can obtain some tail bound for the values of the
process $(\mathcal{W}_{t})_{t\geq 1}$ and its jumps. Below, we prove points
(ii), (iii) and (iv) of Proposition 3.1.

###### Proof of Proposition 3.1.(ii).

Let $(A_{k})_{k\geq 1}$ be an increasing sequence of positive real numbers
that we will specify later on. Using a union bound and the Markov property at
times $k\geq 1$, we get

$\displaystyle\mathbb{P}\left(\exists k\geq 1,\
\mathcal{W}_{k}>A_{k}\right)\leq\mathbb{P}\left(\mathcal{W}_{1}>A_{1}\right)+\sum_{k=1}^{\infty}\mathbb{P}\left(\mathcal{W}_{k+1}>A_{k+1}\mathrel{}\middle|\mathrel{}\mathcal{W}_{k}\leq
A_{k}\right).$

From Lemma 3.3 we get that

$\displaystyle\mathbb{P}\left(\mathcal{W}_{k+1}>A_{k+1}\mathrel{}\middle|\mathrel{}\mathcal{W}_{k}\leq
A_{k}\right)\leq\mathbb{P}\left(M^{\mathscr{Q}}(\mathopen{(}k\mathclose{}\mathpunct{},k+1\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},A_{k+1}\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)})>(A_{k+1}-A_{k})\right).$

Using a Chernoff bound and Lemma 3.3, for any $k\geq 1$ and $\lambda<k^{2}/2$,

$\displaystyle\mathbb{P}\left(M^{\mathscr{Q}}(\mathopen{(}k\mathclose{}\mathpunct{},k+1\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},A_{k+1}\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)})>(A_{k+1}-A_{k})\right)$
$\displaystyle\leq\exp\left(-\lambda(A_{k+1}-A_{k})+A_{k+1}\cdot\int_{k}^{k+1}F(\lambda,t)\differential{t}\right)$
$\displaystyle\leq\exp\left(-\lambda(A_{k+1}-A_{k})+A_{k+1}\cdot\left(\lambda\log(1+\frac{1}{k})+O\left(\frac{\lambda^{2}}{k^{3}}\right)\right)\right).$
(25)

For our purposes, we fix some $A>0$ and use the sequence $(A_{k})$ defined in
such a way that

$\displaystyle A_{1}=A\qquad\text{and}\qquad
A_{k+1}=A_{k}\left(1+\frac{1}{k}+\frac{1}{k^{\frac{5}{4}}}\right).$

From this, remark that we have $A\cdot k\leq A_{k}\leq C\cdot A\cdot k$ for
all $k\geq 1$ for some constant $C$ and then

$\displaystyle
A_{k+1}-A_{k}=A_{k}\cdot\left(\frac{1}{k}+\frac{1}{k^{\frac{5}{4}}}\right)\geq\frac{A_{k}}{k}+\frac{A}{k^{\frac{1}{4}}}\qquad\text{and}\quad\frac{A_{k+1}}{k}\leq\frac{A_{k}}{k}+\frac{C^{\prime}}{k},$

for some constant $C^{\prime}$. Using the above and writing
$\log(1+\frac{1}{k})=\frac{1}{k}+O\left(\frac{1}{k^{2}}\right)$ we can bound
the term that appears in the exponential in (3.3) as

$\displaystyle-\lambda(A_{k+1}-A_{k})+A_{k+1}\cdot\left(\lambda\log(1+\frac{1}{k})+O\left(\frac{\lambda^{2}}{k^{3}}\right)\right)$
$\displaystyle\leq-\lambda\left(\frac{A_{k}}{k}+\frac{A}{k^{\frac{1}{4}}}\right)+\lambda
A_{k+1}\cdot\left(\frac{1}{k}+O\left(\frac{1}{k^{2}}\right)\right)+O\left(A_{k+1}\cdot\frac{\lambda^{2}}{k^{3}}\right)$
$\displaystyle\leq\lambda\cdot\left(-\frac{A}{k^{\frac{1}{4}}}+O\left(\frac{1}{k}\right)\right)+O\left(\frac{\lambda^{2}}{k^{3}}\right).$

Now taking $\lambda=k^{\frac{5}{4}}$ and $A$ large enough, the bounds (3.3)
are summable in $k$ and the obtained sum is exponentially small in $A$. This
proves that with complementary probability we have $\mathcal{W}_{k}\leq
A_{k}\leq C\cdot A\cdot k$ for all $k\geq 1$; otherwise stated, the quantity

$\displaystyle\mathbb{P}\left(\sup_{k\geq 1}\frac{\mathcal{W}_{k}}{k}\geq
x\right)$

decays exponentially in $x$. The analogous result where the supremum is taken
on all the real numbers $t\geq 1$ follows from the monotonicity of
$\mathcal{W}$. ∎

###### Proof of Proposition 3.1.(iii).

First, we use a union bound to write

$\displaystyle\mathbb{P}\left(Z_{2}\geq A\right)\geq\mathbb{P}\left(Z_{1}\leq
A,\ Z_{2}\geq A\right)+\mathbb{P}\left(Z_{1}>A\right).$

Then

$\displaystyle\mathbb{P}\left(Z_{1}\leq A,\ Z_{2}\geq A\right)$
$\displaystyle\leq\mathbb{P}\left(\mathscr{Q}\cap\left\\{(m,y,t)\mathrel{}\middle|\mathrel{}y\geq
A(1\vee\log^{2}t)/t^{2},\ m\leq At,\ t\geq 1\right\\}\neq\emptyset\right)$
$\displaystyle=1-\exp\left(-\int_{1}^{\infty}\differential
t\int_{A(1\vee\log^{2}t)/t^{2}}^{\infty}\differential
y\int_{0}^{At}\differential
m\frac{\exp\left(-\frac{t^{2}}{2}y\right)}{y^{\frac{3}{2}}\sqrt{2\pi}}\right)$
$\displaystyle\leq 1-\exp\left(-\int_{1}^{\infty}\differential
t\frac{At}{\sqrt{2\pi}}\exp\left(-\frac{A}{2}(1\vee\log^{2}t)\right)\int_{A(1\vee\log^{2}t)/t^{2}}^{\infty}\frac{1}{y^{\frac{3}{2}}}\differential
y\right)$ $\displaystyle\leq 1-\exp\left(-\int_{1}^{\infty}\differential
t\frac{2At^{2}}{(1\vee\log^{2}t)\sqrt{2\pi}}\exp\left(-\frac{A}{2}(1\vee\log^{2}t)\right)\right)$
$\displaystyle\leq\exp(-cA),$

for some constant $c>0$. Since the second term
$\mathbb{P}\left(Z_{1}>A\right)$ also decays exponentially in $A$ thanks to
Proposition 3.1.(ii), this concludes the proof. ∎

###### Proof of Proposition 3.1.(iv).

We write

$\displaystyle\mathbb{E}\left[\mathbf{1}_{\\{Z_{1}\leq A\\}}\sum_{t<s\leq
2t}w_{s}\mathbf{1}_{\\{w_{s}\leq s^{-2-\delta}\\}}\right]$
$\displaystyle\leq\mathbb{E}\left[M^{\mathscr{Q}}(\mathopen{(}t\mathclose{}\mathpunct{},2t\mathclose{]}\times\mathopen{(}0\mathclose{}\mathpunct{},2At\mathclose{]}\times\mathopen{[}0\mathclose{}\mathpunct{},t^{-2-\delta}\mathclose{]})\right]$
$\displaystyle\leq\int_{t}^{2t}\differential s\int_{0}^{2At}\differential
m\int_{0}^{t^{-2-\delta}}y\frac{\exp(-\frac{s^{2}}{2}y)}{\sqrt{2\pi}y^{3/2}}\differential
y$
$\displaystyle\leq\operatorname{cst}At\int_{t}^{2t}t^{-1-\delta/2}\differential
s$ $\displaystyle\leq\operatorname{cst}At^{1-\delta/2}.$

Using Markov’s inequality, summing over a sequence $t_{n}=2^{n}$ and using the
Borel-Cantelli lemma we get that almost surely for any choice of $A$ we have

$\displaystyle\mathbf{1}_{\\{Z_{1}\leq A\\}}\sum_{1<s\leq
t}w_{s}\mathbf{1}_{\\{w_{s}\leq s^{-2-\delta}\\}}=O(t^{1-\delta/4})$

This proves the claim because the random variable $Z_{1}$ only finite values
almost surely. ∎

###  3.4 Martingale associated to the weight process

We prove the following lemma, which directly ensures that Proposition 3.1.(i)
holds.

######  Lemma .

The process $\left(\frac{\mathcal{W}_{t}}{t}\right)_{t\geq 1}$ is a positive
$L^{2}$ martingale. We have

$\displaystyle\frac{\mathcal{W}_{t}}{t}\underset{t\rightarrow\infty}{\rightarrow}Y,$
(26)

almost surely and in $L^{2}$. The limiting random variable $Y$ is positive
almost surely.

###### Proof of Lemma 3.4.

The proof is made of two parts: a first and a second moment computation.
First moment computation. Using that the expression (21) with $F(x)=x$ is a
martingale we get for any $t\geq s$,

$\displaystyle\mathbb{E}\left[\mathcal{W}_{t}-\mathcal{W}_{s}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]=\mathbb{E}\left[\int_{s}^{t}\mathcal{W}_{u-}\cdot\int_{0}^{\infty}v\cdot\frac{\exp\left(-\frac{u^{2}}{2}v\right)}{v^{\frac{3}{2}}\sqrt{2\pi}}\differential
v\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]=\int_{s}^{t}\frac{\mathbb{E}\left[\mathcal{W}_{u}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]}{u}\differential
u,$

where we used (18) in the second equality. Taking derivatives, we get that
$\frac{\differential}{\differential
t}\mathbb{E}\left[\mathcal{W}_{t}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]=\frac{\mathbb{E}\left[\mathcal{W}_{t}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]}{t}$,
so that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\mathbb{E}\left[\mathcal{W}_{t}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]}{t}\right)=\frac{1}{t^{2}}\cdot\left(\frac{\mathbb{E}\left[\mathcal{W}_{t}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]}{t}\cdot
t-\mathbb{E}\left[\mathcal{W}_{t}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]\right)=0.$

This ensures in that for any $t\geq s\geq 1$ we have
$\mathbb{E}\left[\frac{\mathcal{W}_{t}}{t}\mathrel{}\middle|\mathrel{}\mathcal{W}_{s}\right]=\frac{\mathcal{W}_{s}}{s}$,
so the the process $(\frac{\mathcal{W}_{t}}{t})_{t\geq t}$ is a martingale.
Being a positive martingale, it converges almost surely to a limit $Y$.
Second moment computation. Using that the expression (21) with $F(x)=x^{2}$ is
a martingale we get that

$\displaystyle\mathbb{E}\left[\mathcal{W}_{t}^{2}-\mathcal{W}_{s}^{2}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]$
$\displaystyle=\mathbb{E}\left[\int_{s}^{t}\differential
u\int_{0}^{\infty}(2y\mathcal{W}_{u-}+y^{2})(\mathcal{W}_{u-})\frac{\exp\left(-\frac{u^{2}}{2}y\right)}{y^{\frac{3}{2}}\sqrt{2\pi}}\mathrm{d}y\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]$
$\displaystyle=\int_{s}^{t}\left(2\mathbb{E}\left[\mathcal{W}_{u}^{2}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]\cdot\frac{1}{u}+\mathbb{E}\left[\mathcal{W}_{u}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]\cdot\frac{1}{u^{3}}\right)\differential
u$

so that $\frac{\differential}{\differential
t}\mathbb{E}\left[\mathcal{W}_{t}^{2}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]=2\mathbb{E}\left[\mathcal{W}_{t}^{2}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]\cdot\frac{1}{t}+\mathbb{E}\left[\mathcal{W}_{t}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]\cdot\frac{1}{t^{3}}$
and

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathbb{E}\left[\frac{\mathcal{W}_{t}^{2}}{t^{2}}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]$
$\displaystyle=\frac{1}{t^{2}}\left(2\mathbb{E}\left[\mathcal{W}_{t}^{2}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]\cdot\frac{1}{t}+\mathbb{E}\left[\mathcal{W}_{t}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]\cdot\frac{1}{t^{3}}\right)-\frac{2}{t^{3}}\mathbb{E}\left[\mathcal{W}_{t}^{2}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]$
$\displaystyle=\mathbb{E}\left[\mathcal{W}_{t}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]\cdot\frac{1}{t^{5}}$
$\displaystyle=\frac{\mathcal{W}_{s}}{s}\cdot\frac{1}{t^{4}}$

And so integrating between $s$ and $t$, we get

$\displaystyle\mathbb{E}\left[\frac{\mathcal{W}_{t}^{2}}{t^{2}}\mathrel{}\middle|\mathrel{}\mathcal{F}_{s}\right]=\frac{\mathcal{W}_{s}^{2}}{s^{2}}+\frac{\mathcal{W}_{s}}{s}\left(\frac{1}{3s^{3}}-\frac{1}{3t^{3}}\right)\underset{t\rightarrow\infty}{\rightarrow}\frac{\mathcal{W}_{s}^{2}}{s^{2}}+\frac{\mathcal{W}_{s}}{3s^{4}}.$

The process $\left(\frac{\mathcal{W}_{t}}{t},\ t\geq 1\right)$ is a positive
martingale that is bounded in $L^{2}$ and hence converges in $L^{2}$ to the
non-negative limit $Y$. From what we computed above, we have

$\displaystyle\mathbb{E}\left[Y\mathrel{}\middle|\mathrel{}\mathcal{W}_{t}\right]=\frac{\mathcal{W}_{t}}{t}\quad\text{
and
}\quad\mathrm{Var}\left(Y\mathrel{}\middle|\mathrel{}\mathcal{W}_{t}\right)=\frac{\mathcal{W}_{t}}{3t^{4}}.$

Then using the Chebychev inequality and the fact that $\mathcal{W}_{t}$ is
non-decreasing in $t$ almost surely we get

$\displaystyle\mathbb{P}\left(Y=0\mathrel{}\middle|\mathrel{}\mathcal{W}_{t}\right)\leq\frac{\mathrm{Var}\left(Y\mathrel{}\middle|\mathrel{}\mathcal{W}_{t}\right)}{\mathbb{E}\left[Y\mathrel{}\middle|\mathrel{}\mathcal{W}_{t}\right]^{2}}\leq\frac{1}{3t^{3}\mathcal{W}_{t}}$

which decreases to $0$ almost surely as $t\rightarrow\infty$. Hence writing
$\mathbb{P}\left(Y=0\right)=\mathbb{E}\left[\mathbb{P}\left(Y=0\mathrel{}\middle|\mathrel{}\mathcal{W}_{t}\right)\right]$
and using monotone convergence we get that $\mathbb{P}\left(Y=0\right)=0$. ∎

Note that it would be possible to do the same computation we $F(x)=x^{k}$ to
obtain information about the value of higher moments of $Y$ recursively.

##  4 The root component of the WMSF of the PWIT and its scaling limit

In this section, we present in detail the construction of the object
$\mathrm{M}^{\infty}$ that appears in Theorem 1 as the local weak limit of
$\mathrm{M}_{n}$. We then interpret this construction in the framework of
chains and aggregation processes, in order to prove the convergence of this
object to $\mathcal{M}^{\infty}$, which is the content of Theorem 1.2.

Table 2: Table of Notation for Section 4 About Poisson–Galton–Watson trees
---
$\operatorname{PGW}(\lambda)$ | : |  Poisson–Galton–Watson distribution on trees, with parameter $\lambda$
$\absolutevalue{\operatorname{PGW}(\lambda)}$ | : |  by abuse of notation, a random variable with the distribution of the total number of vertices of a Poisson–Galton–Watson tree with parameter $\lambda$
$\theta(\lambda)$ | : |  the probability $\mathbb{P}\left(\absolutevalue{\operatorname{PGW}(\lambda)}=\infty\right)$
$\lambda^{*}$ | : |  the dual parameter associated to $\lambda>1$, defined by the relation $\lambda e^{-\lambda}=\lambda^{*}e^{-\lambda^{*}}$
Objects related to the structure of $\mathrm{M}^{\infty}$
$\alpha_{1},\alpha_{2},\dots$ | : |  forward maximal weights in decreasing order
$\mathscr{A}$ | : |  set of forward maximal weights
$\mathrm{T}_{\alpha}^{\bullet}$ | : |  pond located just before the forward maximal edge of weight $\alpha$
$\mathrm{M}_{\alpha}^{\bullet}$ | : |  obtained as the result of the Poisson–Galton–Watson Aggregation Process (PGWAP) started from $\mathrm{T}_{\alpha}^{\bullet}$
$(W^{\alpha}(s))_{s\geq\alpha}$ | : |  process counting the number of vertices during the PGWAP started from $\mathrm{T}_{\alpha}^{\bullet}$
$(W(s))_{s\geq\alpha}$ | : |  Markov process with the same transitions as $W^{\alpha}$
$X_{\alpha}$ | : |  number of vertices of $\mathrm{T}_{\alpha}^{\bullet}$ so that $W^{\alpha}(\alpha)=X_{\alpha}$
$W^{\alpha}(\infty)$ | : |  (finite) limiting value of $(W^{\alpha}(s))_{s\geq\alpha}$, it is the number of vertices in $\mathrm{M}_{\alpha}^{\bullet}$
$\hat{T}$ | : |  version of a tree $T$ endowed with a phantom root
$\hat{\mathrm{M}}^{\bullet}_{\alpha}$ | : |  version of $\mathrm{M}^{\bullet}_{\alpha}$ with an added phantom root
$\widetilde{\mathrm{M}}^{\bullet}_{\alpha}$ | : |  version of $\hat{\mathrm{M}}^{\bullet}_{\alpha}$ with normalized measure
$\eta^{\mathrm{Poi}}$ | : |  family $(\eta^{\mathrm{Poi}}(t,w),\ t>0,w>0)$ of distributions where a tree under $\eta^{\mathrm{Poi}}(t,w)$ is distributed as $\hat{T}$ where $T$ is a Poisson-Galton-Watson tree conditioned on having exactly $\lfloor w\rfloor$ vertices, endowed with a phantom root.
$L_{k}$ | : |  type of the $k$-th vertex along the spine of $\mathrm{T}^{\infty}$
Scaled versions of the discrete objects and their corresponding limits
$\mathscr{A}^{r}$ | : |  rescaled version $\left\\{r^{-1}\cdot(\alpha-1)\mathrel{}\middle|\mathrel{}\alpha\in\mathscr{A}\right\\}$, tending to $\mathscr{P}$
$\mathscr{P}$ | : |  Poisson Point Process of intensity $\frac{\differential t}{t}$ on $\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$
$\left(\mathsf{W}^{r}(t)\right)_{t\geq 1}$ | : |  defined as $\left(r^{2}\cdot W^{1+r}(1+rt)\right)_{t\geq 1}$
$(\mathcal{W}_{t})_{t\geq 1}$ | : |  weight process used in the construction of $\mathcal{M}^{\bullet}$
$\mathbf{A}^{r}$ | : |  generator of the process $\left((t,\mathsf{W}^{r}(t))\right)_{t\geq 1}$
$\mathbf{A}$ | : |  generator of the process $\left((t,\mathcal{W}(t))\right)_{t\geq 1}$
$\mathsf{W}^{r}(1)$ | : |  starting value of $\left(\mathsf{W}^{r}(t)\right)_{t\geq 1}$, also expressed as $r^{-2}\cdot X_{1+r}$
$\mathcal{W}_{1}$ | : |  starting value of $\left(\mathcal{W}_{t}\right)_{t\geq 1}$, also expressed as $X$
$r\cdot C_{0}^{-1}\cdot\mathsf{W}^{r}(\infty)$ | : |  scaled version of the end-value of the process $\left(\mathsf{W}^{r}(t)\right)_{t\geq 1}$
$Y$ | : |  defined as $\lim_{t\rightarrow\infty}\frac{\mathcal{W}_{t}}{t}$
$\eta^{\mathrm{Poi},r}$ | : |  defined from $\eta^{\mathrm{Poi}}$, where a tree sampled under $\eta^{\mathrm{Poi},r}(t,w)$ is distributed as $\mathrm{Scale}(r,r^{2};\hat{T})$, where $\hat{T}\sim\eta^{\mathrm{Poi}}(t,w)$.
$\eta^{\mathrm{Br}}$ | : |  family $(\eta^{\mathrm{Br}}(t,w),\ t>0,w>0)$ of distributions where a tree under $\eta^{\mathrm{Br}}(t,w)$ is a Brownian tree of mass $w$

###  4.1 Description of the root component of the WMSF of the PWIT

Recall the definition of $\mathrm{M}^{\infty}$ from Section 1.1 in the
introduction. In the tree $\mathrm{M}^{\infty}$, there is a unique infinite
path starting from the root. For every vertex $v$ we can define its _type_ as
the maximal weight of edges on the path between $v$ and $\infty$. This is
always a number in the interval
$\mathopen{(}1\mathclose{}\mathpunct{},\infty\mathclose{)}$. In what follows,
we describe the tree $\mathrm{M}^{\infty}$ as a rooted tree whose vertices
have a type and forget about the weights of the edges and the planar
structure. First, we have to recall a few fact about Poisson–Galton–Watson
trees and their probability of being infinite.

####  4.1.1 Around the probability of extinction of Poisson–Galton–Watson
trees

For any $\lambda\geq 1$ we write
$\theta(\lambda)=\mathbb{P}\left(\absolutevalue{\operatorname{PGW}(\lambda)}=\infty\right)$.
The dual parameter $\lambda^{*}$ is such that $\lambda
e^{-\lambda}=\lambda^{*}e^{-\lambda^{*}}$. We state a few useful facts about
those quantities:

  * •

The value $\theta=\theta(\lambda)$ is defined by the equation
$1-\theta=e^{\lambda\theta}$.

  * •

For any $\lambda>1$ we have $\lambda^{*}=\lambda(1-\theta(\lambda))$.

  * •

For any $\lambda>1$ we have

$\displaystyle\theta^{\prime}(\lambda)=\frac{\theta(\lambda)(1-\theta(\lambda))}{1-\lambda^{*}}\qquad\text{and}\qquad\theta^{\prime}(\lambda)\underset{\lambda\rightarrow
1}{\rightarrow}2.$ (27)

  * •

For $\epsilon>0$ we have

$\displaystyle(1+\epsilon)^{*}=1-\epsilon+\frac{2}{3}\epsilon^{2}+O(\epsilon^{3}).$
(28)

  * •

For any $\lambda>1$ we have

$\displaystyle\frac{1-\theta(\lambda)}{1-\lambda^{*}}\leq\frac{1}{\lambda-1}.$
(29)

  * •

For any $\lambda>1$ we have

$\displaystyle(\lambda-1)\frac{1-\theta(\lambda)}{(1-\lambda^{*})^{3}}\leq\frac{1}{(\lambda-1)^{2}}$
(30)

The first four facts can be found in [25] and [31, Corollary 3.19]. We only
prove (29), since (30) is obtained is a very similar way.

###### Proof of (29).

For $\theta=\theta(\lambda)$, since $1-\theta=e^{\lambda\theta}$ we have
$\lambda=\frac{\log(1-\theta)}{\theta}$. Then since
$\lambda^{*}=\lambda(1-\theta)$ we have
$\lambda^{*}=\frac{(1-\theta)\log(1-\theta)}{\theta}$. Proving the inequality
(29) can then be reduced to proving

$\displaystyle\frac{1-\theta}{1-\frac{(1-\theta)\log(1-\theta)}{\theta}}\leq\frac{1}{\frac{\log(1-\theta)}{\theta}-1},$

which in turns corresponds to

$\displaystyle 2(1-\theta)\log(1-\theta)-2\theta+\theta^{2}\leq 0.$

Now we can just check that the right-hand-side is $0$ for $\theta=0$ and the
derivative in $\theta$ is negative for
$\theta\in\mathopen{(}0\mathclose{}\mathpunct{},1\mathclose{)}$. This
concludes the proof. ∎

####  4.1.2 Distribution of $(\mathrm{T}^{\infty},\mathrm{M}^{\infty})$.

As explained in the introduction, we can consider the sequence
$e_{1},e_{2},\dots$ of forward maximal edges in the invasion percolation
process run from the root $\rho$ of the PWIT, and their corresponding weights
$\alpha_{1}>\alpha_{2}>\dots$. Writing
$\mathscr{A}:=\\{\alpha_{1},\alpha_{2},\dots\\}$, if we disconnect
$\mathrm{T}^{\infty}$ (resp. $\mathrm{M}^{\infty})$ by removing all the edges
$e_{1},e_{2},\dots$, we get a sequence of connected component that we call
$\mathrm{T}_{\alpha}^{\bullet}$ (resp. $\mathrm{M}_{\alpha}^{\bullet}$),
indexed by $\alpha\in\mathscr{A}$. Those components are rooted at their lowest
vertex $\rho_{\alpha}$ and pointed at their highest vertex $v_{\alpha}$. Since
the whole structure of $\mathrm{T}^{\infty}$ (resp. $\mathrm{M}^{\infty})$ can
be recovered from the data of $\mathscr{A}$ and
$(\mathrm{T}_{\alpha}^{\bullet})_{\alpha\in\mathscr{A}}$ (resp.
$(\mathrm{M}_{\alpha}^{\bullet})_{\alpha\in\mathscr{A}}$), we just need to
express their joint distribution. From [2, Section 2.2], we get that:

  * •

The sequence $(\alpha_{k})_{k\geq 1}$ is a Markov process in $k$, where

    * –

$\alpha_{1}$ has density $\theta^{\prime}(y)\differential y$ on
$\mathopen{(}1\mathclose{}\mathpunct{},\infty\mathclose{)}$,

    * –

for any $k\geq 1$, conditionally on $\alpha_{k}$, the random variable
$\alpha_{k+1}$ has density
$\frac{\theta^{\prime}(y)}{\theta(\alpha_{k})}\mathbf{1}_{\\{y<\alpha_{k}\\}}\differential
y$.

  * •

Conditionally on $\mathscr{A}$, the
$((\mathrm{T}_{\alpha}^{\bullet},\mathrm{M}_{\alpha}^{\bullet}),\
\alpha\in\mathscr{A})$ are independent and

    * –

The total number $X_{\alpha}$ of vertices of $\mathrm{T}_{\alpha}^{\bullet}$
has the Borel–Tanner distribution with parameter $\alpha$ where for $m\geq 1$,

$\displaystyle\mathbb{P}\left(X_{\alpha}=m\right)=\frac{\theta(\alpha)}{\theta^{\prime}(\alpha)}\frac{e^{-\alpha
m}(\alpha m)^{m-1}}{(m-1)!},$

see [2, Eq. (2.2)].

    * –

Conditionally on $X_{\alpha}$, the tree $\mathrm{T}_{\alpha}^{\bullet}$ has
the distribution of a PGW tree conditioned on having $X_{\alpha}$
vertices222Note that the distribution of a tree under
$\operatorname{PGW}(\lambda)$, conditioned on its number of vertices, does not
depend on $\lambda$., pointed at a uniform random vertex.

    * –

Conditionally on $\mathrm{T}_{\alpha}^{\bullet}$, the tree
$\mathrm{M}_{\alpha}^{\bullet}$ is obtained by running the
Poisson–Galton–Watson Aggregation Process from the seed
$\mathrm{T}_{\alpha}^{\bullet}$.

##### The Poisson–Galton–Watson Aggregation Process.

We recall from [2, Section 2.2] the definition of the Poisson–Galton–Watson
Aggregation Process. We define the process $(A(s))_{s\geq\alpha}$ started from
some time $\alpha>1$ with some finite tree $A(\alpha)$.

At any time $s\geq\alpha$, for any vertex $v$ of $A(s)$, Poisson–Galton–Watson
trees attach themselves at $v$ by having their root linked to $v$ by an edge;
at time $s$, attachments at every vertex occur at rate $(1-\theta(s))$. A tree
attaching to $v$ at time $\lambda$ is distributed as
$\operatorname{PGW}(\lambda^{*})$, independently of everything else. This
process stabilizes after a finite amount of time to some finite tree that we
denote by $A(\infty)$. In the above description, every
$\mathrm{M}^{\bullet}_{\alpha}$ is obtained from the corresponding
$\mathrm{T}^{\bullet}_{\alpha}$ by running a PGW process started at time
$\alpha$ from $A(\alpha)=\mathrm{T}^{\bullet}_{\alpha}$.

##### Interpretation in the $\mathrm{Aggreg}$ construction.

Start the PGWA process at time $\alpha$ as defined above, from the seed tree
$\mathrm{T}^{\bullet}_{\alpha}$. We introduce the weight process
$(W^{\alpha}(s))_{s\geq\alpha}$ as

$W^{\alpha}(s):=\\#\\{\text{vertices in the aggregate at time }s\\}.$

Note that the starting value $W^{\alpha}(\alpha)$ is equal to $X_{\alpha}$,
the number of vertices in the tree $\mathrm{T}^{\bullet}_{\alpha}$. Now for
any $s$ we denote by $w_{s}=W^{\alpha}(s)-W^{\alpha}(s-)$ and we consider the
jump times of this process
$J^{\alpha}=\left\\{s\geq\alpha\mathrel{}\middle|\mathrel{}w_{s}\neq
0\right\\}$. Each of those jumps corresponds to the attachment of a tree to
the aggregate. For any $s\in J^{\alpha}$, we call $T_{s}$ the corresponding
tree. This tree is considered as rooted at the closest vertex to the seed
graph $\mathrm{T}^{\bullet}_{\alpha}$. The total number of vertices of that
tree is given by the size of the jump $w_{s}$, and conditionally on $w_{s}$,
its distribution is that of a $\operatorname{PGW}$-tree conditioned to have
that number of vertices.

Then, conditionally on $(W^{\alpha}(s))_{s\geq\alpha}$, and
$\mathrm{T}^{\bullet}_{\alpha}$ and $(T_{s},\ s\in J^{\alpha})$, all these
trees are glued on each other. From the definition of the process, a rooted
tree $T_{s}$ corresponding to a jump at time $s$ is attached on a uniform
vertex $x_{s}$ on the aggregate at time $s-$ by adding an edge between its
root and $x_{s}$. In order to be in the context of the aggregation process of
Section 2.3, we introduce a slight modification in the way we express this
attachment procedure: for every rooted tree $T$, we define the tree $\hat{T}$
as the tree $T$ with the addition of a parent to the root of $T$, which is
then the root of $\hat{T}$ (which we call the phantom root). When we consider
those trees as endowed with the counting measure, the phantom root has zero
mass. Whenever a gluing happens at time $s$, we can now express that as having
the phantom root of the tree $\hat{T}_{s}$ identified to vertex $x_{s}$
uniformly chosen on the aggregate.

In the end, we can describe this process as an aggregation process in the
sense of Section 2.3, with

  * •

weight process $(W^{\alpha}(s))_{s\geq\alpha}$,

  * •

seed metric space $\mathrm{T}^{\bullet}_{\alpha}$,

  * •

block distributions $\eta^{\mathrm{Poi}}(t,w)$ where for any $t>1$, a tree
under $\eta^{\mathrm{Poi}}(t,w)$ is distributed as a Poisson-Galton-Watson
tree conditioned on having exactly $\lfloor w\rfloor$ vertices, endowed with a
phantom root (and a single point with no mass if $\lfloor w\rfloor=0$).

Additionally, the process $(W^{\alpha}(s))_{s\geq\alpha}$ is Markovian and we
can express its transitions explicitly. This will be done in more details in
Section 4.5.1. The process $(W^{\alpha}(s))_{s\geq\alpha}$ almost surely
stabilizes to some value as $s\rightarrow\infty$ and we write
$W^{\alpha}(\infty):=\lim_{s\rightarrow\infty}W^{\alpha}(s)$. According to
this description we can write $\hat{\mathrm{M}}^{\bullet}_{\alpha}$ as the
result of a PGWA process started from the seed tree
$\hat{\mathrm{T}}^{\bullet}_{\alpha}$ so that

$\displaystyle\hat{\mathrm{M}}^{\bullet}_{\alpha}:=\mathrm{Scale}\left(1,W^{\alpha}(\infty);\widetilde{\mathrm{M}}^{\bullet}_{\alpha}\right),$
(31)

where

$\displaystyle\widetilde{\mathrm{M}}^{\bullet}_{\alpha}:=\mathrm{Aggreg}((W^{\alpha}(s))_{s\geq\alpha}\
,\hat{\mathrm{T}}^{\bullet}_{\alpha}\ ,\eta^{\mathrm{Poi}}),$ (32)

where the scaling part of the expression in (31) just appears from the fact
that in our definition of the aggregation process, the measure carried by the
aggregate is normalized to a probability measure.

####  4.1.3 Alternative description of the process

##### Another description of the distribution of
$\mathrm{T}_{\alpha}^{\bullet}$.

Another way of constructing the tree $\mathrm{T}^{\bullet}_{\alpha}$ for a
given $\alpha$ is to first construct a spine of length given by a
$\mathrm{Geo}(\alpha^{*})$-distributed random variable linking the root
$\rho_{\alpha}$ to the marked point $v_{\alpha}$, and to let every vertex on
the spine have a progeny off the spine distributed as independent
$\mathrm{PGW}(\alpha^{*})$ random trees. This is the content of [5, Theorem
30, Theorem 31].

##### The types along the spine.

The previous paragraph together with the fact that the sequence of weights
$(\alpha_{k})_{k\geq 1}$ of the forward maximal edges has an explicit
description given in Section 4.1.2 allows to give an explicit description of
the sequence $(L_{k})_{k\geq 0}$ that describes the type of the vertex at
height $k$ along the spine of the tree. This process admits a scaling limit,
this is studied in Section 4.3.

##### Description as a branching process with types.

We can describe the trees $\mathrm{T}^{\infty}$ and $\mathrm{M}^{\infty}$
using a branching process in discrete time. This branching process considers
individuals with types in
$\mathopen{(}1\mathclose{}\mathpunct{},\infty\mathclose{)}$. Additionally,
individuals can be normal or special and there are two types of reproduction
events that happen simultaneously: giving birth to individuals of smaller or
equal type, and adopting individuals of larger type.

  * •

A normal individual of type $\alpha$

    * –

gives birth to a Poisson number $\operatorname{Poi}(\alpha^{*})$ of normal
individuals of type $\alpha$,

    * –

adopts normal individuals of larger type that are given by the points of a
Poisson process with intensity
$(1-\theta(y))\mathbf{1}_{\\{y>\alpha\\}}\mathrm{d}y$

  * •

A special individual of type $\alpha$

    * –

gives birth to and adopts normal individuals with the same distribution as
normal individuals do,

    * –

gives birth to exactly one special individual. This special child has the same
type $\alpha$ as the parent with probability $\alpha^{*}$ and has a different
type with probability $(1-\alpha^{*})$. In that case its type is taken in the
interval $\mathopen{(}1\mathclose{}\mathpunct{},\alpha\mathclose{)}$ with a
distribution that has a density given by
$\mathbf{1}_{\mathopen{(}1\mathclose{}\mathpunct{},\alpha\mathclose{)}}(y)\frac{\theta^{\prime}(y)}{\theta(\alpha)}\mathrm{d}y$.

The tree $\mathrm{M}^{\infty}$, seen as a rooted tree with types, has the same
distribution as the genealogy of this process started with one special
individual with type taken under the distribution with density
$\theta^{\prime}(y)\mathrm{d}y$ on the interval
$\mathopen{(}1\mathclose{}\mathpunct{},\infty\mathclose{)}$. The tree
$\mathrm{T}^{\infty}\subset\mathrm{M}^{\infty}$ corresponds to the tree
obtained by removing the adoption mechanism in this process.

###  4.2 Analogy between the discrete construction and the continuous one: an
outline of the proof of Theorem 1.2

From the description of $\mathrm{M}^{\infty}$, we have

$\displaystyle\hat{\mathrm{M}}^{\infty}=\mathrm{Chain}\left(\hat{\mathrm{M}}_{\alpha}^{\bullet},\
\alpha\in\mathscr{A}\right),$ (33)

where we use to phantom root construction as in the previous section: for a
discrete rooted tree $T$ (seen as a measured metric space), we define
$\hat{T}$ as the tree $T$ to which we add a parent $\hat{\rho}$ of the root
$\rho$ of $T$. The measure associated to $\hat{T}$ does not charge
$\hat{\rho}$ and $\hat{T}$ is rooted at $\hat{\rho}$. We can easily check that
the assumptions (Ch1), (Ch2) and (Ch3) hold here so that the construction (33)
makes sense.

We want to use the result of Proposition 2.2.2, which allows to have the
convergence of metric spaces defined as chains to spaces of the same type. For
that we write

$\displaystyle\mathscr{A}^{r}:=\left\\{r^{-1}\cdot(\alpha-1)\mathrel{}\middle|\mathrel{}\alpha\in\mathscr{A}\right\\}$

so that

$\displaystyle\mathrm{Scale}\left(r,\frac{r^{3}}{C_{0}};\hat{\mathrm{M}}^{\infty}\right)$
$\displaystyle=\mathrm{Scale}\left(r,\frac{r^{3}}{C_{0}};\mathrm{Chain}\left(\hat{\mathrm{M}}_{\alpha}^{\bullet},\
\alpha\in\mathscr{A}\right)\right)$
$\displaystyle=\mathrm{Chain}\left(\mathrm{Scale}\left(r,\frac{r^{3}}{C_{0}};\hat{\mathrm{M}}_{1+tr}^{\bullet}\right),\
t\in\mathscr{A}^{r}\right).$ (34)

From this description, recalling the analogous definition of the continuous
object

$\displaystyle\mathcal{M}^{\infty}=\mathrm{Chain}\left(\mathcal{M}_{x}^{\bullet},\
x\in\mathscr{P}\right),$

we want to check that the assumptions of Proposition 2.2.2 are satisfied. For
that, in order to check that (ChConv1) is satisfied we need to prove the
following:

  1. (a)

The point process $\mathscr{A}^{r}$ converges vaguely towards to a Poisson
point process with intensity measure $\frac{\differential t}{t}$ on
$\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$, this will be the
content of Lemma 4.3.

  2. (b)

We have the convergence

$\displaystyle\mathrm{Scale}(r,C_{0}^{-1}\cdot
r^{3};\hat{\mathrm{M}}_{1+r}^{\bullet})\rightarrow\mathcal{M}^{\bullet}$

in distribution as $r\rightarrow 0$, this will be the content of Proposition
4.6.

In order to prove this convergence, we recall the description (32) and rescale
things so that

$\displaystyle\mathrm{Scale}\left(r,1;\widetilde{\mathrm{M}}^{\bullet}_{1+r}\right)=\mathrm{Aggreg}((r^{-2}\cdot
W^{1+r}(1+rt))_{t\geq 1}\
,\mathrm{Scale}(r,r^{2};\hat{\mathrm{T}}^{\bullet}_{1+r})\
,\eta^{\mathrm{Poi},r}).$

where a tree sampled according to $\eta^{\mathrm{Poi},r}(t,w)$ has the
distribution of $\mathrm{Scale}\left(r,r^{2};\hat{\mathrm{T}}_{\lfloor
wr^{-2}\rfloor}\right)$ where $\mathrm{T}_{k}$ is a PGW tree conditioned to
having exactly $k$ vertices. Recall the definition of
$\widetilde{\mathcal{M}}$ as

$\displaystyle\widetilde{\mathcal{M}}^{\bullet}:=\mathrm{Aggreg}((\mathcal{W}_{t})_{t\geq
1},\mathcal{T}^{\bullet},\eta^{\mathrm{Br}}).$

Then the convergence of
$\mathrm{Scale}\left(r,1;\widetilde{\mathrm{M}}^{\bullet}_{1+r}\right)$ to
$\widetilde{\mathcal{M}}$ consists in applying Proposition 2.3.3 for which
three main ingredients are required:

     * •

convergence of the seed
$\mathrm{Scale}(r,r^{2};\hat{\mathrm{T}}^{\bullet}_{1+r})\rightarrow\mathcal{T}^{\bullet}$,
this will be done in Section 4.4,

     * •

convergence of the weight process $(r^{-2}\cdot W^{1+r}(1+rt))_{t\geq
1}\rightarrow(\mathcal{W}_{t})_{t\geq 1}$,

     * •

convergence of $\eta^{\mathrm{Poi},r}$ towards $\eta^{\mathrm{Br}}$, which
follows from well-known scaling limit theorems, see [7].

The last step in order to get the convergence
$\mathrm{Scale}(r,C_{0}^{-1}\cdot
r^{3};\hat{\mathrm{M}}_{1+r}^{\bullet})\rightarrow\mathcal{M}^{\bullet}$ is to
write $\mathrm{Scale}(r,C_{0}^{-1}\cdot
r^{3};\hat{\mathrm{M}}_{1+r}^{\bullet})=\mathrm{Scale}(r,C_{0}^{-1}\cdot
r^{3}\cdot W^{1+r}(\infty);\widetilde{\mathrm{M}}_{1+r}^{\bullet})$ and just
invoke the fact that $C_{0}^{-1}\cdot r^{3}\cdot
W^{1+r}(\infty)\underset{r\rightarrow
0}{\rightarrow}Y=\lim_{t\rightarrow\infty}\frac{\mathcal{W}_{t}}{t}$ in
distribution, and that convergence takes place jointly with the previous ones.

  3. (c)

We also need to check that some technical assumptions hold at several steps of
the way, namely (AGConv4) and (AGConv5) for the application of Proposition
2.3.3 in the convergence towards $\mathcal{M}^{\bullet}$. This will be
contained in Section 4.5.4.

  4. (d)

We also need to check that the assumptions (ChConv2) and (ChConv3) indeed hold
to be able to apply Proposition 2.2.2. This will be done in Section 4.7.

In the end, the proof of Theorem 1.2 uses everything above and is found in
Section 4.8.

###  4.3 Convergence of the set of weights of forward maximal edges

The goal of Section 4.3 is to prove the following lemma.

######  Lemma .

As $r\rightarrow 0$, we have

$\displaystyle\mathscr{A}^{r}\rightarrow\mathscr{P},$

in distribution for the vague topology on
$\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$, where
$\mathscr{P}$ is a Poisson process of intensity $\frac{\differential t}{t}$.

In order to prove the convergence stated in the lemma, we use the fact that
the distribution of $\mathscr{P}$ is characterized by the following: for any
$K>0$, if we enumerate
$\mathscr{P}\cap\mathopen{(}0\mathclose{}\mathpunct{},K\mathclose{)}$ in
decreasing order as $X_{1}>X_{2}>\dots$, then $(X_{1},X_{2},\dots)$ has the
same distribution as

$\displaystyle(KU_{1},KU_{1}U_{2},\dots),$ (35)

where $(U_{i})_{i\geq 1}$ is a sequence of i.i.d. uniform random variables on
$\mathopen{(}0\mathclose{}\mathpunct{},1\mathclose{)}$.

###### Proof of Lemma 4.3.

First, recall the random variable $\alpha_{1}$ with density
$\theta^{\prime}(y)\mathbf{1}_{\\{y>1\\}}$. Because of (27), we can get that
for $x\rightarrow 1$, we have the following convergence in distribution

$\displaystyle\mathrm{Law}\left(\frac{\alpha_{1}-1}{x-1}\ \bigg{|}\
\alpha_{1}\leq x\right)\underset{x\rightarrow
1}{\rightarrow}\mathrm{Unif}\mathopen{(}0\mathclose{}\mathpunct{},1\mathclose{)}.$
(36)

Now, note that the transition of the Markov chain $(\alpha_{k})_{k\geq 1}$ is
so that conditionally on $\\{\alpha_{k}=x\\}$, the term $\alpha_{k+1}$ has the
same distribution as $\alpha_{1}$ conditionally on $\\{\alpha_{1}\leq x\\}$.
This entails that, for $T_{x}=\inf\left\\{k\geq
1\mathrel{}\middle|\mathrel{}\alpha_{k}\leq x\right\\}$333Note that we can
easily see that $T_{x}$ is almost surely finite from the description above.,
then $\alpha_{T_{x}}$ has the distribution of $\alpha_{1}$ conditional on
$\\{\alpha_{1}\leq x\\}$. Indeed, for any $1\leq y\leq x$,

$\displaystyle\mathbb{P}\left(\alpha_{T_{x}}\leq y\right)$
$\displaystyle=\sum_{k=1}^{\infty}\mathbb{P}\left(T_{x}=k\right)\mathbb{P}\left(\alpha_{k}\leq
y\mathrel{}\middle|\mathrel{}T_{x}=k\right)$
$\displaystyle=\sum_{k=1}^{\infty}\mathbb{P}\left(T_{x}=k\right)\mathbb{P}\left(\alpha_{k}\leq
y\mathrel{}\middle|\mathrel{}\alpha_{k-1}>x,\alpha_{k}\leq x\right)$
$\displaystyle=\sum_{k=1}^{\infty}\mathbb{P}\left(T_{x}=k\right)\mathbb{P}\left(\alpha_{1}\leq
y\mathrel{}\middle|\mathrel{}\alpha_{1}\leq
x\right)=\mathbb{P}\left(\alpha_{1}\leq
y\mathrel{}\middle|\mathrel{}\alpha_{1}<x\right).$

Now we have $\mathscr{A}=\\{\alpha_{1},\alpha_{2},\dots\\}$ and
$\mathscr{A}^{r}=r^{-1}\cdot\\{\alpha_{1}-1,\alpha_{2}-1,\dots\\}$. Fix some
$K>0$. Applying the above for $x=1+rK$, the highest point in
$\mathscr{A}^{r}\cap\mathopen{(}0\mathclose{}\mathpunct{},K\mathclose{)}$ is
$r^{-1}\cdot(\alpha_{T_{1+rK}}-1)$, and has the distribution
$\mathrm{Law}\left(\frac{\alpha_{1}-1}{r}\bigg{|}\ \alpha_{1}\leq 1+rK\right)$
so that thanks to (36), we have
$r^{-1}\cdot(\alpha_{T_{1+rK}}-1)\underset{r\rightarrow 0}{\rightarrow}KU_{1}$
with
$U_{1}\sim\mathrm{Unif}\mathopen{(}0\mathclose{}\mathpunct{},1\mathclose{)}$.
Using (36) iteratively, we can then easily get the finite dimensional
convergence of
$r^{-1}(\alpha_{T_{1+rK}}-1,\alpha_{T_{1+rK}+1}-1,\alpha_{T_{1+rK}+2}-2\dots)$
towards $(KU_{1},KU_{1}U_{2},\dots)$ that appears in (35). This concludes the
proof of the lemma. ∎

##### Connection with the lower envelope process.

For completeness, in order to connect our current work the approach of [10],
we can also consider the process $(L_{k})_{k\geq 0}$ of the types of vertices
along the spine of $\mathrm{T}^{\infty}$. From the considerations of Section
4.1.3, we get that $(L_{k})_{k\geq 0}$ is a Markov process where $L_{0}$ has
density $\theta^{\prime}(y)$ on
$\mathopen{(}1\mathclose{}\mathpunct{},\infty\mathclose{)}$ and the
transitions are given by

$\displaystyle\mathbb{P}\left(L_{k+1}=L_{k}\mathrel{}\middle|\mathrel{}L_{k}=\alpha\right)$
$\displaystyle=\alpha^{*},$
$\displaystyle\mathbb{P}\left(L_{k+1}\in\mathrm{d}y\mathrel{}\middle|\mathrel{}L_{k}=\alpha\right)$
$\displaystyle=(1-\alpha^{*})\frac{\theta^{\prime}(y)}{\theta(\alpha)}\mathbf{1}_{\\{y\leq\alpha\\}}\differential
y.$

Using [10, Proposition 3.3] which treats a very similar case, we get the
following convergence in the Skorokhod topology for any positive $\epsilon$,

$\displaystyle(k(L_{\lceil
kt\rceil}-1))_{t\geq\epsilon}\underset{k\rightarrow\infty}{\rightarrow}(\mathcal{L}(t))_{t\geq\epsilon},$
(37)

where $(\mathcal{L}(t))_{t>0}$ is called the Poisson Lower Envelope Process.
It is defined as such: Let $\mathscr{R}$ denote the Poisson point process on
the positive quadrant with intensity 1. Then
$\mathcal{L}:\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}\rightarrow\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$
is defined by

$\displaystyle\mathcal{L}(t):=\min\left\\{y>0\mathrel{}\middle|\mathrel{}(x,y)\in\mathscr{R}\text{
for some }x\leq t\right\\}.$

The process $t\mapsto\mathcal{L}(t)$ is positive and non-increasing and tends
to $\infty$ at $0$ and to $0$ at $\infty$. This process is intimately linked
with a Poisson Point Process of intensity $\frac{\differential t}{t}$, as the
next lemma asserts.

######  Lemma .

The set of values
$\mathscr{P}:=\left\\{\mathcal{L}(t)\mathrel{}\middle|\mathrel{}t\in\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}\right\\}$,
seen as a point process on
$\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$, is distributed as
a Poisson point process with intensity measure $\frac{\mathrm{d}t}{t}$.

###### Proof.

This fact comes easily from the characterization (35), the fact that
$t\mapsto\mathcal{L}(t)$ decreases from infinity at times $0$ and the fact
that at very jump time $t>0$ of the process, the value $\mathcal{L}(t)$
conditionally on the trajectory on the interval
$\mathopen{(}0\mathclose{}\mathpunct{},t\mathclose{)}$, has the distribution
of $\mathcal{L}(t-)\cdot U$, where
$U\sim\mathrm{Unif}\mathopen{(}0\mathclose{}\mathpunct{},1\mathclose{)}$. The
details are left to the reader. ∎

###  4.4 Convergence of the seed
$\mathrm{Scale}(r,r^{2};\mathrm{T}^{\bullet}_{1+r})$

From Section 4.1.2, a way to sample $\mathrm{T}^{\bullet}_{\alpha}$ is to
first sample a random variable $X_{\alpha}$ with the Borel–Tanner distribution
with parameter $\alpha$, which is such that for all $m\geq 1$

$\displaystyle\mathbb{P}\left(X_{\alpha}=m\right)=\frac{\theta(\alpha)}{\theta^{\prime}(\alpha)}\frac{e^{-\alpha
m}(\alpha m)^{m-1}}{(m-1)!},$

and then let $\mathrm{T}^{\bullet}_{\alpha}$ be a PGW tree conditioned to have
$X_{\alpha}$ vertices. The vertex $v_{\alpha}$ is then uniformly picked in
$\mathrm{T}^{\bullet}_{\alpha}$. Using Sterling’s approximation we have, for
$m\rightarrow\infty$

$\displaystyle m!=\sqrt{2\pi
m}\cdot\left(\frac{m}{e}\right)^{m}\cdot(1+o(1)).$

Then, for any $x\in\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$
and $r\rightarrow 0$ we have

$\displaystyle\mathbb{P}\left(X_{1+r}=\lfloor xr^{-2}\rfloor\right)$
$\displaystyle=\frac{\theta(1+r)}{\theta^{\prime}(1+r)}\frac{e^{-r\lfloor
xr^{-2}\rfloor}}{r^{-1}\sqrt{2\pi x}}(1+r)^{\lfloor
xr^{-2}\rfloor}\cdot(1+o(1))$ $\displaystyle=\frac{r^{2}}{\sqrt{2\pi
x}}\cdot\exp\left(-\frac{x}{2}\right)\cdot(1+o(1))$

and this proves that

$\displaystyle r^{2}\cdot X_{1+r}\overset{(d)}{\underset{r\rightarrow
0}{\rightarrow}}X$ (38)

where $X$ has density $\frac{\exp\left(-\frac{x}{2}\right)}{\sqrt{2\pi
x}}\mathrm{d}x$ on
$\mathopen{(}0\mathclose{}\mathpunct{},\infty\mathclose{)}$, which is the
density of $\sqrt{2}\cdot G$ where $G$ is
$\mathrm{Gamma}(\frac{1}{2})$-distributed. In fact the above convergence also
takes place in $L^{2}$ and $\mathbb{E}\left[X\right]=1$ and
$\mathrm{Var}\left(X\right)=2$.

From there and from known results about the convergence of
$\operatorname{PGW}$ trees conditioned on their size towards to Brownian tree,
it is easy to check that we have the convergence

$\displaystyle\mathrm{Scale}(r,r^{2};\mathrm{T}^{\bullet}_{1+r})\underset{r\rightarrow
0}{\rightarrow}\mathcal{T}^{\bullet},$ (39)

where $\mathcal{T}^{\bullet}$ is a Brownian tree with mass
$\sqrt{2}\cdot\mathrm{Gamma}(\frac{1}{2})$-distributed, pointed at a point
sampled at random using a normalized version of its weight measure.

###  4.5 Properties and convergence of the discrete weight process

####  4.5.1 Definition of the weight process as a Markov process

Recall the definition of the PGWA process started from at time $\alpha$ from
some seed tree $S_{\alpha}$. The aggregate starts at $A(\alpha)=S_{\alpha}$
and at any time $s\geq\alpha$, growing event happen at each vertex of the
aggregate at rate $(1-\theta(s))$. Whenever an growing event happens, a tree
with distribution $\operatorname{PGW}(s^{*})$ is attached to the corresponding
vertex. From this description, it is easy to check that the weight process
$(W(s))_{s\geq\alpha}$ defined as

$W(s):=\\#\\{\text{vertices in $A(s)$ at time }s\\},$

is then a time-inhomogeneous Markov process such that at time $s$, a jump
happens with rate $W(s)\cdot(1-\theta(s))$ and the size of the corresponding
jump is distributed as the number of vertices of a
$\operatorname{PGW}(s^{*})$-distributed tree. This entails that the process
$((s,W(s))_{s\geq\alpha}$ is a time-homogeneous Markov process with generator
$A$ characterized by the fact that, given any compactly supported smooth
function $f:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ we have

$\displaystyle Af(s,x)$
$\displaystyle=\partial_{s}f+x\cdot(1-\theta(s))\cdot\sum_{k=1}^{\infty}\left(f(s,x+k)-f(s,x)\right)\cdot\mathbb{P}\left(\absolutevalue{\operatorname{PGW}(s^{*})}=k\right).$
(40)

Note that, by construction, this process satisfies the so-called _branching
property_ : if we denote by $\left(W(s)\right)_{s\geq\alpha}$ and
$\left(W^{\prime}(s)\right)_{s\geq\alpha}$ two independent processes with the
same transitions as above, started from respectively $k_{1}$ and $k_{2}$, then
their sum $\left(W(s)+W^{\prime}(s)\right)_{s\geq\alpha}$ would still be a
Markov process with the same transitions, started at $k_{1}+k_{2}$.

In the rest of the section, we will use $(W(s))_{s\geq\alpha}$ to denote a
general version of this Markov process started at time $\alpha$ and write
$(W^{\alpha}(s))_{s\geq\alpha}$ for the version of this process started at
time $\alpha$ from the value $X_{\alpha}$, the number of vertices of
$\mathrm{T}_{\alpha}^{\bullet}$, which is the version of the process that
appears in the construction of $\mathrm{M}^{\bullet}_{\alpha}$, see Section
4.1.2.

####  4.5.2 Moment computations

In order to compute the moments, we will follow the same strategy as in the
proof of Lemma 3.4, and rely on the fact that for the functions $F$ given by
$x\mapsto x$ and $x\mapsto x^{2}$, one can prove that

$\displaystyle
F(W(t))-F(W(\alpha))-\int_{\alpha}^{t}W(s-)\cdot(1-\theta(s))\cdot\sum_{k=1}^{\infty}\left(F(W(s-)+k)-F(W(s-))\right)\cdot\mathbb{P}\left(\absolutevalue{\operatorname{PGW}(s^{*})}=k\right),$
(41)

is a martingale, in the same way as we prove the analogous statement (21) for
process $(\mathcal{W}_{t})_{t\geq 1}$, by representing $(W(t))_{t\geq\alpha}$
as the solution of an SDE.

##### Expectation of $W^{\alpha}(s)$.

We follow the same steps as in the proof of Lemma 3.4 and use the fact that
the process described in (41) is a martingale. We get, using the same steps,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\mathbb{E}\left[W(s)\mathrel{}\middle|\mathrel{}W(\alpha)\right]$
$\displaystyle=\mathbb{E}\left[W(s)\mathrel{}\middle|\mathrel{}W(\alpha)\right](1-\theta(s))\mathbb{E}\left[\absolutevalue{\operatorname{PGW}(s^{*})}\right]$
$\displaystyle=\mathbb{E}\left[W(s)\mathrel{}\middle|\mathrel{}W(\alpha)\right]\frac{(1-\theta(s))}{1-s^{*}}.$

Integrating the above differential equation yields

$\displaystyle\mathbb{E}\left[W(s)\mathrel{}\middle|\mathrel{}W(\alpha)\right]=W(\alpha)\cdot\exp\left(\int_{\alpha}^{s}\frac{1-\theta(\lambda)}{1-\lambda^{*}}\mathrm{d}\lambda\right).$
(42)

Since the right-hand side tends to a finite limit as $s\rightarrow\infty$ we
can write, denoting $W(\infty)$ as the limit of the increasing process
$s\mapsto W(s)$ and using monotone convergence

$\displaystyle\mathbb{E}\left[W(\infty)\mathrel{}\middle|\mathrel{}W(\alpha)\right]=W(\alpha)\cdot\exp\left(\int_{\alpha}^{\infty}\frac{(1-\theta(\lambda))}{1-\lambda^{*}}\mathrm{d}\lambda\right).$
(43)

Assuming that $\alpha\leq 2$, (we are interested in what happens when
$\alpha\rightarrow 1$), we can write the second term in the last display as

$\displaystyle\exp\left(\int_{\alpha}^{\infty}\frac{(1-\theta(\lambda))}{1-\lambda^{*}}\mathrm{d}\lambda\right)$
$\displaystyle=\exp\left(\int_{\alpha}^{2}\frac{1}{\lambda-1}\mathrm{d}\lambda\right)\cdot\exp\left(\int_{\alpha}^{\infty}\frac{(1-\theta(\lambda))}{1-\lambda^{*}}-\frac{\mathbf{1}_{\\{\lambda\leq
2\\}}}{\lambda-1}\mathrm{d}\lambda\right)$
$\displaystyle=\frac{1}{\alpha-1}\cdot\exp\left(\int_{\alpha}^{\infty}\frac{(1-\theta(\lambda))}{1-\lambda^{*}}-\frac{\mathbf{1}_{\\{\lambda\leq
2\\}}}{\lambda-1}\mathrm{d}\lambda\right),$ (44)

where the last integrand is integrable over
$\mathopen{[}1\mathclose{}\mathpunct{},\infty\mathclose{)}$ and so the second
term in the above product is bounded above by the value

$\displaystyle
C_{0}:=\exp\left(\int_{1}^{\infty}\frac{(1-\theta(\lambda))}{1-\lambda^{*}}-\frac{\mathbf{1}_{\\{\lambda\leq
2\\}}}{\lambda-1}\mathrm{d}\lambda\right),$ (45)

and converges to that value when $\alpha\rightarrow 1$.

######  Remark .

From the computations above, we also get that for any $s\geq\beta\geq\alpha$
we have

$\displaystyle\mathbb{E}\left[\frac{W(s)}{s-1}\mathrel{}\middle|\mathrel{}W(\beta)\right]$
$\displaystyle=\frac{W(\beta)}{\beta-1}\cdot\frac{\beta-1}{s-1}\cdot\exp\left(\int_{\beta}^{s}\frac{1-\theta(\lambda)}{1-\lambda^{*}}\mathrm{d}\lambda\right)$
$\displaystyle=\frac{W(\beta)}{\beta-1}\cdot\exp\left(\int_{\beta}^{s}\frac{1-\theta(\lambda)}{1-\lambda^{*}}-\frac{1}{\lambda-1}\mathrm{d}\lambda\right).$

Since from (29) the integrand in the last display is negative for any
$\lambda>1$, this entails that $(\frac{W(s)}{s-1})_{s\geq\alpha}$ is a
supermartingale in its own filtration. This will be useful later on.

##### Second moment of $W(s)$.

We apply the same method as above with $F(x)=x^{2}$. We use in the computation
below the fact that for any parameter
$\lambda\in\mathopen{[}0\mathclose{}\mathpunct{},1\mathclose{)}$ the second
moment for the size of a Poisson–Galton–Watson tree with parameter $\lambda$
is given by
$\mathbb{E}\left[\absolutevalue{\operatorname{PGW}(\lambda)}^{2}\right]=\frac{1}{(1-\lambda)^{3}}$,
see [1, Eq. (4.1)].

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\mathbb{E}\left[\left(W(s)\right)^{2}\mathrel{}\middle|\mathrel{}W(\alpha)\right]$
$\displaystyle=2\mathbb{E}\left[\left(W(s)\right)^{2}\mathrel{}\middle|\mathrel{}W(\alpha)\right](1-\theta(s))\mathbb{E}\left[\absolutevalue{\operatorname{PGW}(s^{*})}\right]+\mathbb{E}\left[W(s)\mathrel{}\middle|\mathrel{}W(\alpha)\right](1-\theta(s))\mathbb{E}\left[\absolutevalue{\operatorname{PGW}(s^{*})}^{2}\right]$