# Convergence of trees with a given degree sequence and of their associated
laminations.

Gabriel Berzunza Ojeda111 Department of Mathematical Sciences, University of
Liverpool, United Kingdom. E-mail<EMAIL_ADDRESS>,
Cecilia Holmgren222 Department of Mathematics, Uppsala University, Sweden.
E-mail<EMAIL_ADDRESS>and Paul Thévenin333 Department of
Mathematics, Uppsala University, Sweden. E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we study uniform rooted plane trees with given degree sequence.
We show, under some natural hypotheses on the degree sequence, that these
trees converge toward the so-called Inhomogeneous Continuum Random Tree after
renormalisation. Our proof relies on the convergence of a modification of the
well-known $\L$ukasiewicz path. We also give a unified treatment of the limit,
as the number of vertices tends to infinity, of the fragmentation process
derived by cutting-down the edges of a tree with a given degree sequence,
including its geometric representation by a lamination-valued process. The
latter is a collection of nested laminations that are compact subsets of the
unit disk made of non-crossing chords. In particular, we prove an equivalence
between Gromov-weak convergence of discrete trees and the convergence of their
associated lamination-valued processes.

Key words and phrases: Bridge with exchangeable increments, continuum random
tree, fragmentation processes, Inhomogeneous CRT, lamination of the disk,
scaling limits.

MSC 2020 Subject Classifications: 60C05, 60F17, 60G09, 05C05.

###### Contents

  1. 1 Introduction
    1. 1.1 Scaling limits of trees
    2. 1.2 Fragmentations and laminations
    3. 1.3 Organization
  2. 2 Plane trees and their encoding paths
  3. 3 Convergence of the trees with given degrees
    1. 3.1 The exploration process of the Inhomogeneous CRT
    2. 3.2 The modified $\L$ukasiewicz path
    3. 3.3 Gromov-Hausdorff-Prokhorov convergence of a specific model of TGDS
  4. 4 Lamination-valued processes
    1. 4.1 The discrete setting
    2. 4.2 The continuum setting
      1. 4.2.1 Compact continuum trees and excursion-type functions
      2. 4.2.2 Reduced tree and lamination from continuum trees
    3. 4.3 Convergence of the processes of laminations
    4. 4.4 Convergence of the process of masses
  5. 5 Convergence of the fragmentation processes

##  1 Introduction

In his seminal papers [3, 4, 5], Aldous introduced the so-called Brownian
continuum random tree (Brownian CRT) as the limit - after renormalisation - of
a uniform tree with $n$ vertices, and more generally, of critical size-
conditioned Galton–Watson trees with finite offspring variance. The Brownian
CRT has appeared since then as the limit of various random tree-like
structures such as multi-type Galton-Watson trees [43] or unordered binary
trees [40]. Moreover, these results have had plenty of applications in the
study of other random structures, e.g. random planar maps [36], random
dissections of regular polygons [6, 7], fragmentation and coalescent processes
[9], Erdős-Rényi random graphs in the critical window [2], just to mention a
few. Therefore, over the last decade, the study of scaling limits of large
discrete random trees toward a random continuum tree has seen numerous
developments.

We investigate in this paper the scaling limit of trees with given degree
sequence as well as of their associated laminations. To be precise, for
$n\in\mathbb{N}$, a degree sequence $\mathbf{s}_{n}=(N_{i}^{n},i\geq 0)$ is a
sequence of non-negative integers, satisfying $\sum_{i\geq
0}N_{i}^{n}=1+\sum_{i\geq 0}iN_{i}^{n}<\infty$. Then, a random tree with given
degree sequence (TGDS) $\mathbf{s}_{n}$ is a random variable whose law is
uniform on the set $\mathbb{T}_{\mathbf{s}_{n}}$ of rooted plane trees with
$V_{n}\coloneqq\sum_{i\geq 0}N_{i}^{n}$ vertices amongst which $N_{i}^{n}$
have $i$ offspring for every $i\geq 0$, and $E_{n}\coloneqq\sum_{i\geq
0}iN_{i}^{n}$ edges.

###  1.1 Scaling limits of trees

Scaling limits for trees with given degree sequence were first studied by
Broutin & Marckert [20]. Let $\mathbf{s}_{n}$ be a degree sequence and
$\mathbf{t}_{n}$ be a random tree sampled uniformly at random in
$\mathbb{T}_{\mathbf{s}_{n}}$. We see $\mathbf{t}_{n}$ as a rooted metric
measure space $(\mathbf{t}_{n},r_{n}^{\text{gr}},\rho_{n},\mu_{n})$, i.e.
$\mathbf{t}_{n}$ is identified as its set of $V_{n}$ vertices,
$r_{n}^{\text{gr}}$ is the graph-distance on $\mathbf{t}_{n}$, $\rho_{n}$ is
its root, and $\mu_{n}$ is the uniform measure on the set of vertices of
$\mathbf{t}_{n}$. Consider the global variance term
$\sigma_{n}^{2}=\sum_{i\geq 1}i(i-1)N_{i}^{n}$ for the degree sequence
$\mathbf{s}_{n}$, and the maximum degree $\Delta_{n}=\max\\{i\geq
0:N_{i}^{n}>0\\}$ of any tree with degree sequence $\mathbf{s}_{n}$. Under
technical assumptions on $\mathbf{s}_{n}$, in particular
$\sigma_{n}^{2}\sim\sigma^{2}V_{n}$ as $n\rightarrow\infty$ for some
$\sigma\in(0,\infty)$ and
$\lim_{n\rightarrow\infty}\sigma_{n}^{-1}\Delta_{n}=0$, Broutin & Marckert
[20] showed the convergence in distribution

$\displaystyle\left(\mathbf{t}_{n},\frac{\sigma_{n}}{V_{n}}r_{n}^{\text{gr}},\rho_{n},\mu_{n}\right)\xrightarrow[]{d}(\mathcal{T}_{\rm
Br},r_{\rm Br},\rho_{\rm Br},\mu_{\rm Br}),\hskip 8.53581pt\text{as}\hskip
5.69054ptn\rightarrow\infty,$

for the so-called Gromov-Hausdorff-Prokhorov topology, where
$(\mathcal{T}_{\rm Br},r_{\rm Br},\rho_{\rm Br},\mu_{\rm Br})$ is the Brownian
CRT. In particular, $\mu_{\rm Br}$ is a probability measure supported on the
leaves of $\mathcal{T}_{\rm Br}$.

Marzouk [42] has extended the above result under the assumption of no
macroscopic degree only. To be more precise, he proved a weaker convergence,
in the sense of subtrees spanned by finitely many random vertices. Fix $q\geq
1$ and let $u_{1},\dots u_{q}$ be $q$ i.i.d. uniform random vertices of
$\mathbf{t}_{n}$. The reduced tree $\mathbf{t}_{n}^{(q)}$ is obtained by
keeping only the root of $\mathbf{t}_{n}$, these $q$ vertices, the subsequent
branching points (if any), and then connecting by a single edge two of these
vertices if one is the ancestor of the other in $\mathbf{t}_{n}$ and there is
no other vertex of $\mathbf{t}_{n}^{(q)}$ inbetween. We define the length of
an edge $e$ in $\mathbf{t}_{n}^{(q)}$ as the number of edges in
$\mathbf{t}_{n}$ between the endpoints of $e$. In particular, the
combinatorial structure of $\mathbf{t}_{n}^{(q)}$ is that of a rooted plane
tree with at most $q$ leaves, so there are only finitely many possibilities,
and thus there are a bounded number of edge-lengths to record. The space of
such trees, called trees with edge-lengths, is thus endowed with the natural
product topology. For $x_{1},\dots,x_{q}$ i.i.d. random points of the Brownian
CRT $\mathcal{T}_{\rm Br}$ sampled from its mass measure $\mu_{\rm Br}$, one
can construct similarly a discrete tree with edge-lengths
$\mathcal{T}^{(q)}_{\rm Br}$; see [5]. If
$\lim_{n\rightarrow\infty}\sigma_{n}^{-1}\Delta_{n}=0$, Marzouk [42] proved
that for every $q\geq 1$ one has the convergence in distribution

$\displaystyle\frac{\sigma_{n}}{V_{n}}\mathbf{t}_{n}^{(q)}\xrightarrow[]{d}\mathcal{T}^{(q)}_{\rm
Br},\hskip 8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty.$

In this work, we go one step further and, under the existence of at most
countably many macroscopic degrees (see (A.2) and (${\bf B}$)), we prove weak
convergence of $\mathbf{t}_{n}$ toward the associated Inhomogeneous continuum
random tree (Inhomogeneous CRT, which may be different from the Brownian CRT).
The Inhomogeneous CRT has been introduced in [22] and arises as the scaling
limit of another model of random trees called $\mathbf{p}$-trees (or birthday
trees). The simplest description of the Inhomogeneous CRT is via a line-
breaking construction based on a Poisson point process in the plane which can
be found in [11, 22]. The spanning subtree description is set out in [10], and
its description via an exploration process is given in [8]. An Inhomogeneous
CRT $(\mathcal{T}_{\theta},r_{\theta},\rho_{\theta},\mu_{\theta})$ is uniquely
defined by a parameter set $\theta\coloneqq(\theta_{0},\theta_{1},\dots)$ such
that

$\displaystyle\theta_{1}\geq\theta_{2}\geq\cdots\geq 0,\hskip
8.53581pt\theta_{0}\geq 0,\hskip 8.53581pt\sum_{i\geq 0}\theta_{i}^{2}=1\hskip
8.53581pt\text{and either}\hskip 8.53581pt\theta_{0}>0\hskip
8.53581pt\text{or}\hskip 8.53581pt\sum_{i\geq 1}\theta_{i}=\infty;$

see Figure 1, left for a simulation of an ICRT. In the special case
$\theta=(1,0,0,\dots)$, $\mathcal{T}_{\theta}$ is precisely the Brownian CRT.
For $q\geq 1$, let $x_{1},\dots,x_{q}$ be $q$ i.i.d. random points of
$\mathcal{T}_{\theta}$ sampled from the mass measure $\mu_{\theta}$; one can
also construct a discrete tree with edge-lengths $\mathcal{T}_{\theta}^{(q)}$
whose law is described in [10, 11, 22].

######  Theorem 1.

For $n\in\mathbb{N}$, let $\mathbf{s}_{n}=(N_{i}^{n},i\geq 0)$ be a degree
sequence. Let $(d^{n}(i),1\leq i\leq V_{n})$ denote the associated child
sequence, obtained by writing $N_{0}^{n}$ zeros, $N_{1}^{n}$ ones, etc., and
ordering the resulting sequence decreasingly. Assume that, as
$n\rightarrow\infty$,

  1. (A.1)

Size. $V_{n}\rightarrow\infty$;

  2. (A.2)

Hubs. There exists a sequence $(b_{n},n\geq 1)$ with $b_{n}\rightarrow\infty$
such that, for every $i\geq 1$, the sequence $(d^{n}(i)/b_{n},n\geq 1)$
converges to a limit $\beta_{i}\geq 0$;

  3. (A.3)

Degree variance. There exists $\sigma\in[0,\infty)$ such that
$\frac{1}{b_{n}^{2}}\sum_{i\geq
0}(i-1)^{2}N_{i}^{n}\rightarrow\sigma^{2}+\sum_{i\geq 1}\beta_{i}^{2}$.

Suppose further that

$\sigma^{2}>0,\hskip 8.53581pt\sum_{i\geq 1}\beta_{i}<\infty\hskip
8.53581pt\text{and}\hskip 8.53581pt\sigma^{2}+\sum_{i\geq 1}\beta_{i}^{2}=1.$
(${\bf B}$)

Then, we have that, for all $q\geq 1$, in distribution,

$\displaystyle\frac{b_{n}}{V_{n}}\mathbf{t}_{n}^{(q)}\xrightarrow[]{d}\mathcal{T}_{\theta}^{(q)},\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty,$

where $\mathcal{T}_{\theta}$ is an Inhomogeneous CRT with parameter set
$\theta=(\theta_{0},\theta_{1},\dots)$ given by $\theta_{0}=\sigma$ and
$\theta_{i}=\beta_{i}$, for $i\geq 1$.

Observe that the maximum degree $\Delta_{n}=d^{n}(1)$ and thus, (A.2) implies
that $\Delta_{n}/b_{n}\rightarrow\beta_{1}$, as $n\rightarrow\infty$. In
particular, if $\beta_{1}=0$ in (A.2), the hypotheses made in Theorem 1
correspond to the setting studied by Marzouk [42]. Indeed, by (A.3), the
global variance $\sigma_{n}^{2}=\sum_{i\geq 1}i(i-1)N_{i}^{n}$ of the degree
sequence $\mathbf{s}_{n}$ satisfies that $\sigma^{2}_{n}/b_{n}^{2}\rightarrow
1$, as $n\rightarrow\infty$. On the other hand, Theorem 1 together with [28,
Theorem 5] implies that

$\displaystyle\left(\mathbf{t}_{n},\frac{b_{n}}{V_{n}}r_{n}^{\text{gr}},\rho_{n},\mu_{n}\right)\xrightarrow[]{d}(\mathcal{T}_{\theta},r_{\theta},\rho_{\theta},\mu_{\theta}),\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty,$ (1)

for the so-called Gromov-weak topology (often cited as Gromov-Prokhorov
topology); see e.g., Section 4.3 for background.

Figure 1: Simulation of an Inhomogeneous CRT $\mathcal{T}_{\theta}$ and its
associated lamination $\mathbb{L}(\mathcal{T}_{\theta})$, for
$\theta=(1/\sqrt{7},2/\sqrt{7},1/\sqrt{7},1/\sqrt{7},0,\ldots)$.

In the regime of possibly countably many macroscopic degrees (A.2) and (${\bf
B}$), Theorem 1 characterises the possible scaling limits of this model of
random trees. It is then natural to wonder whether (1) can be reinforced to
hold for the stronger Gromov-Hausdorff-Prokhorov topology under the
assumptions (A.1)-(A.3) and (${\bf B}$). To achieve this, one would need to
prove the tightness of the sequence of discrete trees (see e.g., [5, Equation
25]), which requires precise estimates of the height of $\mathbf{t}_{n}$.
However, as pointed out in [42, Section 1.2], there are cases where the
maximal height of the tree can be much larger than $V_{n}/b_{n}$ and thus no
general tightness result as in [20] holds.

Recently, Blanc-Renaudie [18] independently proved the result in Theorem 1;
see [18, Theorem 5 (a)-(b)]. Indeed, [18, Theorem 6 (a)-(b)] shows that, under
more general conditions ([18, Assumptions 1-2]), $\mathbf{t}_{n}$ converges
either toward a $\mathbf{p}$-tree, or after normalization by
$V_{n}/\sigma_{n}$ toward the Inhomogeneous CRT, for the Gromov-weak topology.
Nevertheless, our methods are completely different. While Blanc-Renaudie
introduces a new recursive construction for TGDSs based on a modified Aldous-
Broder algorithm, in this work we consider a classical approach through the
study of the $\L$ukasiewicz path and the height process that are relevant in
their own right. On the other hand, Blanc-Renaudie [18, Theorem 7] implies,
under additional tightness conditions ([18, Assumption 7]), the convergence of
$\mathbf{t}_{n}$ for the Gromov–Hausdorff–Prokhorov topology. To be precise,
Blanc-Renaudie provides a near-optimal upper bound for the height of
$\mathbf{t}_{n}$ and uses similar estimates to control the Gromov-Hausdorff
distance between $\mathbf{t}_{n}^{(q_{n})}$ and $\mathbf{t}_{n}$, for a well-
chosen sequence $(q_{n},n\geq 1)$.

As we mentioned earlier, this height estimate is the ingredient missing in our
approach to obtain the convergence for the Gromov–Hausdorff–Prokhorov
topology. Indeed, a close inspection to the proof of [18, Lemma 25] together
with [18, Lemma 14] shows [5, equation (25)], which implies that the sequence
of discrete height processes of the trees $(\mathbf{t}_{n},n\geq 1)$ is tight
after normalization by $V_{n}/\sigma_{n}$; see [5, proof of Theorem 20 (ii)].
In particular, under (A.1)-(A.3), (${\bf B}$) and [18, Assumption 7], the
convergence of the finite-dimensional distributions (Theorem 6) and the
tightness prove the weak convergence of the discrete height process toward the
height process of the Inhomogeneous CRT. This implies the convergence of the
contour function and thus the convergence for the Gromov–Hausdorff–Prokhorov
topology; see [35, Section 1.6] and [1]. We do not include here the precise
statement and proof details to avoid repeating and using the arguments
introduced in [18].

Probably, the proof of the Theorem 1 could be extended to the case
$\sum_{i\geq 1}\beta_{i}=\infty$. However, this broader setting brings
additional technical complications that we decide not to consider in this
article. For example, one would first have to define the height process of the
Inhomogeneous CRT for this case, which has not yet been defined properly.

Finally, we expect that similar results also hold for forests with given
degree sequence. This has only been investigated under the assumption of no
macroscopic degrees by Lei [38] and Marzouk [42]. In particular, they viewed
the forest as a single tree by attaching all the roots to an extra root
vertex. In this framework, the limit is a different continuum tree that is
encoded by a certain Brownian first-passage bridge. We believe that our
approach could be used to extend this result and the results in [42] on random
planar maps with a given degree sequence.

###  1.2 Fragmentations and laminations

Aldous, Evans and Pitman [9, 25, 45] initiated the study of fragmentation
processes derived by deleting one by one the edges of tree-like structures
uniformly at random. As time passes, the deletion of edges creates more and
more connected components whose sequence of sizes is called the fragmentation
process of the tree. Aldous, Evans and Pitman studied the case of a uniform
random tree with $n$ labelled vertices and showed that the associated
fragmentation process, suitably rescaled, converges to the fragmentation
process of the Brownian CRT, as $n\rightarrow\infty$; see also [21, 41]. This
latter is connected to the standard additive coalescent via a deterministic
time-change and it is constructed by cutting down the skeleton of the Brownian
CRT in a Poisson manner. Aldous and Pitman [11] (see also [25]) established a
similar result in the broader context of $\mathbf{p}$-trees. They showed that
this fragmentation process converges after rescaling to the fragmentation
process of the limiting Inhomogeneous CRT. Recently, the authors [16] studied
the case of critical Galton-Watson trees conditioned on having $n$ vertices,
whose offspring distribution $\mu$ belongs to the domain of attraction of a
stable law of index $\alpha\in(1,2]$ ($\alpha$-stable Galton-Watson trees). In
this case, the limit is the fragmentation process of the so-called
$\alpha$-stable Lévy tree constructed by cutting down its skeleton in a
Poisson manner.

It turns out that the fragmentation process of a tree can be coded by a non-
decreasing process of subsets of the unit disk called laminations. A
lamination is a closed subset of the closed unit disk $\bar{\mathbb{D}}$ made
of the union of the unit circle $\mathbb{S}^{1}$ and a set of chords that do
not intersect in the open unit disk $\mathbb{D}$. A face of a lamination $L$
is a connected component of the complement of $L$ in $\bar{\mathbb{D}}$.
Laminations appear for instance in topology and hyperbolic geometry, see [19]
and references therein. We denote by $\mathbb{L}(\bar{\mathbb{D}})$ the set of
laminations of $\bar{\mathbb{D}}$ and equip it with the usual Hausdorff
topology on the compact subsets of $\bar{\mathbb{D}}$.

The idea of coding (random) tree-like structures by (random) laminations of
$\bar{\mathbb{D}}$ goes back to Aldous [6, 7] in his study of a uniform
triangulation of a large polygon. Since then, laminations have appeared in
different contexts, as limits of discrete structures [34, 23, 47] or in the
theory of random maps [37]. Roughly speaking, each chord of the lamination
corresponds to an edge of the tree. Then, by adding chords one by one in the
order in which they are removed, we code the fragmentation of the tree by a
random process taking its values in the set of laminations
$\mathbb{L}(\bar{\mathbb{D}})$. See Section 4 for a rigorous definition of
this process. Furthermore, at any given time in the process, there is a one-
to-one correspondence between faces of the lamination and connected components
of the fragmented tree. Indeed, it can be shown that the rescaled size of a
component in the fragmentation process is equal to the mass of the
corresponding face, that is, $(2\pi)^{-1}$ times the fraction of its perimeter
lying on the unit circle. In the case of $\alpha$-stable Galton-Watson trees,
the third author proves in [47] the convergence of this lamination-valued
process, toward a limiting process that can be constructed directly from the
corresponding $\alpha$-stable Lévy tree and encodes its fragmentation process.

A natural way of extending the previous investigations is to study the
asymptotic behaviour of the fragmentation process and the lamination-valued
process derived by cutting-down a rooted plane tree, and in particular a tree
with given degree sequence. This is the second goal of this paper. To present
the main result of this section, we need some notation and background that
some readers may not be familiar with. We refer to Section 4 for proper
definitions. Let $(\mathbb{L}_{t}(\tau),t\geq 0)$ be the lamination-valued
process associated to $\tau$ (that is, for all $t\geq 0$,
$\mathbb{L}_{t}(\tau)$ is obtained by removing the first $\lfloor
t\rfloor\wedge(\zeta(\tau)-1)$ edges from $\tau$); see Definition 1. Let
$\mathbf{D}(I,\mathbb{M})$ be the space of càdlàg functions (that is, right-
continuous with left limits) from an interval $I\subseteq\mathbb{R}_{+}$ to a
separable, complete metric space $\mathbb{M}$. We equip
$\mathbf{D}(I,\mathbb{M})$ with the $J_{1}$ Skorohod topology; see e.g. [17,
Chapter 3] or [31, Chapter VI] for details on this space. We denote by
$\mathcal{T}=(\mathcal{T},r,\rho,\mu)$ a continuum tree, that is,
$(\mathcal{T},r)$ is a metric space, $\rho$ is a distinguished element of
$\mathcal{T}$ called the root and $\mu$ is a probability measure on the set of
leaves of $\mathcal{T}$; see Definition 5. As in the case of finite trees, it
is possible to define from $\mathcal{T}$ a lamination-valued process
$(\mathbb{L}_{t}(\mathcal{T}),t\geq 0)$ obtained by cutting $\mathcal{T}$ in a
Poissonian way and associating to each cutpoint a chord in the disk; see
Definition 7. For $n\geq 1$, we let $\tau_{n}$ be a rooted plane tree and we
see it as a rooted metric measure space
$(\tau_{n},r_{n}^{\text{gr}},\emptyset_{n},\mu_{n})$, i.e., $\tau_{n}$ is
identified as its set of $\zeta(\tau_{n})$ vertices, $r_{n}^{\text{gr}}$ is
the graph distance on $\tau_{n}$, $\emptyset_{n}$ is the root of $\tau_{n}$
and $\mu_{n}$ is the uniform measure on the set of vertices of $\tau_{n}$. The
following result, which in particular can be applied to trees with given
degree sequence, states the equivalence of the Gromov-weak convergence of a
sequence of plane trees and the convergence its associated lamination-valued
processes.

######  Theorem 2.

Let $(\tau_{n},n\geq 1)$ be a sequence of rooted plane trees, $\mathcal{T}$ be
a continuum tree and $(a_{n},n\geq 1)$ be a sequence of non-negative real
numbers satisfying $a_{n}\rightarrow\infty$ and
$\zeta(\tau_{n})/a_{n}\rightarrow\infty$, as $n\rightarrow\infty$. Then, the
following assertions are equivalent:

  1. (C.1)

The following holds in
$\mathbf{D}(\mathbb{R}_{+},\mathbb{L}(\bar{\mathbb{D}}))$:

$\displaystyle(\mathbb{L}_{ta_{n}}(\tau_{n}),t\geq
0)\xrightarrow[]{d}(\mathbb{L}_{t}(\mathcal{T}),t\geq 0),\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty,\hskip 5.69054pt.$

  2. (C.2)

$\displaystyle\left(\tau_{n},\frac{a_{n}}{\zeta(\tau_{n})}r_{n}^{\rm
gr},\emptyset_{n},\mu_{n}\right)\rightarrow(\mathcal{T},r,\rho,\mu)$, as
$n\rightarrow\infty$, for the Gromov-weak topology.

To prove Theorem 2, we develop a general approach that is based on the notion
of reduced laminations that may be of independent interest. These reduced
laminations are constructed considering reduced trees obtained by sampling
only a finite number of vertices in the tree. In particular, the more vertices
are sampled, the closer one is to the lamination-valued process associated
with the entire tree.

Theorem 1 and Theorem 2 immediately entail the following result about the
lamination-valued process associated with the fragmentation of a tree with
given degree sequence $\mathbf{t}_{n}$.

######  Corollary 1.

Suppose that $\mathbf{s}_{n}$ satisfies (A.1)-(A.3) and (${\bf B}$). Let
$\mathcal{T}_{\theta}$ be an Inhomogeneous CRT with parameter set
$\theta=(\theta_{0},\theta_{1},\dots)$ given by $\theta_{0}=\sigma$ and
$\theta_{i}=\beta_{i}$, for $i\geq 1$. Then, jointly with the convergence of
Theorem 1, we have that

$\displaystyle(\mathbb{L}_{tb_{n}}(\mathbf{t}_{n}),t\geq
0)\xrightarrow[]{d}(\mathbb{L}_{t}(\mathcal{T}_{\theta}),t\geq 0),\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty,\hskip
5.69054pt\text{in}\hskip
5.69054pt\mathbf{D}(\mathbb{R}_{+},\mathbb{L}(\bar{\mathbb{D}})).$

In Figure 1, one can see a simulation of
$\mathbb{L}_{\infty}(\mathcal{T}_{\theta})\coloneqq\lim_{t\rightarrow\infty}\mathbb{L}_{t}(\mathcal{T}_{\theta})$,
for a given parameter set $\theta$. Indeed, Theorem 2 and [18, Theorem 5 (b)]
imply the convergence of lamination-valued processes associated even to trees
with given degree sequence $\mathbf{s}_{n}$ satisfying [18, Assumption 2]. In
this case, the limit is the lamination-valued process associated to an
Inhomogeneous CRT. Moreover, by the well-known convergence of suitably
rescaled $\mathbf{p}$-trees ([11, 22, 18]), one can apply Theorem 2 to
establish the convergence of the corresponding lamination-valued processes.
Theorem 2 also provides another proof to [47, Theorems 1.2 and 3.3], in the
case of $\alpha$-stable Galton-Watson trees.

Theorem 2 (or Corollary 1) does not a priori imply the convergence, after a
proper rescaling, of the underlying fragmentation process associated to a tree
with given degree sequence $\mathbf{t}_{n}$. However, this convergence can be
independently obtained without relying on the scaling limit of
$\mathbf{t}_{n}$. For convenience, let us define a slightly different version
of the fragmentation process, where edges are removed at i.i.d. times and not
at integer times. It turns out that, asymptotically, these two fragmentation
processes have the same behaviour. Equip the edges of $\mathbf{t}_{n}$ with
i.i.d. random weights uniform on $[0,1]$ (independent of $\mathbf{t}_{n}$).
For $u\in[0,1]$, consider the forest $\mathbf{f}_{n}(u)$ of connected
components created by keeping the edges of $\mathbf{t}_{n}$ with weight
smaller than or equal to $u$ and by discarding the others. Let
$\mathbf{F}_{n}=(\mathbf{F}_{n}(u),u\in[0,1])$ be the process given by

$\displaystyle\mathbf{F}_{n}(u)=(F_{n,1}(1-u),F_{n,2}(1-u),\dots),\hskip
8.53581pt\text{for}\hskip 5.69054ptu\in[0,1],$

the sequence of sizes (number of vertices) of the connected components of the
forest $\mathbf{f}_{n}(1-u)$, ranked in non-increasing order. We strategically
view the sequence of sizes as an infinite sequence, by completing it with an
infinite number of zero terms. Plainly as time passes more and more connected
components are created, and thus, the process $\mathbf{F}_{n}$ evolves as a
fragmentation process. Note also that $\mathbf{F}_{n}(0)=(V_{n},0,0,\dots)$
and that $\mathbf{F}_{n}(1)=(1,1,\dots,1,0,0,\dots)$ are infinite sequences
(where the first $V_{n}$ terms are $1$’s in $\mathbf{F}_{n}(1)$). We call
$\mathbf{F}_{n}$ the dynamical fragmentation process of $\mathbf{t}_{n}$.
Consider the infinite ordered set

$\displaystyle\bm{\Delta}\coloneqq\Big{\\{}\mathbf{x}=(x_{1},x_{2},\dots):x_{1}\geq
x_{2}\geq\cdots\geq 0\hskip 5.69054pt\text{and}\hskip
5.69054pt\sum_{i=1}^{\infty}x_{i}<\infty\Big{\\}},$

endowed with the $\ell^{1}$-norm,
$\|\mathbf{x}\|_{1}=\sum_{i=1}^{\infty}|x_{i}|$ for
$\mathbf{x}\in\bm{\Delta}$.

######  Theorem 3.

Suppose that $\mathbf{s}_{n}$ satisfies (A.1)-(A.3) and

  1. (A.4)

Unbounded variation. Either $\sigma^{2}>0$ or $\sum_{i\geq
1}\beta_{i}=\infty$.

Then, there exists a process $(\mathbf{F}(t),t\geq 0)$ such that

$\displaystyle\left(\frac{1}{V_{n}}\mathbf{F}_{n}\left(t\frac{b_{n}}{V_{n}}\right),t\geq
0\right)\xrightarrow[]{d}(\mathbf{F}(t),t\geq 0),\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty,\hskip
5.69054pt\text{in}\hskip 5.69054pt\mathbf{D}(\mathbb{R}_{+},\bm{\Delta}).$

The limiting process $(\mathbf{F}(t),t\geq 0)$ corresponds precisely to the
fragmentation process constructed by Bertoin [14, 15] via partitions of the
unit interval induced by certain bridges with exchangeable increments. For a
precise construction, we refer to Section 5 in which we actually prove a more
general result about the convergence of the fragmentation process of rooted
plane trees. On the other hand, if $\sigma^{2}+\sum_{i\geq 1}\beta_{i}^{2}=1$
then $(\mathbf{F}(t),t\geq 0)$ coincides with the fragmentation process of the
Inhomogeneous CRT, with parameter set $\theta=(\theta_{0},\theta_{1},\dots)$
given by $\theta_{0}=\sigma$ and $\theta_{i}=\beta_{i}$, for $i\geq 1$; see
[11].

Finally, as a consequence of Theorem 3, we relate the sequence of masses of
the faces of $(\mathbb{L}_{t}(\mathcal{T}_{\theta}),t\geq 0)$ with the
fragmentation process $(\mathbf{F}(t),t\geq 0)$. For any lamination
$L\in\mathbb{L}(\bar{\mathbb{D}})$, let $\mathbb{M}{\rm ass}[L]$ denote the
sequence of the masses of its faces, sorted in non-increasing order.

######  Corollary 2.

Suppose that $\mathbf{s}_{n}$ satisfies (A.1)-(A.4) with
$\sigma^{2}+\sum_{i\geq 1}\beta_{i}^{2}=1$. Then,

$\displaystyle(\mathbb{M}{\rm ass}[\mathbb{L}_{tb_{n}}(\mathbf{t}_{n})],t\geq
0)\xrightarrow[]{d}\left(\mathbf{F}(t),t\geq 0\right),\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty,\hskip
5.69054pt\text{in}\hskip 5.69054pt\mathbf{D}(\mathbb{R}_{+},\bm{\Delta}).$

Furthermore, $(\mathbb{M}{\rm ass}[\mathbb{L}_{t}(\mathcal{T}_{\theta})],t\geq
0)=\left(\mathbf{F}(t),t\geq 0\right)$ almost surely, where
$\mathcal{T}_{\theta}$ is an Inhomogeneous CRT with parameter set
$\theta=(\theta_{0},\theta_{1},\dots)$ given by $\theta_{0}=\sigma$,
$\theta_{i}=\beta_{i}$, for $i\geq 1$.

Let us finish with some remarks on the assumptions that we make on the degree
sequences. The hypotheses size (A.1), hubs (A.2), degree variance (A.3) and
unbounded variation (A.4) are exactly those made in [13] to study the profile
of a TGDS. They are necessary to apply the characterization and convergence
results for exchangeable increments processes of [32] that are crucial to
understand the shape of a TGDS via its $\L$ukasiewicz path and up to some
extent its height process. Let us also point out that the hypotheses
(A.1)-(A.4) are included in [18, Assumption 2].

###  1.3 Organization

In Section 2, we first recall the definition of rooted plane trees and their
encoding by paths. In Section 3, we prove Theorem 1 by studying the behaviour
of a modified version of the $\L$ukasiewicz paths associated with trees with
given degree sequence. Section 4 is devoted to the study of lamination-valued
processes of discrete trees and continuum trees; we prove in particular
Theorem 2 and Corollary 2 about the convergence of the geometric
representation of trees by laminations. Finally, in Section 5, we prove
Theorem 3, showing the convergence of the associated fragmentation processes.

##### Notation.

For $f\in\mathbf{D}(I,\mathbb{R})$, we denote by $f(t)$ the value of $f$ at
$t\in I$, by $f(t-)$ its left-hand limit at time $t$ (with the convention
$f(0-)=f(0)$ if $0\in I$) and by $\Delta f(t)=f(t)-f(t-)$ the size of the jump
(if any) at $t$.

We write $\xrightarrow[]{d}$, $\xrightarrow[]{\mathbb{P}}$ and
$\xrightarrow[]{a.s.}$ to denote convergence in distribution, probability and
almost surely, respectively.

Let $(A_{n},n\geq 1)$ and $(B_{n},n\geq 1)$ be two sequences of real random
variables such that $B_{n}>0$, for all $n\geq 1$. We say that $A_{n}=o(B_{n})$
in probability if $\lim_{n\rightarrow\infty}|A_{n}|/B_{n}=0$ in probability.

##  2 Plane trees and their encoding paths

We provide here some background on finite rooted trees and recall how they can
be coded by different integer-valued paths.

Following [44], let $\mathbb{N}=\\{1,2,\dots\\}$ be the set of positive
integers, set $\mathbb{N}^{0}=\\{\emptyset\\}$ and consider the set of labels
$\mathbb{U}=\bigcup_{n\geq 0}\mathbb{N}^{n}$. An element $u\in\mathbb{U}$ is a
sequence $u=(u_{1},\dots,u_{n})$ of positive integers. If
$v=(v_{1},\dots,v_{m})\in\mathbb{U}$, we let
$uv=(u_{1},\dots,u_{n},v_{1},\dots,v_{m})\in\mathbb{U}$ be the concatenation
of $u$ and $v$. By a slight abuse of notation, if $z\in\mathbb{N}$, we let
$uz=(u_{1},\dots,u_{n},z)$. A rooted plane tree is a non-empty, finite subset
$\tau\subset\mathbb{U}$ such that: (i) $\emptyset\in\tau$; (ii) if $v\in\tau$
and $v=uz$ for some $z\in\mathbb{N}$, then $u\in\tau$; (iii) if $u\in\tau$,
then there exists an integer $k_{u}\geq 0$ such that $ui\in\tau$ if and only
if $1\leq i\leq k_{u}$. We view each vertex $u$ of $\tau$ as an individual of
a population whose genealogical tree is $\tau$. The vertex $\emptyset$ is
called the root of the tree. For every $u=(u_{1},\dots,u_{n})\in\tau$, the
vertex $pr(u)=(u_{1},\dots,u_{n-1})$ is its parent, $k_{u}$ represents the
number of children of $u$ (if $k_{u}=0$, then $u$ is called a leaf, otherwise,
$u$ is called an internal vertex), and $|u|=n$ represents the length (or
generation, or height) of $u$. We let $\chi_{u}\in\\{1,\dots,k_{pr(u)}\\}$ be
the only index such that $u=pr(u)\chi_{u}$, which is the relative position of
$u$ amongst its siblings. The total progeny (or size) of $\tau$ will be
denoted by $\zeta(\tau)=\text{Card}(\tau)$ (i.e., the number of vertices of
$\tau$). In the following, by tree, we will always mean a rooted plane tree
and we denote the set of all trees by $\mathbb{T}$. By a slight abuse, we
sometimes consider a tree $\tau$ as a metric space, by drawing an edge of
length $1$ between each non-root vertex $u$ and its parent $pr(u)$.

We will use three different orderings of the vertices of a tree
$\tau\in\mathbb{T}$:

  * (i)

Lexicographical ordering. Given $v,w\in\tau$, we write $v\prec_{\text{lex}}w$
if there exists $z\in\tau$ such that $v=z(v_{1},\dots,v_{n})$,
$w=z(w_{1},\dots,w_{m})$ and $v_{1}<w_{1}$.

  * (ii)

Reverse-lexicographical ordering. Given $v,w\in\tau$, we write
$v\prec_{\text{rev}}w$ if $w\prec_{\text{lex}}v$.

  * (iii)

Prim ordering. Let $\textbf{edge}(\tau)$ be the set of edges of $\tau$ and
consider a sequence of distinct and positive weights
$\mathbf{w}=(w_{e}:e\in\textbf{edge}(\tau))$ (i.e., each edge $e$ of $\tau$ is
marked with a different and positive weight $w_{e}$). Given two distinct
vertices $u,v\in\tau$, we write $\\{u,v\\}$ for the edge connecting $u$ and
$v$ in $\tau$ \- if it exists. Let us describe the Prim order
$\prec_{\text{prim}}$ of the vertices in $\tau$, that is,
$\emptyset=u(0)\prec_{\text{prim}}u(1)\prec_{\text{prim}}\dots\prec_{\text{prim}}u(\zeta(\tau)-1)$.
First set $u(0)=\emptyset$ and $V_{1}=\\{u(0)\\}$. Suppose that for some
$1\leq i\leq\zeta(\tau)-1$, the vertices $u(0),\dots,u(i-1)$ have been
defined. We will use the notation $V_{i}$ for the set
$\\{u(0),\dots,u(i-1)\\}$, for $0\leq i\leq\zeta(\tau)-1$. Consider the
minimum of the set of weights $\\{w_{\\{u,v\\}}:u\in V_{i},v\not\in V_{i}\\}$
of the edges between a vertex of $V_{i}$ and another outside of $V_{i}$. Since
all the weights are distinct, this minimum is reached at a unique edge
$\\{\tilde{u},\tilde{v}\\}$ where $\tilde{u}\in V_{i}$ and $\tilde{v}\not\in
V_{i}$. Then set $u(i)=\tilde{v}$. This iterative procedure completely
determines the Prim order $\prec_{\text{prim}}$.

#####  $\L$ukasiewicz path, reverse-$\L$ukasiewicz path and Prim path.

Fix $\tau\in\mathbb{T}$, and for $\ast\in\\{{\rm lex},{\rm rev},{\rm
prim}\\}$, associate to the ordering
$\emptyset=u(0)\prec_{\ast}u(1)\prec_{\ast}\dots\prec_{\ast}u(\zeta(\tau)-1)$
of its vertices a path $W_{\tau}^{\ast}=(W_{\tau}^{\ast}(i),0\leq
i\leq\zeta(\tau))$, by letting $W_{\tau}^{\ast}(0)=0$ and for $0\leq
i\leq\zeta(\tau)-1$, $W_{\tau}^{\ast}(i+1)=W_{\tau}^{\ast}(i)+k_{u(i)}-1$.
Observe that $W_{\tau}^{\ast}(i+1)-W_{\tau}^{\ast}(i)=k_{u(i)}-1\geq-1$ for
every $0\leq i\leq\zeta(\tau)-1$, with equality if and only if $u(i)$ is a
leaf of $\tau$. Note also that $W_{\tau}^{\ast}(i)\geq 0$ for every $0\leq
i\leq\zeta(\tau)-1$, but $W_{\tau}^{\ast}(\zeta(\tau))=-1$. We shall think of
such a path as the step function on $[0,\zeta(\tau)]$ given by $s\mapsto
W_{\tau}^{\ast}(\lfloor s\rfloor)$. The path $W^{\text{lex}}_{\tau}$ is
usually called the $\L$ukasiewicz path of $\tau$ and we will refer to
$W^{\text{rev}}_{\tau}$ and $W^{\text{prim}}_{\tau}$ as the
reverse-$\L$ukasiewicz path and the Prim path of $\tau$, respectively.

For $u,v\in\tau$, we denote by $\llbracket u,v\rrbracket$ the unique geodesic
path between $u$ and $v$ in $\tau$, and $\llbracket u,v\llbracket=\llbracket
u,v\rrbracket\backslash\\{v\\}$. In particular, we write
$\llbracket\varnothing,u\llbracket$ for the ancestral line or branch of $u$.
For $u\in\tau$, let us denote by ${\rm L}(u)$ and ${\rm R}(u)$ respectively
the number of vertices whose parent is a strict ancestor of $u$ and which lie
strictly to the left (respectively to the right) of the ancestral line
$\llbracket\emptyset,u\llbracket$. Then, we set ${\rm LR}(u)={\rm L}(u)+{\rm
R}(u)$, the total number individuals branching-off the ancestral line of $u$.
In particular, let
$\emptyset=u(0)\prec_{\text{lex}}u(1)\prec_{\text{lex}}\dots\prec_{\text{lex}}u(\zeta(\tau)-1)$
be the sequence of vertices of $\tau$ in lexicographical order. Then, one
readily sees that

$\displaystyle{\rm R}(u(i))=W^{\text{lex}}_{\tau}(i),\hskip
11.38109pt\text{for}\hskip 5.69054pt0\leq i\leq\zeta(\tau)-1.$ (2)

Similarly, if
$\emptyset=u(0)\prec_{\text{rev}}u(1)\prec_{\text{rev}}\dots\prec_{\text{rev}}u(\zeta(\tau)-1)$
are listed in reverse-lexicographical order, then

$\displaystyle{\rm L}(u(i))=W^{\text{rev}}_{\tau}(i),\hskip
11.38109pt\text{for}\hskip 5.69054pt0\leq i\leq\zeta(\tau)-1.$ (3)

##### Height process.

Let
$\emptyset=u(0)\prec_{\text{lex}}u(1)\prec_{\text{lex}}\dots\prec_{\text{lex}}u(\zeta(\tau)-1)$
be the sequence of vertices of $\tau\in\mathbb{T}$ in lexicographical order.
The height process $H_{\tau}=(H_{\tau}(i):0\leq i\leq\zeta(\tau))$ of $\tau$
is defined by letting $H_{\tau}(i)=|u(i)|$, for every
$i\in\\{0,\dots,\zeta(\tau)-1\\}$, and $H_{\tau}(\zeta(\tau))=0$. We sometimes
think of $H_{\tau}$ as a continuous function on $[0,\zeta(\tau)]$, obtained by
linear interpolation.

##### Contour function.

The contour function $C_{\tau}=(C_{\tau}(t),s\in[0,2\zeta(\tau)])$ of
$\tau\in\mathbb{T}$ is defined as follows. Imagine a particle exploring $\tau$
from left to right at unit speed, going backwards when it reaches a leaf. For
all $s\in[0,2\zeta(\tau)-2]$, let $C_{\tau}(s)$ be the distance from the
particle to the root $\emptyset$ at time $s$. By convention, we set
$C_{\tau}(s)=0$ for $s\in[2\zeta(\tau)-2,2\zeta(\tau)]$. In particular,
$C_{\tau}$ is continuous and $C_{\tau}(0)=C_{\tau}(2\zeta(\tau))=0$.

##  3 Convergence of the trees with given degrees

In this section, we prove Theorem 1 which states the convergence of trees with
a given degree sequence toward the Inhomogeneous CRT. In Section 3.1, we first
recall the definition of the exploration process that encodes the
Inhomogeneous CRT. In Section 3.2, we introduce the discrete version of the
above exploration process, the so-called modified $\L$ukasiewicz path, which
encodes a TGDS. We then prove that this modified $\L$ukasiewicz path, suitably
rescaled, converges to the exploration process of the Inhomogeneous CRT, which
implies Theorem 1. Finally, in Section 3.3, we consider a specific case in
which we can prove the convergence of the trees for the Gromov-Hausdorff-
Prokhorov topology.

###  3.1 The exploration process of the Inhomogeneous CRT

Let us start with some definitions. A bridge with exchangeable increments
(abridged EI process) is a continuous-time stochastic process $X^{\rm
bg}=(X^{\rm bg}(t),t\in[0,1])$ with paths in $\mathbf{D}([0,1],\mathbb{R})$
and of the form

$\displaystyle X^{\rm bg}(t)=\sigma B^{\rm
bg}(t)+\sum_{i=1}^{\infty}\beta_{i}(\mathbf{1}_{\\{U_{i}\leq t\\}}-t),\hskip
11.38109ptt\in[0,1],$

where $B^{\rm bg}=(B^{\rm bg}(t),t\in[0,1])$ is a Brownian bridge on $[0,1]$,
$(U_{i},i\geq 1)$ are i.i.d. random variables with uniform law on $[0,1]$
independent of $B^{\rm bg}$, and $\sigma\in\mathbb{R}_{+}$,
$\beta_{1}\geq\beta_{2}\geq\cdots\geq 0$ are constants such that $\sum_{i\geq
1}\beta_{i}^{2}<\infty$. We say that $X^{\rm bg}$ is an EI process with
parameters $(\sigma,(\beta_{i},i\geq 1))$.

The so-called Vervaat transform (or Vervaat excursion) was introduced by
Takács [46] and used by Vervaat [48] to change a bridge-type process with
paths in $\mathbf{D}([0,1],\mathbb{R})$ into an excursion-type process, i.e.,
a non-negative process that is equal to $0$ at times $0$ and $1$. More
precisely, let $X=(X(t),t\in[0,1])$ be a stochastic process with paths in
$\mathbf{D}([0,1],\mathbb{R})$ such that $X(0)=X(1)=X(1-)=0$. Assume that $X$
reaches its infimum value uniquely and continuously at $\rho$. The Vervaat
transform of $X$ is the stochastic process $V(X)=(V(t),t\in[0,1])$ with paths
in $\mathbf{D}([0,1],\mathbb{R})$, defined by $V(t)=X(\\{t+\rho\\})-X(\rho)$,
for $t\in[0,1]$, where $\\{s\\}$ is the fractional part of $s$. In particular,
$V(X)$ is a non-negative process on $[0,1]$.

Through this manuscript, unless otherwise specified, we always consider EI
processes $X^{\rm bg}$ with parameters $(\sigma,(\beta_{i},i\geq 1))$
satisfying (A.4), i.e. either $\sigma^{2}>0$ or $\sum_{i\geq
1}\beta_{i}=\infty$. This is a necessary and sufficient condition for $X^{\rm
bg}$ to have paths of infinite variation. More importantly, by [33] or [15,
Proof of Lemma 6], it is well-known that under this condition $X^{\rm bg}$
almost surely achieves its infimum at a unique time $\rho$ and continuously.
Then, we let $X^{\rm exc}=(X^{\rm exc}(t),t\in[0,1])$ be the excursion-type
process associated to $X^{\rm bg}$ via its Vervaat transform.

The excursion process $X^{\rm exc}$ is not necessarily continuous. However,
following [8, Section 2], one can also associate to $X^{\rm bg}$ a continuous
excursion process $H^{\rm exc}$. For all $i\geq 1$, write
$t_{i}=\\{U_{i}-\rho\\}$ (the fractional part of $U_{i}-\rho$) for the
location of the jump with size $\beta_{i}$ in $X^{\rm exc}$. For each $i\geq
1$ such that $\beta_{i}>0$, write $T_{i}=\inf\\{t\in(t_{i},1]:X^{\rm
exc}(t)=X^{\rm exc}(t_{i}-)\\}$, which exists since the process $X^{\rm exc}$
has no negative jumps and goes back to $0$ at time $1$. In particular, all
$t_{i}$’s and $T_{i}$’s are distinct almost surely. For $u\in[0,1]$ and $i\geq
1$ such that $\beta_{i}>0$, let $R_{i}=(R_{i}(u),u\in[0,1])$ be the process
defined by

$\displaystyle R_{i}(u)=\left\\{\begin{array}[]{lcl}\inf_{t_{i}\leq s\leq
u}X^{\rm exc}(s)-X^{\rm exc}(t_{i}-)&\mbox{ if }&u\in[t_{i},T_{i}],\\\
0&&\mbox{otherwise}.\\\ \end{array}\right.$ (6)

If $\beta_{i}=0$ then let $R_{i}$ be the null process on $[0,1]$. Define the
process $H^{\rm exc}=(H^{\rm exc}(u),u\in[0,1])$ by

$\displaystyle H^{\rm exc}(u)=X^{\rm exc}(u)-\sum_{i\geq 1}R_{i}(u),\hskip
11.38109pt\text{for}\hskip 5.69054ptu\in[0,1].$

As explained in [8, Section 2], $H^{\rm exc}$ is a well-defined continuous
excursion-type process. Furthermore, Aldous, Miermont and Pitman [8, Theorem
1] also showed that $H^{\rm exc}$ can be used to define an exploration process
that encodes the structure of an Inhomogeneous CRT with parameter set
$\theta=(\theta_{0},\theta_{1},\dots)$ satisfying
$\theta_{1}\geq\theta_{2}\geq\cdots\geq 0$, $\theta_{0}>0$, $\sum_{i\geq
0}\theta_{i}^{2}=1$ and $\sum_{i\geq 1}\theta_{i}<\infty$.

###  3.2 The modified $\L$ukasiewicz path

For $n\in\mathbb{N}$, let $\mathbf{s}_{n}$ be a degree sequence,
$\mathbf{t}_{n}$ be a tree sampled uniformly at random in
$\mathbb{T}_{\mathbf{s}_{n}}$ and $W_{n}^{\rm lex}=(W_{n}^{\rm
lex}(V_{n}u),u\in[0,1])$ denote its time-rescaled $\L$ukasiewicz path; recall
that $V_{n}$ denotes the number of vertices in $\mathbf{t}_{n}$.

######  Theorem 4.

Suppose that $\mathbf{s}_{n}$ satisfies (A.1)-(A.4). Then,

$\displaystyle(b_{n}^{-1}W_{n}^{\rm
lex}(V_{n}u),u\in[0,1])\xrightarrow[]{d}(X^{\rm exc}(u),u\in[0,1]),\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty,\hskip
5.69054pt\text{in}\hskip 5.69054pt\mathbf{D}([0,1],\mathbb{R}).$

###### Proof.

It follows as in the proof of [13, Proposition 1], where a similar result is
proved for the so-called breadth-first walk of $\mathbf{t}_{n}$, which has the
same law as its $\L$ukasiewicz path. ∎

We now describe a modification of $W^{\text{lex}}_{n}$ from which we construct
the discrete analogue of $H^{\rm exc}$. For $1\leq i\leq V_{n}$, write
$t_{i}^{n}$ for the location of the jump with size $d^{n}(i)-1$ in
$W^{\text{lex}}_{n}$. In the case where there are $1\leq i<j$ such that
$\beta_{i}=\beta_{k}>0$, for $k=i,\dots,j$ (and $\beta_{k}\neq\beta_{i}$
otherwise), we sort them from left to right by letting
$t_{k}^{n}=t_{(k-i+1)}^{n}$, where $t_{(k-i+1)}^{n}$ is the $(k-i+1)$-th
smallest element of the set $\\{t_{i}^{n},\dots,t_{j}^{n}\\}$. We also set
$T_{i}^{n}=\inf\\{u\in(t_{i}^{n},1]:W^{\text{lex}}_{n}(V_{n}u)-W^{\text{lex}}_{n}(V_{n}t_{i}^{n}-)=-1\\}$.
Let $R_{i}^{n}=(R_{i}^{n}(u),u\in[0,1])$ be the process given by

$\displaystyle R_{i}^{n}(u)=\left\\{\begin{array}[]{lcl}\inf_{t_{i}^{n}\leq
s\leq u}W^{\text{lex}}_{n}(V_{n}s)-W^{\text{lex}}_{n}(V_{n}t_{i}^{n}-)&\mbox{
if }&u\in[t_{i}^{n},T_{i}^{n}],\\\ 0&\mbox{otherwise}.&\\\ \end{array}\right.$

Consider a sequence $(I_{n},n\geq 1)$ of non-negative numbers such that
$I_{n}\uparrow\infty$, as $n\rightarrow\infty$. Then, define the modified
$\L$ukasiewicz path $G_{n}=(G_{n}(V_{n}u),u\in[0,1])$ of $\mathbf{t}_{n}$ by
letting

$\displaystyle G_{n}(V_{n}u)=\left\\{\begin{array}[]{lcl}W^{\rm
lex}_{n}(V_{n}u)&\mbox{ if }&\beta_{1}=0,\\\ W^{\rm
lex}_{n}(V_{n}u)-\sum_{i=1}^{I_{n}}R_{i}^{n}(u)&\mbox{otherwise.}&\\\
\end{array}\right.$

We henceforth denote by $(H^{\rm exc}(u),u\in[0,1])$ the continuous excursion-
type process associated to an EI process with parameters
$(\sigma,(\beta_{i},i\geq 1))$ satisfying (${\bf B}$), i.e., $\sigma^{2}>0$,
$\sum_{i\geq 1}\beta_{i}<\infty$ and $\sigma^{2}+\sum_{i\geq
1}\beta_{i}^{2}=1$; see Section 3.1. Let $\mathbf{C}([0,1],\mathbb{R})$ be the
space of real-valued continuous functions on $[0,1]$ equipped with the uniform
topology.

######  Theorem 5.

Suppose that $\mathbf{s}_{n}$ satisfies (A.1)-(A.3) and (${\bf B}$). Then,
there exists a non-decreasing sequence $(I_{n},n\geq 1)$ of non-negative
numbers such that

$\lim_{n\rightarrow\infty}I_{n}/b_{n}=0,\hskip
8.53581pt\lim_{n\rightarrow\infty}I_{n}=\infty\hskip 8.53581pt\text{and}\hskip
8.53581pt\lim_{I\rightarrow\infty}\limsup_{n\rightarrow\infty}b_{n}^{-1}\sum_{I\leq
i\leq I_{n}}d^{n}(i)=0.$ (${\bf S}$)

Moreover, for $G_{n}$ constructed from such a sequence $(I_{n},n\geq 1)$, we
have that

$\displaystyle\left(b_{n}^{-1}G_{n}(V_{n}u),b_{n}^{-1}W^{{\rm
lex}}_{n}(V_{n}u):u\in[0,1]\right)\xrightarrow[]{d}(H^{\rm exc}(u),X^{\rm
exc}(u):u\in[0,1]),\hskip 8.53581pt\text{as}\hskip
5.69054ptn\rightarrow\infty,$

in $\mathbf{D}([0,1],\mathbb{R})\otimes\mathbf{D}([0,1],\mathbb{R})$. In
particular,$(b_{n}^{-1}G_{n}(V_{n}u),u\in[0,1])$ converges to $H^{\rm exc}$
uniformly on $[0,1]$.

###### Proof.

First, we prove the existence of a non-decreasing sequence $(I_{n},n\geq 1)$
that satisfies (${\bf S}$). Since $\sum_{i\geq 1}\beta_{i}<\infty$ by (${\bf
B}$), there exists an increasing sequence $(j_{k},k\geq 1)$ such that
$\sum_{i\geq j_{k}}\beta_{i}<k^{-2}$, for every $k\geq 1$. Now, by (${\bf B}$)
again, for every fixed $k\geq 1$ and $j\geq j_{k}$, we have that
$b_{n}^{-1}\sum_{j_{k}\leq i\leq j}d^{n}(i)<k^{-2}$ for $n$ large enough.
Hence, there exists $N(k,j)$ such that for any $n\geq N(k,j)$ we have that
$b_{n}^{-1}\sum_{j_{k}\leq i\leq j}d^{n}(i)<k^{-2}$. So, for every fixed
$k\geq 1$, there exists a non-decreasing sequence $(J_{k,n},n\geq 1)$ such
that $\lim_{n\rightarrow\infty}J_{k,n}=\infty$ and for $n$ large enough
(depending on $k$), $b_{n}^{-1}\sum_{j_{k}\leq i\leq J_{k,n}}d^{n}(i)<k^{-2}$.
Finally, the sequence $(I_{n},n\geq 1)$ defined by $I_{1}=j_{1}$,
$I_{n}=J_{1,n}$ until $J_{1,n}>j_{2}$, $I_{n}=J_{2,n}$ until $J_{2,n}\geq
j_{3}$, and so on, satisfies (${\bf S}$); it is clear that we can construct
$(I_{n},n\geq 1)$ such that $I_{n}/b_{n}\rightarrow 0$, as
$n\rightarrow\infty$.

We now prove the second claim of Theorem 5. Recall that the modified
$\L$ukasiewicz path $G_{n}$ is defined with $(I_{n},n\geq 1)$ such that (${\bf
S}$) is fulfilled. If $\beta_{1}=0$, our claim follows from Theorem 4. Suppose
now that $\beta_{1}>0$. By the Skorohod representation theorem, we can and
will assume that the convergence of Theorem 4 holds almost surely. Consider
$I\geq 1$ such that $\beta_{I}>\beta_{I+1}$. For $1\leq i\leq I$, recall that
$t_{i}$ denotes the location of the jump with size $\beta_{i}$ of $X^{{\rm
exc}}$ and that $T_{i}=\inf\\{t\in(t_{i},1]:X^{\rm exc}(t)=X^{\rm
exc}(t_{i}-)\\}$. Recall also that $R_{i}=(R_{i}(u),u\in[0,1])$ denotes the
process defined in (6). We claim that almost surely, for each $1\leq i\leq I$,

  * (i)

$\lim_{n\rightarrow\infty}t_{i}^{n}=t_{i}$,

  * (ii)

$\lim_{n\rightarrow\infty}T_{i}^{n}=T_{i}$ and

  * (iii)

$\left(b_{n}^{-1}R_{n}^{i}(u),u\in[0,1]\right)\rightarrow(R_{i}(u),u\in[0,1])$,
as $n\rightarrow\infty$, in $\mathbf{D}([0,1],\mathbb{R})$.

Assume that this holds, and let $J(X^{{\rm exc}})=\\{u\in[0,1]:\Delta X^{{\rm
exc}}(u)=0\\}$ be the set of continuity points of $X^{{\rm exc}}$. For $1\leq
i\leq I$, the set of continuity points of $R_{i}$ is
$[0,1]\setminus\\{t_{i}\\}$. Then, it follows from [31, Proposition 2.1 in
Chapter VI] that for each $u\in J(X^{{\rm exc}})$ there is a sequence
$(u_{n},n\geq 0)$ such that $\lim_{n\rightarrow\infty}u_{n}=u$ and

$\displaystyle\lim_{n\rightarrow\infty}b_{n}^{-1}\Delta
R_{n}^{i}(u_{n})=\Delta R_{i}(u)=0=\Delta X^{{\rm
exc}}(u)=\lim_{n\rightarrow\infty}b_{n}^{-1}\Delta W^{{\rm
lex}}_{n}(V_{n}u_{n}).$

On the other hand, for $1\leq i\leq I$, it follows from (A.2) that

$\displaystyle\lim_{n\rightarrow\infty}b_{n}^{-1}\Delta
R_{n}^{i}(t_{i}^{n})=\Delta R_{i}(t_{i})=\beta_{i}=\Delta X^{{\rm
exc}}(t_{i})=\lim_{n\rightarrow\infty}b_{n}^{-1}\Delta W^{{\rm
lex}}_{n}(V_{n}t_{i}^{n}).$

Moreover, for $1\leq j\leq I$ such that $j\neq i$, (A.2) also implies that

$\displaystyle\lim_{n\rightarrow\infty}b_{n}^{-1}\Delta
R_{n}^{i}(t_{j}^{n})=\Delta R_{i}(t_{j})=0.$

Denote by $\Theta_{1}$ the class of strictly increasing, continuous mappings
of $[0,1]$ onto itself. Then (iii), [31, Proposition 2.2 in Chapter VI] and
[31, Theorem 1.14 in Chapter VI] imply that there exists a sequence of
functions $(\theta_{n},n\geq 1)\in\Theta_{1}$ such that

$\displaystyle\lim_{n\rightarrow\infty}\sup_{u\in[0,1]}|\theta_{n}(u)-u|=0\hskip
5.69054pt\text{and}\hskip
5.69054pt\lim_{n\rightarrow\infty}\sup_{u\in[0,1]}\left|\frac{1}{b_{n}}W^{{\rm
lex}}_{n}(V_{n}u)-\frac{1}{b_{n}}\sum_{1\leq i\leq
I}R_{n}^{i}(\theta_{n}(u))-X^{\rm exc}(u)+\sum_{1\leq i\leq
I}R_{i}(u)\right|=0.$

Therefore, our claim in Theorem 5 follows from the above convergence, [31,
Theorem 1.14 in Chapter VI] and the triangle inequality provided that

$\displaystyle\lim_{I\rightarrow\infty}\limsup_{n\rightarrow\infty}\sup_{u\in[0,1]}\left|\frac{1}{b_{n}}\sum_{I\leq
i\leq I_{n}}R_{n}^{i}(\theta_{n}(u))\right|=0\hskip 5.69054pt\text{and}\hskip
5.69054pt\lim_{I\rightarrow\infty}\sup_{u\in[0,1]}\left|\sum_{i\geq
I}R_{i}(u)\right|=0.$

This follows from (${\bf B}$) and (${\bf S}$) since
$\sup_{u\in[0,1]}|R_{n}^{i}(\theta_{n}(u))|\leq|d^{n}(i)-1|$ and
$\sup_{u\in[0,1]}|R_{n}^{i}(u)|\leq\beta_{i}$. The “in particular” follows
from the continuity of $H^{\rm exc}$ and [31, Proposition 1.17 in Chapter VI].

We now proceed to prove (i), (ii) and (iii). We start with the proof of (i).
Assume, without loss of generality, that $0=t_{0}<t_{1}<\cdots<t_{I}<1$. If
$0<u<t_{1}$ then $u\in J(X^{{\rm exc}})$, $\sup_{s\in[0,u]}\Delta X^{{\rm
exc}}(s)=0$ and thus [31, Proposition 2.4 in Chapter VI] shows that almost
surely

$\lim_{n\rightarrow\infty}\sup_{s\in[0,u]}b_{n}^{-1}\Delta W^{{\rm
lex}}_{n}(V_{n}s)=\sup_{s\in[0,u]}\Delta X^{{\rm exc}}(s)=0.$

So, for all $n$ large enough $u<t_{1}^{n}$. Since $J(X^{{\rm exc}})$ is dense
in $[0,1]$, we obtain that $\liminf_{n\rightarrow\infty}t_{1}^{n}\geq t_{1}$.
If $u>t_{1}$ such that $u\in J(X^{{\rm exc}})$, then $\sup_{s\in[0,u]}|\Delta
X^{{\rm exc}}(s)|=\beta_{1}$, and [31, Proposition 2.4 in Chapter VI] implies
that almost surely

$\lim_{n\rightarrow\infty}\sup_{s\in[0,u]}b_{n}^{-1}|\Delta W^{{\rm
lex}}_{n}(V_{n}s)|=\sup_{s\in[0,u]}|\Delta X^{{\rm exc}}(s)|=\beta_{1}.$

So, for $n$ large enough $u\geq t_{1}^{n}$ which implies that
$\limsup_{n\rightarrow\infty}t_{1}^{n}\leq t_{1}$ and thus,
$\lim_{n\rightarrow\infty}t_{1}^{n}=t_{1}$ almost surely. For $u\in[0,1]$,
consider the processes

$\displaystyle\hat{W}^{{\rm lex}}_{n}(V_{n}u)=W^{{\rm lex}}_{n}(V_{n}u)-\Delta
W^{{\rm lex}}_{n}(V_{n}t_{1}^{n})\mathbf{1}_{\\{t_{1}^{n}\leq u\\}}\hskip
5.69054pt\text{and}\hskip 5.69054pt\hat{X}^{{\rm exc}}(u)=X^{{\rm
exc}}(s)-\Delta X^{{\rm exc}}(t_{1})\mathbf{1}_{\\{t_{1}\leq u\\}}.$

Theorem 4 and [31, Proposition 2.1 in Chapter VI] imply that

$\displaystyle(b_{n}^{-1}\hat{W}^{{\rm
lex}}_{n}(V_{n}u),u\in[0,1])\xrightarrow[]{a.s.}(\hat{X}^{{\rm
exc}}(u),u\in[0,1]),\hskip 5.69054pt\text{as}\hskip
5.69054ptn\rightarrow\infty,$ (9)

in $\mathbf{D}([0,1],\mathbb{R})$. If $0<u<t_{2}$ such that $u\in J(X^{{\rm
exc}})$, then $\sup_{s\in[0,u]}\Delta\hat{X}^{{\rm exc}}(s)=0$ and thus, (9)
and [31, Proposition 2.4 in Chapter VI] shows that almost surely

$\lim_{n\rightarrow\infty}\sup_{s\in[0,u]}b_{n}^{-1}\Delta\hat{W}^{{\rm
lex}}_{n}(V_{n}s)=\sup_{s\in[0,u]}\Delta\hat{X}^{{\rm exc}}(s)=0.$

So, for all $n$ large enough $u<t_{2}^{n}$ which implies that
$\liminf_{n\rightarrow\infty}t_{2}^{n}\geq t_{2}$. If $u>t_{2}$ such that
$u\in J(X^{{\rm exc}})$, then $\sup_{s\in[0,u]}|\Delta\hat{X}^{{\rm
exc}}(s)|=\beta_{2}$, and (9) and [31, Proposition 2.4 in Chapter VI] implies
that almost surely

$\lim_{n\rightarrow\infty}\sup_{s\in[0,u]}b_{n}^{-1}|\Delta\hat{W}^{{\rm
lex}}_{n}(V_{n}s)|=\sup_{s\in[0,u]}|\Delta\hat{X}^{{\rm exc}}(s)|=\beta_{2}.$

So, for all $n$ large enough $u\geq t_{2}^{n}$. We deduce that
$\limsup_{n\rightarrow\infty}t_{2}^{n}\leq t_{2}$ and thus,
$\lim_{n\rightarrow\infty}t_{2}^{n}=t_{2}$ almost surely. Then, an inductive
argument proves (i).

Next, we prove (ii). If $u\in[t_{i},T_{i})\cap J(X^{{\rm exc}})$, then (i) and
[31, Proposition 2.1 in Chapter VI] imply that almost surely
$\lim_{n\rightarrow\infty}b_{n}^{-1}R_{n}^{i}(u)=R_{i}(u)>0$. So, for all $n$
large enough $u\in[t_{i}^{n},T_{i}^{n})$, which implies that
$\liminf_{n\rightarrow\infty}T_{i}^{n}\geq T_{i}$. Next, suppose that
$t^{\prime}=\limsup_{n\rightarrow\infty}T_{i}^{n}>T_{i}$ and up to extraction
suppose that $t^{\prime}$ is actually the limit of $(T_{i}^{n},n\geq 1)$.
Since (i) and [31, Proposition 2.1 in Chapter VI] imply that
$\lim_{n\rightarrow\infty}b_{n}^{-1}W^{\text{lex}}_{n}(V_{n}T_{i}^{n})=X^{{\rm
exc}}(t_{i}-)$, we would find that $t^{\prime}>T_{i}$ with $X^{\rm
exc}_{t^{\prime}}=X^{{\rm exc}}(t_{i}-)$ and $X^{\rm exc}_{s}\geq X^{{\rm
exc}}(t_{i}-)$, for $s\in[T_{i},t^{\prime}]$. This shows that $X^{{\rm
exc}}(t_{i}-)$ is a local minimum of $X^{{\rm exc}}$, attained at time
$T_{i}$, which is almost surely impossible by [8, Lemma 1]. Therefore,
$\lim_{n\rightarrow\infty}T_{i}^{n}=T_{i}$ which proves (ii).

Finally, we prove (iii). For simplicity, we assume that $I=1$ and we leave the
general case to the reader. Recall that $\Theta_{1}$ denotes the class of
strictly increasing, continuous mappings of $[0,1]$ onto itself. By [31,
Theorem 1.14 in Chapter VI], it is enough to show that there exists a sequence
of functions $(\theta_{n},n\geq 1)\in\Theta_{1}$ such that

$\displaystyle\lim_{n\rightarrow\infty}\sup_{u\in[0,1]}|\theta_{n}(u)-u|=0\hskip
8.53581pt\text{and}\hskip
8.53581pt\lim_{n\rightarrow\infty}\sup_{u\in[0,1]}|b_{n}^{-1}R_{n}^{1}(\theta_{n}(u))-R_{1}(u)|=0.$

For $n\geq 1$, define $\theta_{n}$ by letting $\theta_{n}(0)=0$,
$\theta_{n}(t_{1})=t_{1}^{n}$, $\theta_{n}(T_{1})=T_{1}^{n}$ and
$\theta_{n}(1)=1$ such that $\theta_{n}$ is linear on $[0,t_{1}]$, on
$[t_{1},T_{1}]$ and on $[T_{1},1]$. Clearly, (i) and (ii) imply that
$\theta_{n}$ converges uniformly to the identity mapping on $[0,1]$, as
$n\rightarrow\infty$. For $u\in[t_{1},T_{1}]$, we see that

$\displaystyle\left|\inf_{t_{1}^{n}\leq
s\leq\theta_{n}(u)}b_{n}^{-1}W^{\text{lex}}_{n}(V_{n}s)-\inf_{t_{1}\leq s\leq
u}X^{\text{exc}}(s)\right|\leq\sup_{t_{1}\leq s\leq
u}\left|b_{n}^{-1}W^{\text{lex}}_{n}(V_{n}\theta_{n}(s))-X^{\text{exc}}(s)\right|.$

Then, the triangle inequality implies that

$\displaystyle\sup_{u\in[0,1]}|b_{n}^{-1}R_{n}^{1}(\theta_{n}(u))-R_{1}(u)|\leq\sup_{t_{1}\leq
u\leq
T_{1}}\left|b_{n}^{-1}W^{\text{lex}}_{n}(V_{n}\theta_{n}(u))-X^{\text{exc}}(u)\right|+\left|b_{n}^{-1}W^{\text{lex}}_{n}(V_{n}t_{i}^{n}-)-X^{\text{exc}}(t_{i}-)\right|.$

On the one hand, the second term on the right-hand side converges, as
$n\rightarrow\infty$, toward zero due to (i) and [31, Proposition 2.1 in
Chapter VI]. On the other hand, the continuity of $X^{\text{exc}}$ on
$[t_{1},T_{1}]$ together with Theorem 4 and [49, Theorem 3.1] implies that the
first term on the right-hand side converges, as $n\rightarrow\infty$, toward
zero. This concludes the proof of (iii). ∎

Let $(H_{n}(V_{n}u),u\in[0,1])$ be the (time-scaled) height process associated
to $\mathbf{t}_{n}$.

######  Theorem 6.

Suppose that $\mathbf{s}_{n}$ satisfies assumptions (A.1)-(A.3) and (${\bf
B}$). Fix $q\geq 1$, let $U_{1},\dots,U_{q}$ be i.i.d. uniform random
variables in $[0,1]$ independent of the rest and denote by
$0=U_{(0)}<U_{(1)}<\cdots<U_{(q)}$ their ordered statistics. Then,

$\displaystyle\frac{\sigma^{2}}{2}\frac{b_{n}}{V_{n}}\left(H_{n}(V_{n}U_{(i)}),\inf_{U_{(i-1)}\leq
u\leq U_{(i)}}H_{n}(V_{n}u)\right)_{1\leq i\leq
q}\xrightarrow[]{d}\left(H^{\rm exc}(U_{(i)}),\inf_{U_{(i-1)}\leq u\leq
U_{(i)}}H^{\rm exc}(u)\right)_{1\leq i\leq q},\hskip 8.53581pt\text{as}\hskip
5.69054ptn\rightarrow\infty,$

holds jointly with that in Theorem 5.

###### Proof of Theorem 1.

It follows from Theorem 6 and [8, Theorem 1]. ∎

As a preparation for the proof of Theorem 6, we need the following property.

######  Lemma 3.1.

Suppose that $\mathbf{s}_{n}$ satisfies assumptions (A.1)-(A.4). Then,
$\lim_{n\rightarrow\infty}V_{n}/b_{n}=\infty$.

###### Proof.

First, suppose that $\sum_{i\geq 1}\beta_{i}=\infty$ in (A.4). Then, our claim
follows from (A.1) and (A.2) since $V_{n}\geq 1+\sum_{i=1}^{k}d^{n}(i)$ for
every fixed $1\leq k\leq V_{n}$. Suppose now that $\sigma^{2}>0$ in (A.4) and
that our claim does not hold, i.e., there exists a constant $C>0$ such that,
along a subsequence, $V_{n}<Cb_{n}$. Then, for any $\varepsilon>0$,

$\displaystyle\frac{1}{b_{n}^{2}}\sum_{i\geq
0}(i-1)^{2}N_{i}^{n}\leq\sum_{i\geq
0}\frac{i^{2}}{b_{n}^{2}}N_{i}^{n}=\sum_{i\leq\lfloor\varepsilon
b_{n}\rfloor}\frac{i^{2}}{b_{n}^{2}}N_{i}^{n}+\sum_{i>\lfloor\varepsilon
b_{n}\rfloor}\frac{i^{2}}{b_{n}^{2}}N_{i}^{n}\leq\varepsilon
C+\sum_{i>\lfloor\varepsilon b_{n}\rfloor}\frac{i^{2}}{b_{n}^{2}}N_{i}^{n},$
(10)

where we have used that $V_{n}=\sum_{i\geq 0}N_{i}=1+\sum_{i\geq 0}iN_{i}$.
Since we have assumed $V_{n}<Cb_{n}$, we necessarily have
$\sum_{i>\lfloor\varepsilon b_{n}\rfloor}N_{i}^{n}\leq\lceil
C/\varepsilon\rceil$ (by considering the number of children of these
vertices). Hence,

$\displaystyle\sum_{i>\lfloor\varepsilon
b_{n}\rfloor}\frac{i^{2}}{b_{n}^{2}}N_{i}^{n}\leq\sum_{i\leq\lceil
C/\epsilon\rceil}\frac{(d^{n}(i))^{2}}{b_{n}^{2}}.$

By (A.2), we get that

$\displaystyle\limsup_{n\rightarrow\infty}\sum_{i>\lfloor\varepsilon
b_{n}\rfloor}\frac{i^{2}}{b_{n}^{2}}N_{i}^{n}\leq\sum_{i\leq\lceil
C/\epsilon\rceil}\beta_{i}^{2}\leq\sum_{i\geq 1}\beta_{i}^{2}.$

Taking $\varepsilon<\sigma^{2}/C$ provides a contradiction between (10) and
(A.3). ∎

###### Proof of Theorem 6.

Let $U$ have the uniform distribution on $[0,1]$ independently of the tree and
let $u_{n}$ be the $\lfloor V_{n}U\rfloor$-th vertex of $\mathbf{t}_{n}$ in
lexicographical order, so that it has the uniform distribution in
$\mathbf{t}_{n}$. Observe that $H_{n}(\lfloor V_{n}U\rfloor)=|u_{n}|$. Let
$G_{n}$ be the modified $\L$ukasiewicz path associated with $\mathbf{t}_{n}$
and defined with a sequence $(I_{n},n\geq 1)$ such that (${\bf S}$) is
fulfilled. If $\beta_{1}=0$, we set $\mathbf{d}_{n}=\varnothing$, while if
$\beta_{1}>0$, we consider the set
$\mathbf{d}_{n}=\\{d^{n}(1),\dots,d^{n}(I_{n})\\}$ of the first $I_{n}$
largest degrees. So, except for vertices that are children of vertices with
degree among those of the set $\mathbf{d}_{n}$, let $\hat{{\rm R}}(u_{n})$ be
the number of individuals branching-off strictly to the right of the ancestral
line $\llbracket\emptyset,u_{n}\llbracket$ in $\mathbf{t}_{n}$. We claim that

$\displaystyle\left|\frac{1}{\hat{\sigma}_{n}}\hat{{\rm
R}}(u_{n})-\frac{\hat{\sigma}_{n}}{2\hat{E}_{n}}|u_{n}|\right|\xrightarrow[]{\mathbb{P}}0,\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty,$ (11)

where $\hat{\sigma}_{n}^{2}=\sum_{k\not\in\mathbf{d}_{n}}k(k-1)N_{k}^{n}$ and
$\hat{E}_{n}=\sum_{k\not\in\mathbf{d}_{n}}kN_{k}^{n}$. According to (2) and
the definition of $G_{n}$, we see that $\hat{{\rm R}}(u_{n})=G_{n}(\lfloor
V_{n}U\rfloor)$. Then (11) and Theorem 5 imply that

$\displaystyle\frac{\sigma^{2}}{2}\frac{b_{n}}{V_{n}}\left(H_{n}(V_{n}U_{(i)})\right)_{1\leq
i\leq q}\xrightarrow[]{d}\left(H^{\rm exc}(U_{(i)})\right)_{1\leq i\leq
q},\hskip 8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty.$

Recall the notation ${\rm LR}(u_{n})={\rm L}(u_{n})+{\rm R}(u_{n})$ for the
total number individuals branching-off the ancestral line of $u_{n}$
introduced in Section 2. Fix $\varepsilon,\eta>0$. By [42, Proposition 4 and
Proposition 5] and our assumptions, we can and will consider $K>0$ such that

$\displaystyle\mathbb{P}\left({\rm LR}(u_{n})\leq Kb_{n}\hskip
2.84526pt\text{and}\hskip 2.84526pt|u_{n}|\leq K\frac{V_{n}}{b_{n}}\right)\geq
1-\eta.$

Then,

$\displaystyle\mathbb{P}\left(\left|\frac{1}{\hat{\sigma}_{n}}\hat{{\rm
R}}(u_{n})-\frac{\hat{\sigma}_{n}}{2\hat{E}_{n}}|u_{n}|\right|>\varepsilon\right)\leq\eta+\mathbb{P}\left(\left|\hat{{\rm
R}}(u_{n})-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}|u_{n}|\right|>\varepsilon\hat{\sigma}_{n},\hskip
2.84526pt{\rm LR}(u_{n})\leq Kb_{n}\hskip 2.84526pt\text{and}\hskip
2.84526pt|u_{n}|\leq K\frac{V_{n}}{b_{n}}\right).$

Suppose that an urn contains initially $kN_{k}^{n}$ balls labelled $k$ for
every $k\geq 1$, so $E_{n}$ balls in total. Let us pick balls repeatedly one
after the other without replacement. For every $1\leq i\leq E_{n}$, we denote
the label of the $i$-th ball by $\xi_{n}(i)$. Conditionally on
$(\xi_{n}(i),1\leq i\leq E_{n})$, let us sample independent random variables
$(\chi_{n}(i),1\leq i\leq E_{n})$ such that each $\chi_{n}(i)$ is uniformly
distributed in $\\{1,\dots,\xi_{n}(i)\\}$. The spinal decomposition obtained
in [42, Lemma 3] (see also [20, Section 3]) with $q=1$ shows that the
probability of the event that $|u_{n}|=h$ and that for all $0\leq i<h$, the
ancestor of $u_{n}$ at generation $i$ has $k_{i}$ offspring and its $j_{i}$-th
is the ancestor of $u_{n}$ at generation $i+1$ is bounded by

$\displaystyle\frac{1+\sum_{1\leq i\leq
h}(k_{i}-1)}{V_{n}}\cdot\mathbb{P}\left(\bigcap_{i\leq
h}\\{(\xi_{n}(i),\chi_{n}(i))=(k_{i},j_{i})\\}\right).$

Note also that, in the previous event, $\hat{{\rm R}}(u_{n})=\sum_{i\leq
h}(k_{i}-j_{i})\mathbf{1}_{\\{k_{i}\not\in\mathbf{d}_{n}\\}}$. Thus, by
decomposing according to the height of $u_{n}$ and taking the worst case (i.e.
the union bound), we then obtain that

$\displaystyle\mathbb{P}\left(\left|\frac{1}{\hat{\sigma}_{n}}\hat{{\rm
R}}(u_{n})-\frac{\hat{\sigma}_{n}}{2\hat{E}_{n}}|u_{n}|\right|>\varepsilon\right)\leq\eta+K^{2}\sup_{h\leq
KV_{n}/b_{n}}\mathbb{P}\left(\left|\sum_{i\leq
h}(\xi_{n}(i)-\chi_{n}(i))\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}-\frac{\hat{\sigma}_{n}^{2}}{2E_{n}}h\right|>\varepsilon\hat{\sigma}_{n}\right).$

From the triangle inequality, the last probability on the right is bounded
above by

$\displaystyle\mathbb{P}\left(\left|\sum_{i\leq
h}\left(\xi_{n}(i)-\chi_{n}(i)-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\right|>\frac{\varepsilon\hat{\sigma}_{n}}{2}\right)+\mathbb{P}\left(\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\sum_{i\leq
h}\mathbf{1}_{\\{\xi_{n}(i)\in\mathbf{d}_{n}\\}}>\frac{\varepsilon\hat{\sigma}_{n}}{2}\right).$

Since the $\xi_{n}(i)$’s are identically distributed, the Markov inequality
then yields for every $h\leq KV_{n}/b_{n}$

$\displaystyle\mathbb{P}\left(\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\sum_{i\leq
h}\mathbf{1}_{\\{\xi_{n}(i)\in\mathbf{d}_{n}\\}}>\frac{\varepsilon\hat{\sigma}_{n}}{2}\right)\leq\frac{h\hat{\sigma}_{n}}{\varepsilon\hat{E}_{n}}\sum_{1\leq
k\leq
I_{n}}\frac{d^{n}(k)}{E_{n}}\leq\frac{V_{n}\hat{\sigma}_{n}}{\varepsilon\hat{E}_{n}E_{n}b_{n}}\sum_{1\leq
k\leq I_{n}}d^{n}(k),$

(with the convention $\sum_{1\leq k\leq I_{n}}d^{n}(k)=0$ if
$\mathbf{d}_{n}=\varnothing$) which converges to $0$ by (A.1)-(A.3), (${\bf
B}$), (${\bf S}$) and Lemma 3.1. Hence

$\displaystyle\mathbb{P}\left(\left|\frac{1}{\hat{\sigma}_{n}}\hat{{\rm
R}}(u_{n})-\frac{\hat{\sigma}_{n}}{2\hat{E}_{n}}|u_{n}|\right|>\varepsilon\right)$
$\displaystyle\leq\eta+\frac{V_{n}\hat{\sigma}_{n}}{\varepsilon\hat{E}_{n}E_{n}b_{n}}\sum_{1\leq
k\leq I_{n}}d^{n}(k)$ $\displaystyle\hskip 14.22636pt+K^{2}\sup_{h\leq
KV_{n}/b_{n}}\mathbb{P}\left(\left|\sum_{i\leq
h}\left(\xi_{n}(i)-\chi_{n}(i)-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\right|>\frac{\varepsilon\hat{\sigma}_{n}}{2}\right).$

Another application of the triangle inequality shows that the last probability
on the right is bounded above by

$\displaystyle\mathbb{P}\left(\left|\sum_{i\leq
h}\left(\xi_{n}(i)-\chi_{n}(i)-\frac{\xi_{n}(i)-1}{2}\right)\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\right|>\frac{\varepsilon\hat{\sigma}_{n}}{4}\right)+\mathbb{P}\left(\left|\sum_{i\leq
h}\left(\frac{\xi_{n}(i)-1}{2}-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\right|>\frac{\varepsilon\hat{\sigma}_{n}}{4}\right).$

Note that the pairs $(\xi_{n}(i),\xi_{n}(j))$, for $i\neq j$, are identically
distributed. Then

$\displaystyle\mathbb{E}\left[\left(\sum_{i\leq
h}\left(\frac{\xi_{n}(i)-1}{2}-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\right)^{2}\right]$
$\displaystyle=h\mathbb{E}\left[\left(\frac{\xi_{n}(i)-1}{2}-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)^{2}\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\right]$
$\displaystyle\hskip
14.22636pt+\frac{h(h-1)}{2}\mathbb{E}\left[\left(\frac{\xi_{n}(i)-1}{2}-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\left(\frac{\xi_{n}(j)-1}{2}-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)\mathbf{1}_{\\{\xi_{n}(j)\not\in\mathbf{d}_{n}\\}}\right]$
$\displaystyle=h\mathbb{E}\left[\left(\frac{\hat{\xi}_{n}(i)-1}{2}-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)^{2}\right]\mathbb{P}(\xi_{n}(i)\not\in\mathbf{d}_{n})$
$\displaystyle\hskip
14.22636pt+\frac{h(h-1)}{2}\mathbb{E}\left[\left(\frac{\hat{\xi}_{n}(i)-1}{2}-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)\left(\frac{\hat{\xi}_{n}(j)-1}{2}-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)\right]\mathbb{P}(\xi_{n}(i)\not\in\mathbf{d}_{n},\xi_{n}(j)\not\in\mathbf{d}_{n}),$

where $\hat{\xi}_{n}(i)$ denotes the label of the $i$-th ball picked from an
urn that contains initially $kN_{k}^{n}$ balls labelled $k$ for every $k\geq
1$ and $k\not\in\mathbf{d}_{n}$, so $\hat{E}_{n}$ balls in total. In
particular,

$\displaystyle\mathbb{E}[\hat{\xi}_{n}(i)-1]=\sum_{k\not\in\mathbf{d}_{n}}(k-1)\frac{kN_{k}^{n}}{\hat{E}_{n}}=\frac{\hat{\sigma}_{n}^{2}}{\hat{E}_{n}}\hskip
8.53581pt\text{and so}\hskip 8.53581pt{\rm
Var}(\hat{\xi}_{n}(i)-1)\leq\mathbb{E}[(\hat{\xi}_{n}(i)-1)^{2}]\leq
d^{n}(I_{n}+1)\frac{\hat{\sigma}_{n}^{2}}{\hat{E}_{n}};$

with the convention $d^{n}(I_{n}+1)=d^{n}(1)$ whenever
$\mathbf{d}_{n}=\varnothing$. Furthermore, the random variables
$\hat{\xi}_{n}(i)$’s are obtained by successive picks without replacement in
an urn, and therefore are negatively correlated; see [12, Proposition 20.6].
Thus,

$\displaystyle\mathbb{E}\left[\left(\sum_{i\leq
h}\left(\frac{\xi_{n}(i)-1}{2}-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\right)^{2}\right]\leq
Khd^{n}(I_{n}+1)\frac{\hat{\sigma}_{n}^{2}}{\hat{E}_{n}}.$

The Markov inequality then yields for every $h\leq KV_{n}/b_{n}$,

$\displaystyle\mathbb{P}\left(\left|\sum_{i\leq
h}\left(\frac{\xi_{n}(i)-1}{2}-\frac{\hat{\sigma}_{n}^{2}}{2\hat{E}_{n}}\right)\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\right|>\frac{\varepsilon\hat{\sigma}_{n}}{4}\right)\leq
K\frac{d^{n}(I_{n}+1)V_{n}}{\varepsilon^{2}b_{n}\hat{E}_{n}}.$

which converges to $0$ by (${\bf S}$). Moreover, conditionally on the
$\xi_{n}(i)$’s, the random variables $\xi_{n}(i)-\chi_{n}(i)$ are independent
and uniformly distributed on $\\{0,\dots,\xi_{n}(i)-1\\}$, with mean
$(\xi_{n}(i)-1)/2$ and variance $(\xi_{n}(i)^{2}-1)/12$. Similarly, the Markov
inequality applied conditionally on the $\xi_{n}(i)$’s yields for every $h\leq
KV_{n}/b_{n}$:

$\displaystyle\mathbb{P}\left(\left|\sum_{i\leq
h}\left(\xi_{n}(i)-\chi_{n}(i)-\frac{\xi_{n}(i)-1}{2}\right)\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\right|>\frac{\varepsilon\hat{\sigma}_{n}}{4}\right)$
$\displaystyle\leq$
$\displaystyle\frac{16}{\varepsilon^{2}\hat{\sigma}_{n}^{2}}\mathbb{E}\left[\sum_{i\leq
h}\frac{\xi_{n}(i)^{2}-1}{12}\mathbf{1}_{\\{\xi_{n}(i)\not\in\mathbf{d}_{n}\\}}\right]$
$\displaystyle\leq$ $\displaystyle
K\frac{4(d^{n}(I_{n}+1)+1)V_{n}}{3\varepsilon^{2}b_{n}E_{n}},$

which converges to $0$ by (${\bf S}$). Thus, a combination of the previous
estimates proves (11).

The full statement of the Theorem follows from similar computations and the
argument used at the end of the proof of [42, Theorem 5]. ∎

###  3.3 Gromov-Hausdorff-Prokhorov convergence of a specific model of TGDS

In this section, we consider a specific case in which the convergence of trees
with given degree sequence holds for the Gromov-Hausdorff-Prokhorov topology.

######  Proposition 1.

Suppose that $\mathbf{s}_{n}$ satisfies (A.1)-(A.3) and (${\bf B}$). If
moreover, $V_{n}^{-1}b_{n}^{2}\rightarrow 1$, as $n\rightarrow\infty$, and
$\sup_{n\geq 1}V_{n}^{-1}N_{1}^{n}<1$, then

$\displaystyle\left(\mathbf{t}_{n},b_{n}^{-1}r_{n}^{{\rm
gr}},\rho_{n},\mu_{n}\right)\xrightarrow[]{d}(\mathcal{T}_{\theta},r_{\theta},\rho_{\theta},\mu_{\theta}),\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty,$

for the Gromov-Hausdorff-Prokhorov topology, where $\mathcal{T}_{\theta}$ is
an Inhomogeneous CRT with parameter set $\theta=(\theta_{0},\theta_{1},\dots)$
given by $\theta_{0}=\sigma$, $\theta_{i}=\beta_{i}$, for $i\geq 1$.

Although Blanc-Renaudie [18, Theorem 7 (a)-(b)] provided necessary conditions
for the Gromov-Hausdorff-Prokhorov convergence, these conditions do not seem
to be easy to verify, as indicated in [18]. So we decided to include this
particular case as the proof is different and may be of interest. The proof of
Proposition 1 is just a simple adaptation of the argument used in the proof of
[20, Theorems 1 and 3]. The idea is to show that the $\L$ukasiewicz path and
the height process are asymptotically proportional whenever the degree
sequence satisfies the additional conditions in the statement of Proposition
1. Therefore, we only sketch the main ideas and leave the details to the
interested reader.

For a rooted plane tree $\tau$, denote by $u\wedge v$ the (deepest) first
common ancestor of $u,v\in\tau$ and write $u\preccurlyeq v$ to mean that $u$
is an ancestor of $v$ in $\tau$ ($u=v$ is allowed). Let
$\emptyset=u(0)\prec_{\text{lex}}u(1)\prec_{\text{lex}}\dots\prec_{\text{lex}}u(\zeta(\tau)-1)$
be the sequence of vertices of $\tau$ in lexicographical order. For
$0<\delta<1$, if $|i-j|<\delta\zeta(\tau)$ we say that $u(i)$ and $u(j)$ are
within distance $\delta\zeta(\tau)$ in lexicographical order. By [20, equation
(11)], we see that

$\displaystyle\underset{|i-j|\leq\delta\zeta(\tau)}{\sup}|H_{\tau}(i)-H_{\tau}(j)|\leq
2+2\underset{u(j)\preccurlyeq
u(i),|i-j|\leq\delta\zeta(\tau)}{\sup}|H_{\tau}(i)-H_{\tau}(j)|.$ (12)

For $n\in\mathbb{N}$, let $\mathbf{s}_{n}$ be a degree sequence satisfying
assumptions in Proposition 1 and $\mathbf{t}_{n}$ be a tree sampled uniformly
at random in $\mathbb{T}_{\mathbf{s}_{n}}$. Let $G_{n}^{\rm rev}=(G_{n}^{\rm
rev}(V_{n}u),u\in[0,1])$ be the version of the modified-$\L$ukasiewicz path
defined as before but with the (time-scaled) reverse-$\L$ukasiewicz path
$W_{n}^{\rm rev}=(W_{n}^{\rm rev}(V_{n}u),u\in[0,1])$ of $\mathbf{t}_{n}$
instead of $W_{n}^{\rm lex}$. In particular, Theorem 5 remains valid if we
replace $W_{n}^{\rm lex}$ and $G_{n}$ by $W_{n}^{\rm rev}$ and $G_{n}^{\rm
rev}$, respectively.

###### Proof of Proposition 1.

Following the exact same argument as in the proof of [20, Theorems 1 and 3]
(and in particular, [20, Proposition 5]), our claim follows from [1,
Proposition 3.3], Theorem 5 and (11) provided that the family of process
$((b_{n}^{-1}H_{n}(V_{n}u),u\in[0,1]):n\geq 1)$ is tight. Since $H_{n}(0)=0$,
it is enough to check that for any $\varepsilon,\varepsilon^{\prime}>0$, there
exists $0<\delta<1$ such that

$\displaystyle\limsup_{n\rightarrow\infty}\mathbb{P}(\omega_{\delta}(H_{n})\geq\varepsilon
b_{n})\leq\varepsilon^{\prime},$ (13)

where $\omega_{\delta}(g)=\sup_{|u-u^{\prime}|\leq\delta}|g(u)-g(u^{\prime})|$
for $g\in\mathbf{D}([0,1],\mathbb{R})$; see [17, Theorem 2.7.3].

If $\beta_{1}=0$, we set $\mathbf{d}_{n}=\varnothing$, otherwise, we consider
the set $\mathbf{d}_{n}=\\{d^{n}(1),\dots,d^{n}(I_{n})\\}$ of the first
$I_{n}$ largest degrees. Let $\mathbf{v}_{n}$ be the set of vertices of
$\mathbf{t}_{n}$ with degrees among those of the set $\mathbf{d}_{n}$. For
$i,j\in\\{0,\dots,V_{n}-1\\}$ such that $u(j)\preccurlyeq u(i)$, we see that
every $v\in\llbracket u(j),u(i)\rrbracket$ which has degree more than one
contributes at least one to the number of vertices branching-off the path
$\llbracket u(j),u(i)\rrbracket$. So,

$\displaystyle 1+I_{n}+\sum_{u(j)\preccurlyeq v\preccurlyeq
u(i)}(k_{v}-1)\mathbf{1}_{\\{v\not\in\mathbf{v}_{n}\\}}\geq
H_{\mathbf{t}_{n}}(i)-H_{\mathbf{t}_{n}}(j)+\sum_{u(j)\preccurlyeq
v\preccurlyeq u(i)}\mathbf{1}_{\\{k_{v}=1\\}}.$ (14)

On the other hand, it follows from (2) and (3) that

$\displaystyle\sum_{u(j)\preccurlyeq v\preccurlyeq
u(i)}(k_{v}-1)\mathbf{1}_{\\{v\not\in\mathbf{v}_{n}\\}}\leq
G_{n}(i)-G_{n}(j)-G_{n}^{\rm rev}(i)-G_{n}^{\rm
rev}(j)+2k_{u(j)}\mathbf{1}_{\\{u(j)\not\in\mathbf{v}_{n}\\}}.$ (15)

The same reasoning as in [20, equation (13)] allows us to deduce from (12),
(14) and (15) that

$\displaystyle\omega_{\delta}(H_{n})\leq
4+2I_{n}+2\omega_{\delta}(G_{n})+2\omega_{\delta}(G_{n}^{\rm
rev})+2d^{n}(I_{n}+1)+\sup_{|i-j|\leq\delta V_{n}}\sum_{u(j)\preccurlyeq
v\preccurlyeq u(i)}\mathbf{1}_{\\{k_{v}=1\\}},$

for $0<\delta<1$. Finally, as in the proof [20, Lemma 8], we obtain (13) from
the previous inequality. ∎

##  4 Lamination-valued processes

This section is devoted to the proofs of Theorem 2 and Corollary 2, about the
convergence of the lamination-valued processes associated to plane trees. In
Sections 4.1 and 4.2, we start by rigorously defining the laminations
associated to discrete and continuum trees, respectively. Theorem 2 and
Corollary 2 are then proved in Sections 4.3 and 4.4, respectively.

In this section, we denote by $d_{\rm H}$ the Hausdorff distance on the
compact subsets of $\bar{\mathbb{D}}$. In particular,
$(\mathbb{L}(\bar{\mathbb{D}}),d_{\rm H})$ is a Polish metric space. We also
denote by $d_{\rm Sk}^{\mathbb{L}}$ the $J_{1}$ Skorohod distance on
$\mathbf{D}(\mathbb{R}_{+},\mathbb{L}(\bar{\mathbb{D}}))$; see e.g., [17,
Chapter 3] or [31, Chapter VI] for a precise definition.

###  4.1 The discrete setting

We start by considering rooted plane trees. In this setting, as for
fragmentation processes, there are two natural ways to define a lamination-
valued process: either the one obtained from removing edges one by one at
integer times, or the one that we get when putting i.i.d. variables on edges
and removing those whose variable is smaller than a given value.

######  Definition 1 (Discrete lamination-valued process).

Let $\tau$ be a rooted plane tree with contour function $C_{\tau}$ and let
$(e_{1},\dots,e_{\zeta(\tau)-1})$ be a uniform ordering of its edges. For
$k=1,\dots,\zeta(\tau)-1$, let $g_{k}$ and $d_{k}$ be the first and last times
at which the contour function $C_{\tau}$ visits the endpoint of the edge
$e_{k}$ further from the root $\emptyset$. Associate to $e_{k}$ the chord
$c_{k}\coloneqq[e^{-2\pi ig_{k}/2\zeta(\tau)},e^{-2\pi
id_{k}/2\zeta(\tau)}]\subset\bar{\mathbb{D}}$. We define the lamination-valued
process $(\mathbb{L}_{t}(\tau),t\geq 0)$ associated to $\tau$ by letting

$\displaystyle\mathbb{L}_{t}(\tau)\coloneqq\mathbb{S}^{1}\cup\bigcup_{k=1}^{\lfloor
t\rfloor\wedge(\zeta(\tau)-1)}c_{k},\hskip 8.53581pt\text{for}\hskip
5.69054ptt\in[0,\infty].$

In particular, the process $(\mathbb{L}_{t}(\tau),t\geq 0)$ interpolates
between $\mathbb{S}^{1}$ and
$\mathbb{L}(\tau)\coloneqq\mathbb{L}_{\infty}(\tau)$; see Figure 2. We also
consider a dynamic continuous-time version of the lamination-valued process
$(\mathbb{L}_{t}(\tau),t\geq 0)$.

| 12345678910111213141230 | 012345678910111213
---|---|---
Figure 2: A tree $\tau$, its contour function $C_{\tau}$, and the associated
lamination $\mathbb{L}(\tau)$.

######  Definition 2 (Dynamic discrete lamination-valued process).

Let $\tau$ be a rooted plane tree with contour function $C_{\tau}$ and denote
by $(e_{1},\dots,e_{\zeta(\tau)-1})$ its edges (their ordering is irrelevant).
Equip the edges of $\tau$ with i.i.d. exponential random variables of
parameter $1$, say $(\gamma_{1},\dots,\gamma_{\zeta(\tau)-1})$. For
$k=1,\dots,\zeta(\tau)-1$, let $g_{k}$ and $d_{k}$ be the first and last times
at which the contour function $C_{\tau}$ visits the endpoint of the edge
$e_{k}$ further from the root $\emptyset$. For $t\geq 0$, we associate to
$e_{k}$ the chord $c_{k}(t)\coloneqq[e^{-2\pi ig_{k}/2\zeta(\tau)},e^{-2\pi
id_{k}/2\zeta(\tau)}]\subset\bar{\mathbb{D}}$ whenever $\gamma_{k}\leq t$, and
otherwise we set $c_{k}(t)=\mathbb{S}^{1}$. We define the dynamic lamination-
valued process $(\mathbb{L}_{t}^{\rm d}(\tau),t\geq 0)$ by letting

$\displaystyle\mathbb{L}^{\rm
d}_{t}(\tau)\coloneqq\mathbb{S}^{1}\cup\bigcup_{k=1}^{\zeta(\tau)-1}c_{k}(t),\hskip
8.53581pt\text{for}\hskip 5.69054ptt\in[0,\infty].$

The process $(\mathbb{L}^{\rm d}_{t}(\tau),t\geq 0)$ also interpolates between
$\mathbb{S}^{1}$ and $\mathbb{L}(\tau)\coloneqq\mathbb{L}^{\rm
d}_{\infty}(\tau)$. It turns out however that, under mild conditions, these
two processes $(\mathbb{L}_{t}(\tau),t\geq 0)$ are $(\mathbb{L}_{t}^{\rm
d}(\tau),t\geq 0)$ are asymptotically close.

######  Proposition 2.

Let $(\tau_{n},n\geq 1)$ be a sequence of rooted plane trees and $(a_{n},n\geq
1)$ be a sequence of positive real numbers such that $a_{n}\rightarrow\infty$
and $\zeta(\tau_{n})/a_{n}\rightarrow\infty$, as $n\rightarrow\infty$. Then,
for any $\varepsilon>0$,

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\left(d_{\rm
Sk}^{\mathbb{L}}\left(\left(\mathbb{L}_{\frac{ta_{n}}{\zeta(\tau_{n})}}^{\rm
d}(\tau_{n}),t\geq 0\right),(\mathbb{L}_{ta_{n}}(\tau_{n}),t\geq
0)\right)>\varepsilon\right)=0.$

###### Proof.

For $1\leq k\leq\zeta(\tau_{n})-1$, let $\kappa_{n,k}$ be the time at which
the $k$-th chord is added in $\left(\mathbb{L}_{ta_{n}/\zeta(\tau_{n})}^{\rm
d}(\tau_{n}),t\geq 0\right)$. Then,

$\displaystyle\mathbb{L}_{ta_{n}}(\tau_{n})=\mathbb{L}_{\frac{\kappa_{n,\lfloor
ta_{n}\rfloor}a_{n}}{\zeta(\tau_{n})}}^{\rm d}(\tau_{n}),\hskip
8.53581pt\text{for all}\hskip 5.69054ptt\geq 0.$

Thus, our claim follows from [17, Theorem 3.9], [49, Theorem 3.1] and [31,
Theorem 1.14 in Chapter VI] provided that, for each $t\geq 0$,

$\sup_{s\in[0,t]}\left|\kappa_{n,\lfloor
sa_{n}\rfloor}-s\right|\xrightarrow[]{\mathbb{P}}0,\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty.$ (16)

Let $E_{1},\dots,E_{\zeta(\tau_{n})-1}$ be i.i.d. exponential random variables
of parameter $1$ and define the process

$\displaystyle
N_{n}(u)=\frac{1}{a_{n}}\sum_{i=1}^{\zeta(\tau_{n})-1}\mathbf{1}_{\\{E_{i}\leq
ua_{n}/\zeta(\tau_{n})\\}},\hskip 5.69054pt\hskip 5.69054ptu\geq 0.$

An application of the Chebyshev inequality shows that, for $s\geq 0$,
$N_{n}(s)\rightarrow s$, in probability, as $n\rightarrow\infty$. On the other
hand, observe that $\kappa_{n,\lfloor sa_{n}\rfloor}=\inf\\{u\geq
0:a_{n}N_{n}(u)\geq\lfloor sa_{n}\rfloor\\}$. Then, by inversion, we have that
$\kappa_{n,\lfloor sa_{n}\rfloor}\rightarrow s$, in probability, as
$n\rightarrow\infty$. Moreover, since $\kappa_{n,\lfloor sa_{n}\rfloor}$ is
non-decreasing as a function of $s$, we obtain (16); see e.g., [30, Lemma 2.2
in Chapter 5]. ∎

Fix $q\geq 1$ and let $u_{1},\dots,u_{q}$ be $q$ i.i.d. uniform random
vertices of a rooted plane tree $\tau$. The reduced tree $\tau^{(q)}$ of
$\tau$ is obtained by keeping only the root $\emptyset$ of $\tau$, these $q$
vertices and the subsequent branching points (if any), i.e. the vertices
$w\in\tau$ such that
$\llbracket\emptyset,u_{i}\rrbracket\cap\llbracket\emptyset,u_{j}\rrbracket=\llbracket\emptyset,w\rrbracket$
for some $1\leq i<j\leq q$. Then one puts an edge between two vertices of
$\tau^{(q)}$ if one is the ancestor of the other in $\tau$, and there is no
other vertex of $\tau^{(q)}$ inbetween. The length of an edge $e$ in
$\tau^{(q)}$ is defined as the number of edges between the vertices of $\tau$
corresponding to the endpoints of $e$. The tree $\tau^{(q)}$ is rooted at
$\emptyset$ and has a plane structure induced by that of $\tau$; see Figure 3.
Note that its number of vertices is a priori random.

|  | $1$$1$$1$$2$$2$$1$
---|---|---
Figure 3: A tree $\tau$ with $4$ marked vertices, and the reduced tree
$\tau^{(4)}$ with its edge lengths.

The notion of reduced trees naturally translates in the lamination setting
into the notion of reduced lamination. Suppose that $\tau^{(q)}$ has exactly
$q$ leaves. Let $u_{0,q}\coloneqq\emptyset$ and $u_{1,q},\dots,u_{q,q}$ be the
$q$ leaves of $\tau^{(q)}$ listed in lexicographical order. Let also
$a_{1,q},\dots,a_{q,q}$ be $q$ i.i.d. uniform random points on the unit circle
$\mathbb{S}^{1}$ such that $1=a_{0,q},a_{1,q},\dots,a_{q,q}$ are sorted in
clockwise order (starting from $a_{0,q}$). Set
$A_{q}=\\{a_{0,q},a_{1,q},\dots,a_{q,q}\\}$ and
$E_{q}=\\{u_{0,q},u_{1,q},\dots,u_{q,q}\\}$. Observe that removing any edge of
$\tau^{(q)}$ splits $E_{q}$ into two subsets, which are made of consecutive
elements of $E_{q}$ in lexicographical order (up to cyclic shift),
corresponding to two subsets of $A_{q}$ of consecutive points. We associate to
a reduced tree $\tau^{(q)}$ a set of laminations $\mathbf{H}(\tau^{(q)})$ as
follows:

######  Definition 3 (Discrete reduced laminations).

By convention, if $\tau^{(q)}$ does not have exactly $q$ leaves, by convention
we say that its set of reduced laminations is the singleton
$\\{\mathbb{S}^{1}\\}$. Otherwise, we associate to $\tau^{(q)}$ a set
$\mathbf{H}(\tau^{(q)})$ of laminations by saying that a lamination $L$
belongs to $\mathbf{H}(\tau^{(q)})$ if the following property holds: for any
$0\leq i<j\leq q$, there exists a chord in $L$ between the open arcs
$(a_{i,q},a_{i+1,q})$ and $(a_{j,q},a_{j+1,q})$ (with the convention that
$a_{q+1,q}=a_{1,q}$), if and only if there exists an edge in $\tau^{(q)}$
splitting the set $E_{q}$ into $\\{u_{i+1,q},\dots,u_{j,q}\\}$ and
$E_{q}\backslash\\{u_{i+1,q},\ldots,u_{j,q}\\}$.

We then associate to $\tau^{(q)}$ a random lamination-valued process. We
henceforth assume that $\tau^{(q)}$ has exactly $q$ leaves. For each edge $e$
of $\tau^{(q)}$, denote by $\ell(e)$ its length. Recall that $\ell(e)$ is
defined as the number of edges between the vertices of $\tau$ corresponding to
the endpoints of $e$. Equip the edges of $\tau$ with i.i.d. exponential random
variables of parameter $1$, and for each edge $e$ of $\tau^{(q)}$, denote by
$\gamma_{e}$ the minimum of the $\ell(e)$ exponential random variables
associated to the edges of $\tau$ between the endpoints of $e$. In particular,
$(\gamma_{e},e\in\tau^{(q)})$ is a sequence of independent exponential random
variables of respective parameter $(\ell(e),e\in\tau^{(q)})$.

######  Definition 4 (Discrete reduced lamination-valued process).

Consider an element $\mathbb{L}(\tau^{(q)})\in\mathbf{H}(\tau^{(q)})$ (the
choice ultimately does not matter). We define the reduced lamination-valued
process $(\mathbb{L}_{t}^{(q)}(\tau),t\geq 0)$ from $\mathbb{L}(\tau^{(q)})$
by letting $\mathbb{L}_{t}^{(q)}(\tau)$ be the union of the unit circle
$\mathbb{S}^{1}$ and the set of chords of $\mathbb{L}(\tau^{(q)})$
corresponding to an edge $e$ if and only if $\gamma_{e}\leq t$.

In particular, this process interpolates between $\mathbb{S}^{1}$ ($t=0$) and
the lamination $\mathbb{L}(\tau^{(q)})$ ($t\rightarrow\infty$).

###  4.2 The continuum setting

In this section, we define lamination-valued processes associated to the so-
called continuum random trees. Let us first recall the notion of
$\mathbb{R}$-tree. A metric space $(\mathcal{T},r)$ is an $\mathbb{R}$-tree,
if for every $x,y\in\mathcal{T}$: (i) there exists a unique isometry
$f_{x,y}:[0,r(x,y)]\rightarrow\mathcal{T}$ such that $f_{x,y}(0)=x$ and
$f_{x,y}(r(x,y))=y$; (ii) for any continuous injective function
$g:[0,1]\rightarrow\mathcal{T}$ such that $g(0)=x$ and $g(1)=y$, we have
$g([0,1])=f_{x,y}([0,r(x,y)])$. The range of the mapping $f_{x,y}$ is the
geodesic between $x,y\in\mathcal{T}$ and is denoted by $\llbracket
x,y\rrbracket$. A point $x\in\mathcal{T}$ is called a leaf if
$\mathcal{T}\backslash\\{x\\}$ is connected, and a branching point if
$\mathcal{T}\backslash\\{x\\}$ has at least three disjoint connected
components. We denote by ${\rm Lf}(\mathcal{T})$ the set of leaves of
$\mathcal{T}$ and by ${\rm Skel}(\mathcal{T})\coloneqq\mathcal{T}\setminus{\rm
Lf}(\mathcal{T})$ its skeleton. The distance $r$ in $\mathcal{T}$ induces a
length measure $\lambda_{r}$ on ${\rm Skel}(\mathcal{T})$ given by
$\lambda_{r}(\llbracket x,y\rrbracket)=r(x,y)$ for all $x,y\in\mathcal{T}$. A
rooted $\mathbb{R}$-tree $(\mathcal{T},r,\rho)$ is a $\mathbb{R}$-tree
$(\mathcal{T},r)$ with a distinguished point $\rho\in\mathcal{T}$ called the
root of $\mathcal{T}$.

######  Definition 5.

A (rooted) continuum tree is a quadruple $(\mathcal{T},r,\rho,\mu)$, where
$(\mathcal{T},r,\rho)$ is a rooted $\mathbb{R}$-tree and $\mu$ is a non-atomic
Borel probability measure on $\mathcal{T}$ such that $\mu({\rm
Lf}(\mathcal{T}))=1$ and for every non-leaf vertex $x\in\mathcal{T}$,
$\mu(\\{y\in\mathcal{T}:\llbracket\rho,y\rrbracket\cap\llbracket\rho,x\rrbracket=\llbracket\rho,x\rrbracket\\})>0$.
We call $\mu$ the mass measure of $\mathcal{T}$.

In [5], Aldous makes slightly different definitions of these quantities which,
in particular, restricts his discussion to binary trees, but the theory can be
easily extended. Note that the definition of a continuum tree implies that the
$\mathbb{R}$-tree $\mathcal{T}$ satisfies certain extra properties; for
example, ${\rm Lf}(\mathcal{T})$ must be uncountable, have no isolated point
and $\lambda_{r}$ is $\sigma$-finite. In what follows, $\mathcal{T}$ will
always denote a continuum tree $(\mathcal{T},r,\rho,\mu)$.

######  Lemma 4.1.

The set of branching points of a continuum tree $\mathcal{T}$ is at most
countable.

###### Proof of Lemma 4.1.

By definition, for any branching point $y$ of $\mathcal{T}$, all connected
components of $\mathcal{T}\backslash\\{y\\}$ have non-zero $\mu$-mass. For all
$k\geq 1$, let $B_{k}$ be the set of branching points of $\mathcal{T}$ such
that at least three connected components of $\mathcal{T}\backslash\\{y\\}$
have $\mu$-mass $>1/k$. Then, the number of points in $B_{k}$ has to be less
than $k$. Our claim follows by taking the union over all $k\geq 1$. ∎

We can equip the continuum tree $\mathcal{T}$ with a total order $\preceq$
that is reminiscent of the lexicographical order in rooted plane trees; see
[24]. To be precise, $\preceq$ is the only total order on $\mathcal{T}$
satisfying the following conditions: (i) for every $x,y\in\mathcal{T}$, if
$x\in\llbracket\rho,y\rrbracket$, then $x\preceq y$ (i.e., $x$ is an ancestor
of $y$); (ii) for every $x,y,z\in\mathcal{T}$, if $x\preceq y\preceq z$, then
$w\in\llbracket\rho,y\rrbracket$, where $w$ is the branching point that
satisfies
$\llbracket\rho,w\rrbracket=\llbracket\rho,x\rrbracket\cap(\llbracket\rho,y\rrbracket\cup\llbracket\rho,z\rrbracket)$.
With a slight abuse of language, we call $\preceq$ the lexicographical order
on $\mathcal{T}$.

######  Definition 6.

Let $\mathcal{T}$ be a continuum tree. For any
$x\in\mathcal{T}\backslash\\{\rho\\}$, let $C_{x}$ be the connected component
of $\mathcal{T}\backslash\\{x\\}$ containing $\rho$ and $C_{x}^{g}$ (resp.
$C_{x}^{d}$) be the subset of $C_{x}$ made of points that are before $x$
(resp. after $x$) in lexicographical order. Let $c_{x}$ be the chord
$[e^{-2\pi is},e^{-2\pi it}]$, where $s\coloneqq\mu(C_{x}^{g})$ and
$1-t\coloneqq\mu(C_{x}^{d})$. The lamination $\mathbb{L}(\mathcal{T})$
associated to $\mathcal{T}$ is defined by

$\displaystyle\mathbb{L}(\mathcal{T})\coloneqq\mathbb{S}^{1}\cup\overline{\bigcup_{x\in\mathcal{T}\backslash\\{\rho\\}}c_{x}}.$

We can also define a stochastic process from this lamination.

######  Definition 7 (Continuum lamination-valued process).

Let $\Pi$ be a Poisson point process on $\mathcal{T}\times\mathbb{R}_{+}$ with
intensity measure $\lambda_{r}\times dt$, where $dt$ is the Lebesgue measure
on $\mathbb{R}_{+}$. For any $t\geq 0$, set
$\Pi_{t}=\\{x\in\mathcal{T}:\,\exists\,s\leq t,\,(x,s)\in\Pi\\}$. We define a
lamination-valued process $(\mathbb{L}_{t}(\mathcal{T}),t\geq 0)$ of
$\mathcal{T}$ by letting, for all $t\geq 0$:

$\displaystyle\mathbb{L}_{t}(\mathcal{T})\coloneqq\mathbb{S}^{1}\cup\overline{\bigcup_{x\in\Pi_{t}}c_{x}},$

where $c_{x}$ is the unique chord associated to $x\in\Pi_{t}$ (as in
Definition 6).

Since $\lambda_{r}$ is supported on ${\rm Skel}(\mathcal{T})$, we see that
$\Pi_{t}\subset{\rm Skel}(\mathcal{T})$, for all $t\geq 0$. Moreover, by Lemma
4.1, almost surely for all $t\geq 0$, $\Pi_{t}$ does not contain branching
points. On the other hand, $(\mathbb{L}_{t}(\mathcal{T}),t\geq 0)$ is non-
decreasing and it interpolates between $\mathbb{S}^{1}$ ($t=0$) and
$\mathbb{L}(\mathcal{T})$ ($t\rightarrow\infty$).

######  Lemma 4.2.

We have that

$\displaystyle\mathbb{L}_{t}(\mathcal{T})\xrightarrow[]{a.s.}\mathbb{L}(\mathcal{T}),\hskip
8.53581pt\text{as}\hskip 5.69054ptt\rightarrow\infty,\hskip
5.69054pt\text{in}\hskip 5.69054pt(\mathbb{L}(\bar{\mathbb{D}}),d_{\rm H}).$

###### Proof of Lemma 4.2.

Fix $\epsilon>0$. By e.g. [47, Lemma $2.3$], we can find a deterministic
integer constant $K_{\varepsilon}>0$ such that there always exists a sub-
lamination $L_{\varepsilon}$ of $\mathbb{L}(\mathcal{T})$ (i.e., a lamination
that is a subset of $\mathbb{L}(\mathcal{T})$) with at most $K_{\varepsilon}$
chords, satisfying $d_{\rm
H}\left(L_{\epsilon},\mathbb{L}(\mathcal{T})\right)<\varepsilon$. In
particular, by construction of $\mathbb{L}(\mathcal{T})$, we can and will
choose $L_{\varepsilon}$ such that its (at most) $K_{\varepsilon}$ chords are
coded by points of ${\rm Skel}(\mathcal{T})\backslash\\{\rho\\}$.

Let $c$ be a chord of $\mathbb{L}(\mathcal{T})$ coded by some point $x\in{\rm
Skel}(\mathcal{T})\backslash\\{\rho\\}$, and consider, for any
$\eta\in(0,\lambda_{r}(\llbracket\rho,x\rrbracket))$, the unique ancestor
$y_{\eta}$ of $x$ such that $r(y_{\eta},x)=\eta$ (or $\rho$ if
$r(\rho,x)<\eta$). Then, we have $\mu(C_{x}\backslash C_{y_{\eta}})\rightarrow
0$ as $\eta\rightarrow 0$, where we recall that $C_{z}$ denotes the connected
component of $\mathcal{T}\backslash\\{z\\}$ containing $\rho\in\mathcal{T}$.
Thus,

$\displaystyle\sup_{z\in\llbracket y_{\eta},x\rrbracket}d_{\rm
H}(c_{z},c_{x})\rightarrow 0\hskip 8.53581pt\text{as}\hskip
5.69054pt\eta\rightarrow 0.$

In particular, we can take
$\eta\in(0,\lambda_{r}(\llbracket\rho,x\rrbracket))$ such that all chords
$c_{z}$, for $z\in\llbracket y_{\eta},x\rrbracket$, are at Hausdorff distance
less than $\varepsilon$ of $c_{x}$. Then, almost surely there exists $t>0$
such that $\Pi_{t}\cap\llbracket y_{\eta},x\rrbracket$ is non-empty.

The result follows by doing this jointly for all $K_{\varepsilon}$ chords of
$L_{\varepsilon}$. ∎

####  4.2.1 Compact continuum trees and excursion-type functions

A common way to construct $\mathbb{R}$-trees is from continuous excursion-type
functions, i.e. continuous functions $f:[0,1]\rightarrow\mathbb{R}_{+}$ such
that $f(0)=f(1)=0$ and $f(x)\geq 0$ for all $0\leq x\leq 1$. Let $f$ be such a
function, consider the pseudo-distance on $[0,1]$,

$\displaystyle r_{f}(x,y)\coloneqq f(x)+f(y)-2\inf_{z\in[x\wedge y,x\vee
y]}f(z),\hskip 8.53581pt\text{for}\hskip 8.53581ptx,y\in[0,1],$

and define an equivalence relation on $[0,1]$ by setting $x\sim_{f}y$ if and
only if $r_{f}(x,y)=0$. The image of the projection
$p_{f}:[0,1]\rightarrow[0,1]\setminus\sim_{f}$ endowed with the pushforward of
$r_{f}$ (again denoted $r_{f}$), i.e.
$\mathcal{T}_{f}=(\mathcal{T}_{f},r_{f},\rho_{f})\coloneqq(p_{f}([0,1]),r_{f},p_{f}(0))$,
is a plane rooted $\mathbb{R}$-tree; see [26, Lemma 3.1]. In particular,
$(\mathcal{T}_{f},r_{f})$ is a compact and connected metric space. Conversely,
it has been noted in [35, Remark following Theorem 2.2] (see also [24,
Corollary 1.2]) that for every compact $\mathbb{R}$-tree $(\mathcal{T},r)$
there exists a continuous excursion-type function
$f:[0,1]\rightarrow\mathbb{R}$ such that $(\mathcal{T},r)$ and
$(\mathcal{T}_{f},r_{f})$, are isometric. We can endow $\mathcal{T}_{f}$ with
the probability measure $\mu_{f}$ given by the pushforward of the Lebesgue
measure on $[0,1]$ under the projection $p_{f}$. Suppose furthermore that the
set of one-sided local minima of $f$ has Lebesgue measure $0$ (recall that
$x\in[0,1]$ is a one-sided local minimum of $f$ if there exists
$\varepsilon>0$ such that $f(x)=\inf\\{f(y):x\leq y\leq x+\varepsilon\\}$ or
$f(x)=\inf\\{f(y):x-\varepsilon\leq y\leq x\\}$). Then, $\mu_{f}$ is a non-
atomic measure and $\mu_{f}(\text{Lf}(\mathcal{T}_{f}))=1$; see [5, Proof of
Theorem 13]. Moreover,
$\mathcal{T}_{f}=(\mathcal{T}_{f},r_{f},\rho_{f},\mu_{f})$ is a (rooted)
continuum tree.

In this setting, we can construct a lamination $\mathbb{L}(f)$, and moreover,
a lamination-valued process $(\mathbb{L}_{t}(f),t\geq 0)$ associated to the
continuous excursion-type function $f$ (and thus to $\mathcal{T}_{f}$) that
coincide with our previous definitions. Let us recall the definition of
$(\mathbb{L}_{t}(f),t\geq 0)$ and refer to [47] for further details. First,
define the epigraph of $f$ as the set of points below its graph, that is,

$\displaystyle\mathcal{EG}(f)\coloneqq\left\\{(x,y)\in\mathbb{R}^{2}:x\in(0,1),0\leq
y<f(x)\right\\}.$

To each $(x,y)\in\mathcal{EG}(f)$, associate the chord
$c(x,y)\coloneqq[e^{-2\pi ig(x,y)},e^{-2\pi id(x,y)}]\in\bar{\mathbb{D}}$,
where $g(x,y)\coloneqq\sup\\{z\leq x:f(z)<y\\}$ and
$d(x,y)\coloneqq\inf\\{z\geq x:f(z)<y\\}$. Consider now a Poisson point
process $\mathcal{N}^{f}$ on $\mathbb{R}^{2}\times\mathbb{R}_{+}$ with
intensity measure

$\displaystyle\frac{1}{d(x,y)-g(x,y)}\mathbf{1}_{\\{(x,y)\in\mathcal{EG}(f)\\}}dxdyds,$

where $ds$ denotes the Lebesgue measure on $\mathbb{R}_{+}$. For $t\geq 0$,
consider also the Poisson point process
$\mathcal{N}^{f}_{t}(\cdot)\coloneqq\mathcal{N}^{f}(\cdot\times[0,t])$ on
$\mathcal{EG}(f)$ and construct the lamination-valued process
$(\mathbb{L}_{t}(f),t\geq 0)$ associated to $f$ as follows. For all $t\geq 0$,

$\displaystyle\mathbb{L}_{t}(f)=\mathbb{S}^{1}\cup\overline{\bigcup_{(x,y)\in\mathcal{N}_{t}^{f}}c(s,t)}.$

Clearly, this process is non-decreasing for the inclusion. Moreover, define

$\displaystyle\mathbb{L}_{\infty}(f)\coloneqq\mathbb{L}(f)=\overline{\bigcup_{t\geq
0}\mathbb{L}_{t}(f)}.$

######  Proposition 3.

We have that $(\mathbb{L}_{t}(\mathcal{T}_{f}),t\geq
0)\overset{d}{=}(\mathbb{L}_{t}(f),t\geq 0)$.

###### Proof.

The idea consists in coupling $(\mathbb{L}_{t}(\mathcal{T}_{f}),t\geq 0)$ and
$(\mathbb{L}_{t}(f),t\geq 0)$. For any $(x,y)\in\mathcal{N}^{f}$, let
$w(x,y)\in\mathcal{T}_{f}$ be the equivalence class of $g(x,y)$ with respect
to $\sim_{f}$. Then, the chord $c(x,y)$ is exactly the chord $c_{w(x,y)}$.
Thus, we only need to check that the image of $\mathcal{N}^{f}$ under the
projection $p_{f}$ is a Poisson point process on ${\rm Skel}(\mathcal{T}_{f})$
with the correct intensity. To this end, remark that, for any
$x^{\prime},y^{\prime}\in\mathcal{T}_{f}$, we have that

$\displaystyle\int_{E(x^{\prime},y^{\prime})}\frac{1}{d(x,y)-g(x,y)}\mathbf{1}_{\\{(x,y)\in\mathcal{EG}(f)\\}}dxdy=\lambda_{r}(\llbracket
x^{\prime},y^{\prime}\rrbracket),$

where
$E(x^{\prime},y^{\prime})\coloneqq\\{(x,y)\in\mathbb{R}^{2}:w(x,y)\in\llbracket
x^{\prime},y^{\prime}\rrbracket\\}$. The result follows. ∎

####  4.2.2 Reduced tree and lamination from continuum trees

For $q\geq 1$, let $x_{q},\dots,x_{q}$ be $q$ i.i.d. random leaves of a
continuum tree $\mathcal{T}$ sampled from its mass measure $\mu$. Observe that
they are a.s. all distinct, and set $E_{q}:=\\{x_{q},0\leq i\leq
q\\}\cup\\{\rho\\}$. The reduced tree $\mathcal{T}^{(q)}$ of $\mathcal{T}$ is
the plane rooted tree with edge-lengths whose vertices are the leaves
$x_{1},\ldots,x_{q}$, the root $\rho$ of $\mathcal{T}$ and all subsequent
branching points. The length of an edge is simply the length measure of the
unique geodesic path in $\mathcal{T}$ between the corresponding endpoints.

We can also define the notion of reduced lamination and reduced lamination-
valued process in the continuum setting. For $x\in\mathcal{T}$, let
$L(x)\coloneqq\mu\left(\left\\{y\in{\rm Lf}(\mathcal{T}):y\preceq
x\right\\}\right)$ be the mass of the set of leaves of $\mathcal{T}$ that lie
on the left of $x$. Let $x_{0,q}=\rho$ be the root of $\mathcal{T}^{(q)}$ and
$x_{1,q},\dots x_{q,q}$ be its $q$ leaves listed in lexicographical order. Set
$a_{j,q}\coloneqq e^{-2i\pi L(x_{j,q})}$, for $0\leq j\leq q$ (in particular,
$a_{0,q}=1$). The following result must be clear since $\mu$ is non-atomic.

######  Lemma 4.3.

If $x\in{\rm Lf}(\mathcal{T})$ is distributed according to $\mu$, then $L(x)$
is uniformly distributed on $[0,1]$.

As a consequence, for all $q\geq 1$, if the $q$ leaves of $\mathcal{T}$ are
sampled in an i.i.d. way according to $\mu$, then $a_{1,q},\ldots,a_{q,q}$ are
the order statistics of $q$ i.i.d. uniform variables on the unit circle.

######  Definition 8 (Continuum reduced lamination).

We associate to $\mathcal{T}^{(q)}$ a set $\mathbf{H}(\mathcal{T}^{(q)})$ of
laminations by saying that a lamination $L$ belongs to
$\mathbf{H}(\mathcal{T}^{(q)})$ if the following property holds: for $0\leq
i<j\leq q$, there exists a chord in $L$ between open arcs
$(a_{i,q},a_{i+1,q})$ and $(a_{j,q},a_{j+1,q})$ (with the convention that
$a_{q+1,q}=a_{0,q}$) if and only if there exists an edge in
$\mathcal{T}^{(q)}$ splitting $E_{q}\coloneqq\\{x_{0,q},\dots,x_{q,q}\\}$ into
$\\{x_{i+1,q},\dots,x_{j,q}\\}$ and
$E_{q}\backslash\\{x_{i+1,q},\dots,x_{j,q}\\}$.

Let us now state and prove a result which will be useful in what follows.

######  Lemma 4.4.

For all $a,a^{\prime}\in\mathbb{S}^{1}$, let $d(a,a^{\prime})$ denote the
length of the shortest arc from $a$ to $a^{\prime}$ in $\mathbb{S}^{1}$. For
all $\varepsilon>0$,

$\displaystyle\lim_{q\rightarrow\infty}\mathbb{P}\left(\sup_{1\leq j\leq
q}d\left(a_{j,q},e^{-2\pi ij/q}\right)<\varepsilon\right)=1.$

###### Proof.

Fix $\varepsilon>0$, choose an integer $K\geq 1$ such that $2/K<\varepsilon$
and take $q\geq K^{8}$. Let $A_{q}:=\\{a_{j,q},0\leq j\leq q\\}$. Split the
unit circle into the $K$ arcs of the form, $(e^{-2\pi i(k-1)/K},e^{-2\pi
ik/K})$ for $1\leq k\leq K$. For $1\leq j\leq q$, almost surely no point of
the form $a_{j,q}$ is one of the endpoints of these arcs. For any $q\geq 2$
and $1\leq k\leq K$, denote by $A_{q}(k)$ the set $A_{q}\cap(e^{-2\pi
i(k-1)/K},e^{-2\pi ik/K})$, and $M_{q}(k)$ the number of points in $A_{q}(k)$.
In particular, $M_{q}(k)$ is distributed as a binomial random variable with
parameters $(q,1/K)$. So, Hoeffding’s inequality implies that

$\displaystyle\mathbb{P}(|M_{q}(k)-q/K|\geq q^{3/4})\leq 2e^{-2\sqrt{q}}.$

Hence, the probability that $|M_{q}(k)-q/K|\leq q^{3/4}$, for all $1\leq k\leq
K$, is at least $1-2Ke^{-2\sqrt{q}}$ and our claim follows by choosing $K$
large enough (depending on $\varepsilon$). Indeed, suppose that
$|M_{q}(k)-q/K|\leq q^{3/4}$, for all $1\leq k\leq K$. Then, for any $1\leq
k\leq K$ and any integer $(k-1)q/K<j\leq kq/K$, we necessarily have that
$a_{j,q}$ is in $A_{q}(k-1),A_{q}(k)$ or $A_{q}(k+1)$ (with the convention
that $A_{q}(0)=A_{q}(K+1)=A_{q}(1)$) and since $e^{-2\pi ij/q}\in A_{q}(k)$,
the result in Lemma 4.4 holds. ∎

We now associate to $\mathcal{T}^{(q)}$ a lamination-valued process. Let
$\gamma_{e}$ be the first time at which a point of $\Pi$ falls on the geodesic
of $\mathcal{T}$ that corresponds to the edge $e$. In particular, $\gamma_{e}$
is an exponential random variable of parameter $\ell(e)$ the length of $e$,
and $(\gamma_{e},e\in\mathcal{T}^{(q)})$ is a collection of independent random
variables.

######  Definition 9 (Continuum reduced lamination-valued process).

Consider an element
$\mathbb{L}(\mathcal{T}^{(q)})\in\mathbf{H}(\mathcal{T}^{(q)})$. We define the
process $(\mathbb{L}_{t}^{(q)}(\mathcal{T}),t\geq 0)$ from
$\mathbb{L}(\mathcal{T}^{(q)})$ by letting $\mathbb{L}_{t}^{(q)}(\mathcal{T})$
be the union of the unit circle $\mathbb{S}^{1}$ and the set of chords of
$\mathbb{L}(\mathcal{T}^{(q)})$ corresponding to an edge $e$ if and only if
$\gamma_{e}\leq t$.

It turns out that these reduced processes actually approximate the usual
lamination-valued process $(\mathbb{L}_{t}(\mathcal{T}),t\geq 0)$.

######  Proposition 4.

The following convergence holds in distribution:

$\displaystyle(\mathbb{L}_{t}^{(q)}(\mathcal{T}),t\geq
0)\xrightarrow[]{d}(\mathbb{L}_{t}(\mathcal{T}),t\geq 0),\hskip
8.53581pt\text{as}\hskip 5.69054ptq\rightarrow\infty,\hskip
5.69054pt\text{in}\hskip
5.69054pt\mathbf{D}(\mathbb{R}_{+},\mathbb{L}(\bar{\mathbb{D}})).$

###### Proof.

We only need to prove that, for all $\varepsilon>0$ and for every $M>0$,

$\displaystyle\lim_{q\rightarrow\infty}\mathbb{P}\left(\sup_{t\in[0,M]}d_{\rm
H}\left(\mathbb{L}_{t}^{(q)}(\mathcal{T}),\mathbb{L}_{t}(\mathcal{T})\right)<\varepsilon\right)=1.$

Since the support of $\lambda_{r}$ is ${\rm Skel}(\mathcal{T})$, we only
consider chords $c_{x}$ that are coded by a point $x\in{\rm
Skel}(\mathcal{T})$. It follows from Lemmas 4.3 and 4.4 that for fixed
$\varepsilon^{\prime}>0$, we can and will choose $Q>0$ large enough such that,
with probability $\geq 1-\varepsilon^{\prime}$, for any $q\geq Q$,

$\displaystyle\sup_{1\leq j\leq q}d\left(a_{j,q},e^{-2\pi
ij/q}\right)<\varepsilon^{\prime}.$ (17)

Fix $M>0$ and consider a point $x\in\Pi_{M}$. There are two cases: either $x$
falls in the geodesic of $\mathcal{T}$ that corresponds to an edge of
$\mathcal{T}^{(q)}$, or not. If $x$ does not fall in such a geodesic, then
removing $x$ does not split the set of leaves of $\mathcal{T}^{(q)}$ and thus
necessarily $d_{\rm H}(c_{x},\mathbb{S}^{1})<2\pi(2\varepsilon^{\prime}+1/Q)$
by (17). Now, suppose that $x$ falls in such a geodesic. By definition, since
$x\in\Pi_{M}$, there exists a chord
$c^{\prime}\in\mathbb{L}_{M}^{(q)}(\mathcal{T})$ corresponding to a point in
the same edge of $\mathcal{T}^{(q)}$ as $x$. Thus, there exist two arcs
$A_{1},A_{2}$ between clockwise consecutive $a_{j,q}$’s such that $c_{x}$ and
$c^{\prime}$ connect $A_{1}$ and $A_{2}$. Hence, by (17) and e.g. [27, Lemma
$5.2$ (i)], $d_{\rm H}(c^{\prime},c_{x})<2\pi(2\varepsilon^{\prime}+1/Q)$.
Finally, our claim follows by choosing $\varepsilon^{\prime}$ so that
$2\pi(2\varepsilon^{\prime}+1/Q)<\varepsilon$. ∎

###  4.3 Convergence of the processes of laminations

In this section, we prove Theorem 2, stating the equivalence between the
Gromov-weak convergence of trees and the convergence of their associated
lamination-valued processes. We first need to introduce some notation, recall
the definition of the Gromov-weak topology and establish some additional
geometric properties of reduced trees.

A rooted metric measure space is a quadruple
$\mathcal{X}=(\mathcal{X},r,\rho,\mu)$, where $(\mathcal{X},r)$ is a metric
space such that $(\text{supp}(\mu),r)$ is complete and separable, the so-
called sampling measure $\mu$ is a finite measure on $(\mathcal{X},r)$ and
$\rho\in\mathcal{X}$ is a distinguished point which is referred to as the
root; the support $\text{supp}(\mu)$ of $\mu$ is defined as the smallest
closed set $\mathcal{X}_{0}\subseteq\mathcal{X}$ such that
$\mu(\mathcal{X}_{0})=\mu(\mathcal{X})$. Two rooted metric measure spaces
$(\mathcal{X},r,\rho,\mu)$ and
$(\mathcal{X}^{\prime},r^{\prime},\rho^{\prime},\mu^{\prime})$ are said to be
equivalent if there exists an isometry $\phi:{\rm
supp}(\mathcal{\mu})\rightarrow{\rm supp}(\mu^{\prime})$ such that
$\phi(\rho)=\rho^{\prime}$ and $\phi_{\ast}\mu=\mu^{\prime}$, where
$\phi_{\ast}\mu$ is the pushforward of $\mu$ under $\phi$. We denote by
$\mathbb{K}_{0}$ the space of rooted metric measure spaces.

We consider that $\mathbb{K}_{0}$ is equipped with the Gromov-weak topology on
$\mathbb{H}_{n}$; see Gromov’s book [29] or [28, 39]. In particular, the
Gromov-weak topology is metrized by the so-called pointed Gromov-Prokhorov
metric $d_{\rm pGP}$. Moreover, $(\mathbb{K}_{0},d_{\rm pGP})$ is a complete
and separable metric space; see [39, Proposition 2.6]. Let us give a simple
characterization for convergence in the Gromov-weak topology, see e.g. [28,
39]. For each $n\in\mathbb{N}\cup\\{\infty\\}$, consider a rooted metric
measure space $\mathcal{X}_{n}=(\mathcal{X}_{n},r_{n},\rho_{n},\mu_{n})$, set
$\xi_{n}(0)=\rho_{n}$ and $(\xi_{n}(i),i\geq 1)$ i.i.d. random variables
sampled according to $\mu_{n}$. The convergence
$\mathcal{X}_{n}\rightarrow\mathcal{X}_{\infty}$, as $n\rightarrow\infty$, for
the Gromov-weak topology is equivalent to the convergence in distribution of
the matrices

$\displaystyle(r_{n}(\xi_{n}(i),\xi_{n}(j)):0\leq i,j\leq
q)\xrightarrow[]{d}(r_{\infty}(\xi_{\infty}(i),\xi_{\infty}(j)):0\leq i,j\leq
q),\hskip 8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty,$ (18)

for every integer $q\geq 1$ fixed. By the Gromov’s reconstruction theorem [29,
Subsection 3$\frac{1}{2}$.7], the distribution of
$(r_{n}(\xi_{n}(i),\xi_{n}(j)):i,j\geq 0)$ characterizes (the equivalence
class of) $\mathcal{X}_{n}$.

A continuum tree $\mathcal{T}=(\mathcal{T},r,\rho,\mu)$ is a particular case
of rooted metric measure space. For $q\geq 1$, let $x_{1},\dots,x_{q}$ be $q$
i.i.d. leaves sampled according to $\mu$. Let $x_{0,q}=\rho$ and
$x_{1,q},\dots,x_{q,q}$ be the $q$ leaves $x_{1},\dots,x_{q}$ of $\mathcal{T}$
listed in lexicographical order. Set $E_{q}=\\{x_{0,1},\dots,x_{q,q}\\}$. For
$\omega\subset\\{0,\dots,q\\}$, let $\Xi_{q}(\omega)$ be the set of points (if
any) of $\mathcal{T}$ splitting the set $E_{q}$ into
$E_{q}^{\omega,1}=\\{x_{k,q}:k\in\omega\\}$ and
$E_{q}^{\omega,2}=\\{x_{k,q}:k\not\in\omega\\}$.

The following two lemmas show that the sets of points whose removal splits the
set of leaves of the reduced trees into two given subsets are either empty, or
a geodesic corresponding to an edge of the reduced tree.

######  Lemma 4.5.

For $q\geq 1$ and $\omega\subset\\{0,\dots,q\\}$, we have that
$\Xi_{q}(\omega)$ is either empty or a geodesic of $\mathcal{T}$ that
corresponds precisely to an edge $e_{\omega}$ in $\mathcal{T}^{(q)}$ of length
$\ell(e_{\omega})=\lambda_{r}(\Xi_{q}(\omega))>0$. Reciprocally, for every
edge $e$ in $\mathcal{T}^{(q)}$ of length $\ell(e)$ there exists
$\omega_{e}\subset\\{0,\dots,q\\}$ such that $\Xi_{q}(\omega_{e})$ is a
geodesic of $\mathcal{T}$ that corresponds precisely to $e$ such that
$\lambda_{r}(\Xi_{q}(\omega_{e}))=\ell(e)$.

Furthermore, for any points $x_{1},x_{2}\in E_{q}^{\omega,1}$ and
$y_{1},y_{2}\in E_{q}^{\omega,2}$, define

$\displaystyle
f\left(x_{1},x_{2};y_{1},y_{2}\right):=\frac{r(x_{1},y_{1})+r(x_{1},y_{2})+r(x_{2},y_{1})+r(x_{2},y_{2})}{4}-\frac{r(x_{1},x_{2})+r(y_{1},y_{2})}{2}.$

Then $\Xi_{q}(\omega)$ is not empty if and only if
$g(\omega)\coloneqq\min\\{f(x_{1},x_{2};y_{1},y_{2}):\,x_{1},x_{2}\in
E_{q}^{\omega,1},y_{1},y_{2}\in E_{q}^{\omega,2}\\}>0$, in which case
$g(\omega)=\ell(e_{\omega})$.

###### Proof.

For an edge $e$ in $\mathcal{T}^{(q)}$, we denote by $e_{g}$ the geodesic in
$\mathcal{T}$ that corresponds to $e$.

Let us first prove the first part. Consider $\omega\subset\\{0,...,q\\}$ such
that $\Xi_{q}(\omega)$ is not empty and recall that a point
$x\in\Xi_{q}(\omega)$ splits $E_{q}$ into $E_{q}^{\omega,1}$ and
$E_{q}^{\omega,2}$. Observe that $x$ cannnot be a branching point of
$\mathcal{T}^{(q)}$. Then, $x\in e_{g}$ for some edge $e$ of
$\mathcal{T}^{(q)}$. Take $y\in E_{q}^{\omega,1}$ and $y^{\prime}\in
E_{q}^{\omega,2}$. By definition, $x\in\llbracket y,y^{\prime}\rrbracket$
(otherwise, $y$ and $y^{\prime}$ are in the same connected component of
$\mathcal{T}\backslash\\{x\\}$). Since the geodesic $\llbracket
y,y^{\prime}\rrbracket$ is injective and connects two vertices of
$\mathcal{T}^{(q)}$ in $\mathcal{T}$, we have $e_{g}\subset\llbracket
y,y^{\prime}\rrbracket$. In particular, for any $x^{\prime}\in e_{g}$ and for
any $y\in E_{q}^{\omega,1}$ and $y^{\prime}\in E_{q}^{\omega,2}$, we have that
$x^{\prime}\in\llbracket y,y^{\prime}\rrbracket$. Thus,
$e_{g}\subset\Xi_{q}(\omega)$.

Consider $\omega\subset\\{0,\dots,q\\}$ and $x\in\Xi_{q}(\omega)$ as before
and assume that there exists another edge $e^{\prime}\neq e$ such that $y\in
e_{g}^{\prime}$ and $y\in\Xi_{q}(\omega)$. Choose $b\in\llbracket
x,y\rrbracket$ a branching point of $\mathcal{T}^{(q)}$, and let $z\in E_{q}$
such that it is not in the same connected component of
$\mathcal{T}\backslash\\{b\\}$ as $x$ nor as $y$ (if $b=\rho$, we take instead
$z=b$). Assume without loss of generality that $z\in E_{q}^{\omega,1}$. Then,
for any $z^{\prime}\in E_{q}^{\omega,2}$, $x$ and $y$ both belong to
$\llbracket z,z^{\prime}\rrbracket$, which contradicts the fact that
$\llbracket z,z^{\prime}\rrbracket$ is a geodesic. Hence,
$\Xi_{q}(\omega)=e_{g}$.

Conversely, by definition, it is not difficult to see that every edge of
$\mathcal{T}^{(q)}$ corresponds to one $\Xi_{q}(\omega)$, for some
$\omega\subset\\{0,\dots,q\\}$.

We now prove the second part. First, assume that $\Xi_{q}(\omega)$ is a
geodesic that corresponds to an edge $e_{\omega}$ of $\mathcal{T}^{(q)}$ and
let $x,y$ be its endpoints, so that $\ell(e_{\omega})=r(x,y)$. For
$z\in\Xi_{q}(\omega)$, $\mathcal{T}\backslash\\{z\\}$ has two connected
components, one of them containing $x$ and the other containing $y$. Without
loss of generality, suppose that $x$ is in the connected component of all
points of $E_{q}^{\omega,1}$, and $y$ is in the one of all points of
$E_{q}^{\omega,2}$. Then, for any $x^{\prime}\in E_{q}^{\omega,1}$ and
$y^{\prime}\in E_{q}^{\omega,2}$, we have
$r(x^{\prime},y^{\prime})=r(x^{\prime},x)+r(x,y)+r(y,y^{\prime})$. Thus, for
all $x_{1},x_{2}\in E_{q}^{\omega,1}$, $y_{1},y_{2}\in E_{q}^{\omega,2}$, we
get that

$\displaystyle f(x_{1},x_{2};y_{1},y_{2})$
$\displaystyle=\frac{2r(x_{1},x)+2r(x_{1},y)+4r(x,y)+2r(y_{1},y)+2r(y_{2},y)}{4}-\frac{r(x_{1},x_{2})+r(y_{1},y_{2})}{2}$
$\displaystyle=r(x,y)+\frac{r(x_{1},x)+r(x_{2},x)-r(x_{1},x_{2})}{2}+\frac{r(y_{1},y)+r(y_{2},y)-r(y_{1},y_{2})}{2}.$

By the triangle inequality, $f(x_{1},x_{2};y_{1},y_{2})\geq r(x,y)$.
Furthermore, if $\omega$ and $\\{0,\dots,q\\}\backslash\omega$ both have at
least two elements, then $x$ and $y$ are branching points of
$\mathcal{T}^{(q)}$, and we can find $x_{1},x_{2}\in E_{q}^{\omega,1}$,
$y_{1},y_{2}\in E_{q}^{\omega,2}$ such that
$f(x_{1},x_{2};y_{1},y_{2})=r(x,y)$. Otherwise, if $\omega$ is a singleton,
say $\\{i\\}$, then $x=x_{i,q}$ and we can find two elements $y_{1},y_{2}\in
E_{q}^{\omega,2}$ such that $f(x,x;y_{1},y_{2})=r(x,y)$. The case where
$\\{0,\dots,q\\}\backslash\omega$ is a singleton is handled the same way.
Hence, we have proved that if $\Xi(\omega)$ is not empty then
$g(\omega)=\ell(e_{\omega})>0$.

Now we assume that $g(\omega)>0$. If $\omega$ is a singleton, say
$E_{q}^{\omega,1}\coloneqq\\{x\\}$, then for all $y_{1},y_{2}\in
E_{q}^{\omega,2}$ such that $y_{1}\neq y_{2}$ we have

$\displaystyle
f(x,x;y_{1},y_{2})=\frac{r(x,y_{1})+r(x,y_{2})-r(y_{1},y_{2})}{2}\geq
g(\omega).$

Hence, by letting $b$ be the branchpoint of $y_{1},y_{2}$ and $x$ in
$\mathcal{T}^{(q)}$ (that is, $b=\llbracket x,y_{1}\rrbracket\cap\llbracket
x,y_{2}\rrbracket\cap\llbracket y_{1},y_{2}\rrbracket$), we have $b\neq x$ and
$r(x,b)\geq g(\omega)$. Remark that if $x=\emptyset$ then $x\neq b$ since a
point of $\Xi_{q}(\omega)$ should belong to both
$\llbracket\emptyset,y_{1}\rrbracket$ and
$\llbracket\emptyset,y_{2}\rrbracket$. Then if $x$ is a leaf of
$\mathcal{T}^{(q)}$, for any edge $e\in\mathcal{T}^{(q)}$ and $z\in e_{g}$
such that $r(x,z)\in(0,g(\omega))$, we see that $z\in\Xi_{q}(\omega)$. If
$x=\emptyset$, any $z\in\llbracket\emptyset,y\rrbracket$ for some $y\in
E_{q}^{\omega,2}$ such that $r(x,z)\in(0,g(\omega))$, we have that
$z\in\Xi_{q}(\omega)$.

Finally, assume that $\omega$ and $\\{0,\ldots,q\\}\backslash\omega$ are not
singletons. Fix $x_{1}\neq x_{2}\in E_{q}^{\omega,1}$. First, we prove that
all $y\in E_{q}^{\omega,2}$, the branchpoints of $x_{1},x_{2}$ and $y$ in
$\mathcal{T}^{(q)}$ are the same. Indeed, let $y_{1}\neq y_{2}$ be two
elements of $E_{q}^{\omega,2}$ and $z_{1},z_{2}$ their respective branchpoints
with $x_{1}$ and $x_{2}$. Then, if $z_{1}\neq z_{2}$, using the fact that
$z_{1},z_{2}\in\llbracket x_{1},x_{2}\rrbracket$, we get

$\displaystyle f(x_{1},x_{2};y_{1},y_{2})=-r(z_{1},z_{2})<0.$

Let $a(x_{1},x_{2})$ be therefore the branchpoint of $x_{1},x_{2}$ and $y$ in
$\mathcal{T}^{(q)}$, for all $y\in E_{q}^{\omega,2}$. Symmetrically, for any
$y_{1},y_{2}\in E_{q}^{\omega,2}$, let $b(y_{1},y_{2})$ be the branchpoint of
$y_{1},y_{2}$ and $x$ in $\mathcal{T}^{(q)}$, for all $x\in E_{q}^{\omega,1}$.
If there exists $x_{1}\neq x_{2}\in E_{q}^{\omega,1},y_{1}\neq y_{2}\in
E_{q}^{\omega,2}$ such that $a(x_{1},x_{2})=b(y_{1},y_{2})$, then
$f(x_{1},x_{2};y_{1},y_{2})=0$ which contradicts our assumption. Choose
$x_{1},x_{2}\in E_{q}^{\omega,1},y_{1},y_{2}\in E_{q}^{\omega,2}$ such that
$r(a(x_{1},x_{2}),b(y_{1},y_{2}))>0$ is minimum, and take $z\in\llbracket
a(x_{1},x_{2}),b(y_{1},y_{2})\rrbracket\backslash\\{a(x_{1},x_{2}),b(y_{1},y_{2})\\}$
. If $z\notin\Xi_{q}(\omega)$ then without loss of generality there exists
$x\in E_{q}^{\omega,1}$ in the component of $\mathcal{T}\backslash\\{z\\}$
containing $y_{1}$ and $y_{2}$. But in this case $a(x_{1},x)\in\llbracket
a(x_{1},x_{2}),b(y_{1},y_{2})\rrbracket\backslash\\{a(x_{1},x_{2}),b(y_{1},y_{2})\\}$
which contradicts the minimality assumption. Thus, $z\in\Xi_{q}(\omega)$ which
concludes the proof. ∎

The previous lemma admits a discrete counterpart. For $n\geq 1$, recall that a
rooted plane tree $\tau_{n}$ can also be viewed as a rooted metric measure
space $(\tau_{n},r_{n}^{\text{gr}},\emptyset_{n},\mu_{n})$. For $q\geq 1$, let
$u_{1}^{n},\dots,u_{q}^{n}$ be $q$ i.i.d. random vertices sampled according to
$\mu_{n}$. Let $u_{0,q}^{n}=\emptyset_{n}$ and $u_{1,q}^{n},\dots,u_{q,q}^{n}$
be the $q$ vertices $u_{1}^{n},\dots,u_{q}^{n}$ of $\tau_{n}$ listed in
lexicographical order. Set $E_{q,n}=\\{u_{0,q}^{n},\dots,u_{q,q}^{n}\\}$. For
$\omega\subset\\{0,\dots,q\\}$, let $\Xi_{q,n}(\omega)$ be the set of edges
(if any) of $\tau_{n}$ splitting the set $E_{q,n}$ into
$E_{q,n}^{\omega,1}=\\{u_{k,q}^{n}:k\in\omega\\}$ and
$E_{q,n}^{\omega,2}=\\{u_{k,q}^{n}:k\not\in\omega\\}$.

######  Lemma 4.6.

For $q\geq 1$ and $\omega\subset\\{0,\dots,q\\}$, we have that
$\Xi_{q,n}(\omega)$ is either empty or a collection of neighbouring edges in
$\tau_{n}$ that corresponds precisely to an edge $e_{\omega,n}$ in
$\tau_{n}^{(q)}$ of length $\ell(e_{\omega,n})$ given by the number of edges
in the set $\Xi_{q,n}(\omega)$. Reciprocally, for every edge $e_{n}$ in
$\tau_{n}^{(q)}$ of length $\ell(e_{n})$ there exists
$\omega_{e_{n}}\subset\\{0,\dots,q\\}$ such that $\Xi_{q,n}(\omega_{e_{n}})$
is a collection of neighbouring edges in $\tau_{n}$ that corresponds precisely
to $e_{n}$ such that number of edges in the set $\Xi_{q,n}(\omega_{e_{n}})$ is
equal to $\ell(e_{n})$.

Furthermore, for any points $x_{1},x_{2}\in E_{q,n}^{\omega,1}$ and
$y_{1},y_{2}\in E_{q,n}^{\omega,2}$, define

$\displaystyle
f_{n}\left(x_{1},x_{2};y_{1},y_{2}\right)\coloneqq\frac{r_{n}^{\rm
gr}(x_{1},y_{1})+r_{n}^{\rm gr}(x_{1},y_{2})+r_{n}^{\rm
gr}(x_{2},y_{1})+r_{n}^{\rm gr}(x_{2},y_{2})}{4}-\frac{r_{n}^{\rm
gr}(x_{1},x_{2})+r_{n}^{\rm gr}(y_{1},y_{2})}{2}.$

Then $\Xi_{q,n}(\omega)$ is not empty if and only if
$g_{n}(\omega)\coloneqq\min\\{f_{n}(x_{1},x_{2};y_{1},y_{2}):\,x_{1},x_{2}\in
E_{q,n}^{\omega,1},y_{1},y_{2}\in E_{q,n}^{\omega,2}\\}>0$, in which case
$g_{n}(\omega)=\ell(e_{\omega,n})$.

###### Proof.

It follows as in the proof of Lemma 4.5. ∎

If $\Xi_{q,n}(\omega)$ is not empty, let $e_{\omega,n}$ be the corresponding
edge of $\tau_{n}^{(q)}$ that splits $E_{q,n}$ into $E_{q,n}^{\omega,1}$ and
$E_{q,n}^{\omega,2}$. Denote by $\ell(e_{\omega,n})$ the number of edges in
$\Xi_{q,n}(\omega)$, that is, the length of the edge $e_{\omega,n}$, with the
convention $\ell(e_{\omega,n})=0$ whenever $\Xi_{q,n}(\omega)$ is empty. If
$\Xi_{q}(\omega)$ is not empty, let $e_{\omega}$ be the corresponding edge of
$\mathcal{T}^{(q)}$ that splits $E_{q}$ into $E_{q}^{\omega,1}$ and
$E_{q}^{\omega,2}$. Denote by $\ell(e_{\omega})$ the length of the geodesic
$\Xi_{q}(\omega)$; with the convention $\ell(e_{\omega})=0$ whenever
$\Xi_{q}(\omega)$ is empty. The following lemma, whose proof makes use of
Lemmas 4.5 and 4.6, states the equivalence of the convergence of a sequence of
discrete trees and the convergence of the lengths of edges of the reduced
trees.

######  Lemma 4.7.

Let $(\tau_{n},n\geq 1)$ be a sequence of rooted plane trees, $\mathcal{T}$ be
a continuum tree and $(a_{n},n\geq 1)$ a sequence of non-negative real numbers
satisfying $a_{n}\rightarrow\infty$ and
$\zeta(\tau_{n})/a_{n}\rightarrow\infty$, as $n\rightarrow\infty$. Then, the
following assertions are equivalent:

  * (i)

$\displaystyle\left(\tau_{n},\frac{a_{n}}{\zeta(\tau_{n})}r_{n}^{\rm
gr},\emptyset_{n},\mu_{n}\right)\rightarrow(\mathcal{T},r,\rho,\mu)$, as
$n\rightarrow\infty$, for the Gromov-weak topology.

  * (ii)

For every integer $q\geq 1$ fixed,
$\displaystyle\left(\frac{a_{n}}{\zeta(\tau_{n})}\ell(e_{\omega,n}):\omega\subset\\{0,\dots,q\\}\right)\xrightarrow[]{d}(\ell(e_{\omega}):\omega\subset\\{0,\dots,q\\})$,
as $n\rightarrow\infty$.

###### Proof of Lemma 4.7.

For all $0\leq i,j\leq q$, let $\Omega_{n}(i,j)$ be the set of subsets of
$\omega\subset\\{0,\dots,q\\}$ such that $u^{n}_{i,q}\in E_{q,n}^{\omega,1}$
and $u^{n}_{j,q}\in E_{q,n}^{\omega,2}$. Similarly, let $\Omega(i,j)$ be the
set of subsets of $\omega\subset\\{0,\dots,q\\}$ such that $x_{i,q}\in
E_{q}^{\omega,1}$ and $x_{j,q}\in E_{q}^{\omega,2}$. Our claim then follows
from (18), Lemma 4.5, Lemma 4.6 and the identities

$\displaystyle
r_{n}^{\text{gr}}(u^{n}_{i,q},u^{n}_{j,q})=\sum_{\omega\in\Omega_{n}(i,j)}\ell(e_{\omega,n})\qquad\text{
and }\hskip
14.22636ptr(x_{i,q},x_{j,q})=\sum_{\omega\in\Omega(i,j)}\ell(e_{\omega}).$

∎

The rest of the section is devoted to the proof of Theorem 2. Let us start by
proving that (C.2) implies (C.1). As a preparation step, we need the following
proposition.

######  Proposition 5.

In the setting of Theorem 2, suppose that (C.2) is satisfied. Then, we have
that :

  * (i)

there exists a coupling between the lamination-valued processes such that, in
probability,

$\displaystyle\displaystyle\lim_{q\rightarrow\infty}\limsup_{n\rightarrow\infty}\,d_{\rm
Sk}^{\mathbb{L}}\left(\left(\mathbb{L}_{\frac{ta_{n}}{\zeta(\tau_{n})}}^{(q)}(\tau_{n}),t\geq
0\right),\left(\mathbb{L}_{t}^{(q)}(\mathcal{T}),t\geq 0\right)\right)=0;$

  * (ii)

for any $\varepsilon>0$,
$\displaystyle\lim_{q\rightarrow\infty}\limsup_{n\rightarrow\infty}\mathbb{P}\left(d_{\rm
Sk}^{\mathbb{L}}\left(\left(\mathbb{L}_{\frac{ta_{n}}{\zeta(\tau_{n})}}^{(q)}(\tau_{n}),t\geq
0\right),(\mathbb{L}_{ta_{n}}(\tau_{n}),t\geq 0)\right)<\varepsilon\right)=1$.

Roughly speaking, (i) states that, as $q\rightarrow\infty$, the time-scaled
discrete reduced lamination-valued process obtained from sampling $q$ i.i.d.
uniform vertices of $\tau_{n}$ is asymptotically close to its continuum
counterpart $(\mathbb{L}_{t}^{(q)}(\mathcal{T}),t\geq 0)$. On the other hand,
by (ii), the complete process $(\mathbb{L}_{ta_{n}}(\tau_{n}),t\geq 0)$
associated to $\tau_{n}$ can be approximated by the time-scaled discrete
reduced lamination-valued process, whenever $q$ is large enough.

###### Proof of Theorem 2, (C.2) $\Rightarrow$ (C.1).

It is a consequence of Propositions 4 and 5. ∎

###### Proof of Proposition 5 (i).

We can and will assume, by Skorohod’s representation theorem, that (C.2) holds
almost surely. By (C.2), we have that $|u_{n}|=o(\zeta(\tau_{n}))$, in
probability, where $u_{n}$ denotes a uniform vertex of $\tau_{n}$. This
implies that $\tau_{n}^{(q)}$ has $q$ leaves with probability $1-o(1)$, as
$n\rightarrow\infty$.

In the setting of Lemma 4.7, for any $\omega\subset\\{0,\ldots,q\\}$, we
define $\gamma_{\omega,n}$ an exponential variable of parameter
$\ell(e_{\omega,n})$ associated to the edge $e_{\omega,n}$ of $\tau_{n}^{(q)}$
whenever $\Xi_{q,n}(\omega)$ is not empty, otherwise we let
$\gamma_{\omega,n}=\infty$ almost surely. Those exponential random variables
correspond to the ones used in the definition of
$(\mathbb{L}_{t}^{(q)}(\tau_{n}),t\geq 0)$. Similarly, denote by
$\gamma_{\omega}$ the exponential variable of parameter $\ell(e_{\omega})$
associated to the edge $e_{\omega}$ of $\mathcal{T}^{(q)}$ whenever
$\Xi_{q}(\omega)$ is not empty, otherwise $\gamma_{\omega}=\infty$ almost
surely. The latter exponential random variables correspond to the ones used in
the definition of $(\mathbb{L}_{t}^{(q)}(\mathcal{T}),t\geq 0)$. By Lemma 4.7,
it follows, jointly with (C.2), that

$\left(\frac{\zeta(\tau_{n})}{a_{n}}\gamma_{\omega,n}:\omega\subset\\{0,\ldots,q\\}\right)\xrightarrow[]{d}\left(\gamma_{\omega}:\omega\subset\\{0,\ldots,q\\}\right),\hskip
8.53581pt\text{as}\hskip 5.69054ptn\rightarrow\infty.$ (19)

This implies that the “jump times” of the time-scaled discrete reduced
lamination-valued process converge to those of the continuum reduced
lamination-valued process. Here, “jump times” refers to the times a new chord
is added in the corresponding reduced lamination-valued processes. In
particular, if $\gamma_{\omega,n}=\infty$ (resp. $\gamma_{\omega}=\infty$),
then no chord associated to $\omega$ is added in the reduced lamination-valued
processes (no jump), i.e., there is no edge in $\tau_{n}^{(q)}$ (resp.
$\mathcal{T}^{(q)}$) associated to $\omega$. In fact, the “actual jump times ”
that will count toward the limit are those $\gamma_{\omega,n}$’s and
$\gamma_{\omega}$’s for which $\ell(e_{\omega})>0$.

We can now assume, by Skorohod’s representation theorem, that (C.2) and (19)
hold almost surely. Denote by $\Theta_{\infty}$ the class of strictly
increasing, continuous mappings
$\theta:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ with $\theta(0)=0$ and
$\theta(t)\uparrow\infty$, as $t\uparrow\infty$. Then to prove our claim, it
is enough to show that there exists a sequence of functions $(\theta_{n},n\geq
1)\in\Theta_{\infty}$ such that, for all $M\geq 0$,

$\displaystyle\limsup_{n\rightarrow\infty}\sup_{t\geq
0}|\theta_{n}(t)-t|=0\hskip 8.53581pt\text{and}\hskip
8.53581pt\lim_{q\rightarrow\infty}\limsup_{n\rightarrow\infty}\sup_{t\in[0,M]}d_{\rm
H}\left(\mathbb{L}_{\frac{\theta_{n}(t)a_{n}}{\zeta(\tau_{n})}}^{(q)}(\tau_{n}),\mathbb{L}_{t}^{(q)}(\mathcal{T})\right)=0,\hskip
5.69054pt\text{in probability};$

see [31, Theorem 1.14 in Chapter VI] (or [17, Chapter 3]).

Let $(s_{j}:j=1,\dots,J)$ be the sequence of “actual jump times” of the
continuum reduced lamination-valued process arranged in increasing order.
Similarly, let $(s_{j}^{n}:j=1,\dots,J)$ be the corresponding sequence of
“actual jump times” of the time-scaled reduced lamination-valued process
arranged in increasing order. Set $s_{0}=0$ and define $\theta_{n}$ by letting
$\theta_{n}(s_{0})=0$, $\theta_{n}(s_{j})=s_{j}^{n}$, for $j=1,\dots,J$, such
that $\theta_{n}$ is linear on $[s_{j},s_{j+1}]$ for $j=0,1,\dots,J-1$ and
$[s_{J},\infty)$ (with slope $1$ after $s_{J}$). Clearly, (19) implies that
$\theta_{n}$ converges uniformly to the identity mapping on $[0,\infty)$, as
$n\rightarrow\infty$. Thus, it only remains to check that the chords that we
add between “actual jump times” are asymptotically close to each other.
Consider $\omega\subset\\{0,\ldots,q\\}$ such that $\ell(e_{\omega})>0$. Let
$e_{\omega}$ be an edge of $\mathcal{T}^{(q)}$ and let $e_{\omega,n}$ be the
corresponding edge of $\tau_{n}^{(q)}$ (which exists for $n$ large enough).
So, independently of the choice of the elements in
$\mathbf{H}(\tau_{n}^{(q)})$ and $\mathbf{H}(\mathcal{T}^{(q)})$, uniformly in
$\omega$, for any chords $c_{\omega,n}$ and $c_{\omega}$ coding respectively
$e_{\omega,n}$ and $e_{\omega}$, Lemma 4.4 implies that
$\sup_{\omega\subset\\{0,\ldots,q\\}}d_{\rm
H}(c_{\omega,n},c_{\omega})\rightarrow 0$, in probability, as
$q\rightarrow\infty$. This concludes our proof. ∎

###### Proof of Proposition 5 (ii).

By Lemma 4.4, it is enough to prove that, for any $\varepsilon>0$,

$\displaystyle\lim_{q\rightarrow\infty}\limsup_{n\rightarrow\infty}\mathbb{P}\left(d_{\rm
Sk}^{\mathbb{L}}\left(\left(\mathbb{L}_{\frac{ta_{n}}{\zeta(\tau_{n})}}^{(q)}(\tau_{n}),t\geq
0\right),\left(\mathbb{L}^{\rm
d}_{\frac{ta_{n}}{\zeta(\tau_{n})}}(\tau_{n}),t\geq
0\right)\right)<\varepsilon\right)=1.$

Fix $q\geq 1$ and recall that, by (C.2), we can assume that $\tau^{(q)}_{n}$
has exactly $q$ leaves. Consider $\omega\subset\\{0,\ldots,q\\}$ such that
$\Xi_{q,n}(\omega)$ is not empty. Then, the time at which the chord coding
$e_{\omega,n}$ appears in the lamination-valued process
$(\mathbb{L}^{(q)}_{t}(\tau_{n}),t\geq 0)$ is distributed as an exponential
variable of parameter $\ell(e_{\omega,n})$ (which is the minimum of the
$\ell(e_{\omega,n})$ exponential random variables of parameter $1$ associated
to the set of edges $\Xi_{q,n}(\omega)$); see also Definition 4). Hence, we
only need to prove that for all $\varepsilon>0$ there exists $Q$ large enough
so that with probability $>1-\varepsilon$, for all $q\geq Q$, all $n$ large
enough,

  * (a)

for any $\omega\subset\\{0,\ldots,q\\}$ such that $\Xi_{q,n}(\omega)$ is not
empty, and for any edge $e_{n}\in\Xi_{q,n}(\omega)$, $d_{\rm
H}(c_{n},\tilde{c}_{\omega,n})<\varepsilon$, where $c_{n}$ is the chord of
$\mathbb{L}(\tau_{n})$ coding $e_{n}$ and $\tilde{c}_{\omega,n}$ is the chord
of $\mathbb{L}^{(q)}(\tau_{n})$ coding $e_{\omega,n}$;

  * (b)

for any edge $e_{n}\in\tau_{n}$ whose removal does not split the set
$E_{q,n}$, $d_{\rm H}(c_{n},\mathbb{S}^{1})<\varepsilon$;

Fix $\varepsilon>0$, and take $Q$ large enough so that Lemma 4.4 holds with
probability $>1-\varepsilon$ for all $q\geq Q$. Observe that (a) follows
directly from Lemma 4.4. So, it only remains to prove (b). For a vertex
$u\in\tau_{n}$, let $\tau_{n}(u)$ be the subtree of $\tau_{n}$ rooted at $u$,
i.e., $\tau_{n}(u)$ consists of $u$ and all its descendants. Observe that the
edges considered in (b) are of two kinds: either they are in a subtree of the
form $\tau_{n}(u_{k,q}^{n})$ for some $1\leq k\leq q$, or they are in subtrees
branching out of the set of edges that are in the geodesic paths of
$\tau_{n}^{(q)}$.

To deal with the edges of the first kind, observe that with probability
$1-o(1)$, as $n\rightarrow\infty$, all subtrees $(\tau_{n}(u_{k,q}^{n}),1\leq
k\leq q)$ have size $o(\zeta(\tau_{n}))$. This follows from (C.2), since
$|u_{n}|=o(\zeta(\tau_{n}))$, in probability, where $u_{n}$ denotes a uniform
vertex of $\tau_{n}$. Then, it should be clear that, with probability
$1-o(1)$, all chords corresponding to edges of the first kind have length
$o(1)$, as $n\rightarrow\infty$.

To deal with edges of the second kind, we use the definition of the
lamination-valued process from the contour function $C_{\tau_{n}}$ of
$\tau_{n}$; see Definition 1 or Definition 2. First, we recall a way of