%~Mouliné par MaN_auto v.0.40.4 (9e9f6e09) 2026-05-18 13:43:34 \documentclass[AHL,Unicode]{cedram} \usepackage{booktabs} \usepackage{bbm} \usepackage{enumitem} \usepackage{mdframed} \usepackage{mathtools} %\usepackage{crossreftools} \usepackage{makecell} \usepackage[capitalise]{cleveref} \crefname{theo}{Theorem}{Theorems} \crefname{prop}{Proposition}{Propositions} \crefname{lemm}{Lem\-ma}{Lem\-mas} %\DeclareMathOperator*{\diam}{Diam} %\DeclarePairedDelimiter\ceil{\lceil}{\rceil} %\DeclarePairedDelimiter\floor{\lfloor}{\rfloor} %\newcommand\hmmax{0} %\newcommand\bmmax{0} \colorlet{link}{red!60!black} \newlist{compitem}{itemize}{4} \setlist[compitem,1]{nolistsep,label=$\bullet$} \makeatletter \newcommand{\optionaldesc}[2]{ \phantomsection #1\protected@edef\@currentlabel{#1}\label{#2} } \makeatother %\makeatletter %\newcommand{\labeltext}[3][]{ %\@bsphack %\csname phantomsection %\endcsname %\newcommand*\tst{[1]} %\newcommand*\labelmarkup{\textcolor{link}} %\newcommand*\refmarkup{} %\ifx\tst\empty %\newcommand*\@currentlabel{\refmarkup{#2}}{\label{#3}} %\else %\newcommand*\@currentlabel{\refmarkup{[1]}}{\label{#3} %}\fi %\@esphack %\labelmarkup{#2} } %\makeatother %\newcommand{\esp}[1]{\mathbb{E}\left[#1\right]} \newcommand{\proba}[1]{\mathbb{P}\left(#1\right)} \newcommand{\probak}[1]{\mathbb{P}^k\left(#1\right)} \newcommand{\e}{\varepsilon} \newcommand{\cv}[1][n]{\enskip\xrightarrow[#1 \to \infty]{}\enskip} \newcommand{\cvloi}[1][n]{\enskip\xrightarrow[#1 \to \infty]{(d)}\enskip} \newcommand{\cvps}[1][n]{\enskip\xrightarrow[#1 \to \infty]{\text{a.s.}}\enskip} \newcommand{\cvproba}[1][n]{\enskip\xrightarrow[#1 \to \infty]{\P}\enskip} %\DeclareMathOperator*{\limit}{\longrightarrow} \DeclareMathOperator*{\supp}{supp} \DeclareMathOperator{\GH}{GH} \DeclareMathOperator{\GP}{GP} \DeclareMathOperator{\GHP}{GHP} \DeclareMathOperator{\TV}{TV} \newcommand{\rmd}{\mathrm{d}} \newcommand{\rmi}{\mathrm{i}} \newcommand{\K}{\mathbb{K}} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} \renewcommand{\P}{{\mathbb{P}}} \newcommand{\E}{\mathbb{E}} \newcommand{\Lp}{\mathbb{L}} \newcommand{\1}{\mathbbm{1}} \newcommand*{\badim}{\underline{\mathfrak{d}}} \newcommand{\fath}{\mathfrak{f}} \newcommand{\fkm}{\mathfrak{m}} \newcommand{\trail}{\mathcal{S}} \newcommand{\Part}{\mathcal{P}} \newcommand{\T}{\mathcal{T}} \newcommand{\clB}{\mathcal{B}} \newcommand{\clF}{\mathcal{F}} \newcommand{\h}{\mathsf{h}} \newcommand{\sLim}{\mathsf{Lim}} \newcommand{\sTight}{\mathsf{Tight}} \newcommand{\sGP}{\mathsf{GP}} \newcommand{\sGHP}{\mathsf{GHP}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bxi}{\bar{\xi}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \let\originalleft\left \let\originalright\right \renewcommand{\left}{\mathopen{}\mathclose\bgroup\originalleft} \renewcommand{\right}{\aftergroup\egroup\originalright} \renewcommand*{\to}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}} \let\oldmapsto\mapsto \renewcommand*{\mapsto}{\mathchoice{\longmapsto}{\oldmapsto}{\oldmapsto}{\oldmapsto}} \renewcommand*{\implies}{\mathchoice{\Longrightarrow}{\Rightarrow}{\Rightarrow}{\Rightarrow}} \newcounter{setofstep} \newcounter{step}[setofstep] \crefname{step}{Step}{Steps} \newenvironment{step}[1][]{\refstepcounter{step}\begin{proof}[{Step~\thestep\if#1\empty\else: #1\fi}]}{\let\qed\relax\end{proof}} \newcommand{\newstepset}{\stepcounter{setofstep}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Better widebar \makeatletter \let\save@mathaccent\mathaccent \newcommand*\if@single[3]{% \setbox0\hbox{${\mathaccent"0362{#1}}^H$}% \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}% \ifdim\ht0=\ht2 #3\else #2\fi } %The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath: \newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna} %If there's a superscript following the bar, then no negative kern may follow the bar; 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A.~Blanc-Renaudie was supported by ERC consolidator grant 101087572 (SuPerGRandMa)} \CDRGrant[SuPerGRandMa]{101087572} \begin{abstract} We consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalized converge. To this end, we study the paths from random vertices to the root using coalescent processes. As an application, we obtain scaling limits of Bienaymé--Galton--Watson trees in varying environment. \end{abstract} \begin{altabstract} Nous considérons de grands arbres aléatoires uniformes dont les degrés et hauteurs des sommets sont fixés. Nous démontrons, sous des conditions naturelles de convergence du profil, que ces arbres convenablement renormalisés convergent. Pour ce faire, nous étudions les chemins issus de sommets aléatoires vers la racine en utilisant des processus coalescents. En guise d'application, nous obtenons les limites d'échelle d'arbres de Bienaymé--Galton--Watson en environnement variable. \end{altabstract} \datereceived{2024-09-29} \daterevised{2025-10-21} \dateaccepted{2026-03-04} \editors{S.~Gou\"ezel and D.~Aldous} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \maketitle \newpage \section{Introduction} \begin{figure}[!ht] \centering \includegraphics[width=0.64\linewidth]{arbre_dh_fixe_26737_sommets_4.pdf} \qquad \includegraphics[width=0.30\linewidth]{arbre_hd_fixes_26737_sommets_profil_sinus.pdf} \caption{Simulation of a uniform random tree of height $100$ whose profile scales towards $t \mapsto \sin(\pi t)$ with vertices of degrees $0$ and $4$ with some additional vertices with large degrees. The profile is drawn on the right. The low vertices are green while the high vertices are red.} \label{fig:ssimus_intro} \end{figure} \subsection{Model} Trees with fixed degree sequence are universal models of random trees and are related to many other objects: Galton--Watson trees, configuration model, bipartite planar maps with fixed face degrees\dots. The aim of this paper is to study a variant of this model where we fix for each vertex its degree and also its height. More precisely, we are interested in the shortest-path distance on the rooted labelled trees defined below. Consider a set of vertices $\{(i,j)\}_{i\geq 0,\,j\geq 1}$. We consider $(i,j)$ to have height $i$ and degree (number of children) $d_{i,j}^n\geq 0$. We use a superscript $n\in \N$ as we study the behaviour of the trees as the degrees vary. For convenience, we assume that on every line $i\ge 0$ the sequence $(d^n_{i,j})_{j\ge 1}$ is non-increasing. Also, to ensure that every vertex $(i,j)$ of degree $d^n_{i,j}\geq 1$ is connected to the root $(0,1)$, we assume that $d^n_{0,2}=0$ and \begin{equation}\label{eq:CoherenceDegree} \forall i \ge 0, \quad \# \left\{ j \ge1 ; \, d^n_{i+1,j}>0\right\} \le D^n_i \coloneqq \sum_{j\ge 1} d^n_{i,j}<\infty. \end{equation} We construct the plane tree $\T^n$ height by height: $(0,1)$ is the root, and independently and uniformly for every height $i\geq 0$ we attach each vertex in $\{(i+1,j)\}_{1\leq j \leq D_i^n}$ to a vertex in $\{(i,j); \, {j \ge 1}, d^n_{i,j} \ge 1\}$ so that every vertex $(i,j)$ has exactly $d_{i,j}^n$ children (see Figure~\ref{fig:Exemple}). Note that $\T^n$ has height $\h^n \coloneqq \inf \{i\ge 0; \, d^n_{i,1}=0 \}$. The tree $\T^n$ is equipped with the shortest path distance, where each edge has length $1$. \begin{figure} \centering \includegraphics[scale=0.44]{UniDegreeHeight.pdf} \caption{An example of tree $\T$ such that $d_{0,1}=4$, $d_{1,1}=3$, $d_{1,2}=1$, $d_{1,3}=1$, $d_{2,1}=2$, $d_{2,2}=1$, $d_{3,1}=2$ and all the other degrees are null. The vertices of $\T$ are represented in red, the root as a red dot, {and} the other vertices as black dots. Here, $D_0=4, D_1=5, D_2=3, D_3=2, D_4=0$.} \label{fig:Exemple} \end{figure} \subsection{Main results} We now present our assumptions on the degrees for the scaling limit of $\T^n$. Without loss of generality, our assumptions are written so that the random metric space $\T^n/n$ obtained after dividing the distances by $n$ converges in distribution. First we assume that the profile converges weakly: {we assume that} for {some} non-atomic probability measure $\nu$ with compact support $\supp \nu = [0,1]$, we assume, writing $\delta_x$ for the Dirac measure on $x$, \smallskip \begin{mdframed}[linecolor=black!60] \begin{description} \item[\textcolor{link}{\optionaldesc{$(\sLim_{\nu})$}{hyp:naissance}}] $ \sum_{i\geq 0} \delta_{i/n}D^n_{i}/(\sum_{i\ge 0} D^n_i)$ converges weakly towards $\nu$. \end{description} \end{mdframed} \smallskip Note that the above assumption is equivalent to {requiring} a scaling limit for the height of random vertices. It also implies that $\liminf \h^n/n \ge 1$. Then, to show the convergence of $(\T^n/n)$, we look at the \emph{genealogies} of random vertices, \emph{that is their paths to the root}. The rate at which genealogies merge actually only depends on the degrees. We assume informally that this rate converges. We distinguish two cases for such merging: either the paths merge at a vertex with large degree, or they merge among the numerous vertices of small degrees. Concretely, we use a locally finite measure $\rho$ on $(0,1)$ to describe the merging rate due to small vertices. We also describe the limit of large degrees, using non negative parameters $\Theta=((\theta_j(t))_{j\ge 1})_{t\in (0,1)}$ satisfying that: \begin{itemize} \item for every height $t\in (0,1)$, the sequence $(\theta_j(t))_{j\geq 1}$ is non-increasing. \item For every $t\in (0,1)$, we have $ \sum_{j\ge 1} \theta_j(t) \le 1. $ \item The set $\{(j,t):\theta_j(t)\neq 0\}$ is countable. Moreover, for {all} real numbers $00} \delta_{i/n} \binom{d^n_{i,j}}{2}\Big/{\binom{D^n_{i}}{2}} \xrightarrow[n\to \infty]{} \rho + \sum_{j,t:\theta_{j}(t)>0} \theta_j(t)^2\delta_{t}. \] \item[\textcolor{link}{\optionaldesc{$(\sLim_{\Theta})$}{hyp:atomes}}] We have the following {vague} convergence on $((0,1)\times \R_{+}^*,\clB((0,1)\times \R_{+}^*))$ \[ \sum_{1 \le i0} \delta_{\left(i/n, d^n_{i,j}/D^n_i\right)} \cv[n] \sum_{t,j:\theta_{j}(t)>0} \delta_{(t, \theta_j(t))}. \] \item[\textcolor{link}{\optionaldesc{$(\sLim_{\neq})$}{hyp:splitatomes}}] For every $\e>0$ there exists $\delta(\e)>0$ such that for every $n>0$, $i, i'$ with $\e n \e D_i^n$ and $d^n_{i',1}>\e D_{i'}^n$ we either have $i=i'$ or $|i-i'|>\delta(\e) n$. \end{description} \end{mdframed} \smallskip %} We require the technical condition~\ref{hyp:splitatomes} because the merging rate behaves really differently {depending on whether} vertices of large degree are at the same height or not. Morally, we use~\ref{hyp:splitatomes} to ensure that in the limit tree vertices of large degree that are at the same height correspond to vertices at the same height in $\T^n$. Lastly, while the previous assumptions are sufficient to prove the convergence of the distances between random vertices, we want the following tightness assumption to ensure that the limit space is well defined: \smallskip \begin{mdframed}[linecolor=black!60] \begin{description} \item[\textcolor{link}{\optionaldesc{$(\sTight_{\sGP})$}{hyp:tightGP}}] $\nu$ has support $[0,1]$ and for all $00$ or \[ \sum_{j\in \N,t\in [a,b]\,:\,\theta_{j}(t)>0} \theta_j(t)=\infty. \] \end{description} \end{mdframed} \begin{rema} Aldous, Camarri, and Pitman~\cite{introICRT1,introICRT2} used a similar condition to define inhomogeneous continuum random trees. Namely, with their notation $\theta_0>0$ or $\sum_{i\geq 1} \theta_i=\infty$. \end{rema} \begin{theo}\label{th:GP} Assume that for some $\nu, \rho,\Theta$ the conditions~\ref{hyp:naissance}, \ref{hyp:coalescence}, \ref{hyp:atomes}, \ref{hyp:splitatomes}, \ref{hyp:tightGP} hold. Then there exists a random measured metric space $\T(\nu,\rho,\Theta)$ such that the following weak convergence holds for the Gromov--Prokhorov topology (see Appendix~\ref{sec:background} for more details) \[ \T^n/n \cvloi \T(\nu,\rho,\Theta), \] {where we recall that $\T^n/n$ is the metric space obtained by dividing the shortest path distance on $\T^n$ by $n$, which is also equipped} with the uniform probability measure on its vertices. \end{theo} The Gromov--Prokhorov topology essentially describes the distances between random vertices in $\T^n$. To study geometric variables that depend on all the vertices such as the diameter of $\T^n$, one needs a stronger topology. To this end, we as usual strengthen the convergence to the Gromov--Hausdorff--Prokhorov topology (see Appendix~\ref{sec:background}). We use the following tightness assumptions. %mis en emph pour l'instant mais cela ferait bien un enoncé ou un paragraph. Let $\|D^n\|_\infty\coloneqq \max_{0\leq i\leq \h^n} D_i^n$. Then, for every $i\in \N, k>0$, let \begin{equation}\label{eq def tau} \tau^n_{i}(k)\coloneqq\sum_{j\ge 1} \frac{d^n_{i,j}}{D^n_i} \min\left (k\frac{d^n_{i,j}-1}{D^n_i-1},1 \right), \end{equation} with the convention $\tau^n_i(k)=\infty$ when $D^n_i\in \{0,1\}$. Then, for every $a0$, the clusters merge according to the sub-probability measure $(\theta_j(t))_{j \ge 1}$ as follows: independently, each cluster is associated with some random integer $j$ in $\N \cup \{0\}$ with probability $\theta_j(t)$ when $j\ge 1$ and $\theta_0(t)\coloneqq1-\sum_{j'\ge 1} \theta_{j'}(t)$ when $j=0$. Then, the clusters associated to $0$ do not merge whereas for all $j \ge 1$, the clusters associated to $j$ merge together into one cluster. Finally, at time zero, all the clusters which have not merged yet coalesce. Here the birth time of the particle should be seen as the height of random vertices in the tree, and the merging time of two clusters as the height of their most recent common ancestor. And, depending on the degree of those ancestors ($2$ or $\infty$), the coalescences occur differently (using respectively $\rho$ or $\Theta$). Formally, let us use the same notation as for coalescent processes where each particle is associated with an integer, where clusters are described by sets of integers and where each coalescence is performed by replacing two clusters by their union. See e.g.\@~\cite{Pit06} or~\cite{Ber06} for the use of this notation in the classical theory of coalescent and fragmentation processes. We consider three types of independent random events: \begin{description} \item[Birth] for every $r\in \N$, let $H_r$ be a random variable in $[0,1]$ of law $\nu$; \item[{Small merge}] for every $q0$ and $r\in \N$, let $\Theta_{t,r}$ be a random variable of law given by $\proba{\Theta_{t,r}=j}=\theta_j(t)$ for every $j\in \N\cup \{0\}$. We write $S^\Theta(t,j)\coloneqq \{r\in \N,\, \Theta_{t,r}=j \}$ for all $j \ge 1$. \end{description} %???c'est quoi ce terme ici ?? From there we describe a {{càglàd}} (left continuous with right limit) jump process $((\Part_r^k(t))_{1\leq r \leq k})_{1\geq t \geq 0}$ which is called a growth-coalescent process. The process starts at $\Part_1^k(1)=\Part_2^k(1)=\dots =\Part_k^k(1)=\emptyset $ and its jumps occur only at some time $t$ satisfying one of the following conditions: \begin{description} \item[{Birth}] If $t=H_r$ for some $1\leq r \leq k$, then $\Part_r^k(t^+)=\{\emptyset\}$, and $\Part_r^k(t)=\{r\}$; \item[{Small merge}] If $t\in \Gamma_{q,r}$ for some $q0$, then for every $j\in \N$ such that $\# (S^\Theta (t,j) \cap [k])\ge 2$, we write $m^k(t,j)$ for the smallest integer $r\in S^\Theta(t,j)\cap[k]$ such that $\Part_r^k(t^+)\neq \emptyset$. We have: \begin{multline*} \forall r\in (S^\Theta(t,j)\cap[k]) \setminus \{m^k(t,j) \}, \; \Part_r^k(t)=\emptyset,\\ \text{and} \quad \Part_{m^k(t,j)}^k(t)=\bigcup_{r\in S^\Theta(t,j)\cap[k]} \Part_{r}^k(t^+). \end{multline*} \end{description} There is a final jump at time $0$ to reach the configuration $\Part_1^k(0)=\{1,\dots, k\}$ and $\Part_2^k(0)=\dots=\Part_k^k(0)=\emptyset$. \begin{lemm}\label{lemme coalescent continu bien defini} The process $((\Part_r^k(t))_{1\leq r \leq k})_{0\leq t\leq 1}$ is a.s.\@ well defined. \end{lemm} \begin{proof} To see that such a process is well defined, it suffices to focus on the number of possible jumps. Births only occur at $\{H_r\}_{1\leq r \leq k}$ which is a finite subset of $(0,1)$ since $\nu$ has no atoms at zero or one. Note that for $1\leq r \leq k$ and $t>H_r$ we must have $\Part_r^k(t)=\emptyset$ so that the highest jump occurs at time $\max_{1\leq r \leq k} H_r$. Then, for every $a>0$, we prove that there is only a finite amount of possible times for small merges and large merges in $[a, \max_{1 \le r \le k} H_r]$, so that the process $((\Part_r^k(t))_{1\leq r \leq k})_{0\leq t\leq 1}$ is well defined on $[a, 1]$. Since a.s. $\Part^k_1(0)=\{1,\dots, k\}$ and $\Part_2^k(0)=\dots=\Part_k^k(0)=\emptyset$, and since $a$ is arbitrary this will conclude the proof. Small merges can only occur when $t\in \Gamma_{q,r}$ for some $1\leq q0, \quad \lim_{k\to \infty} \mathbb{P} \bigl(d(V_{k+1},\{V_1,V_2,\dots, V_k\})>{5}\delta\bigr)=0. \] Since $H_{k+1}$ (the height of $V_{k+1}$) is sampled according to the probability measure $\nu$ which satisfies $\nu(\{0\})=0$, we may assume for some $a>0$ that $a0$, \begin{align*} \lim_{k\to \infty} \mathbb{P}\left( d(V_{k+1},\{V_1,V_2,\dots, V_k\})>{5}\delta \,\middle|\, \begin{aligned} H_{k+1} &\in [a,a+\delta],\;\forall 1\leq i \leq k,\\ H_i &\in[a-\delta,a] \end{aligned} \right) =0. \end{align*} Then we write $\probak{\cdot}=\P(\cdot|H_{k+1}\in [a,a+\delta], \forall 1\leq i \leq k, \, \, H_i\in[a-\delta,a])$. Define similarly the associated expectation $\E^k$ for later. And we recall that for all $i\neq j$, the random variable $c(V_i,V_j)$ is the coalescence time between (the genealogies of) $V_i$ and $V_j$, that is the height of their most recent common ancestor. So that by~\eqref{eq lien distance coalescent continu}, to prove the above it suffices to show \begin{equation}\label{eq:desired_limit_leaf_tight} \lim_{k\to \infty} \probak{\forall 1\leq i \leq k, \; c(V_i,V_{k+1})x, \;\forall j\in [k]\backslash \{i\}, \; c\left(V_i,V_j\right)H_1,\dots, H_k$, $V_{k+1}$ has the most chance among $V_1,V_2,\dots, V_{k+1}$ to have coalesced with one of the others. In other words, \[ \probak{E_{k+1,k+1}(a-2\delta)} \leq \frac{1}{k+1}\sum_{i=1}^{k+1} \probak{E_{i,k+1}(a-2\delta)} = \frac{1}{k+1} \E^k\left [N_{k+1}(a-2\delta)\right]. \] Note that the left term above is exactly the probability in~\eqref{eq:desired_limit_leaf_tight}, so to define the limit it suffices to prove: \begin{lemm}s\label{lem:end_lim_leaf-tight} For any choice of $a,\delta>0$ as $k\to \infty$, $\E^k[N_{k+1}(a-2\delta)]=o(k)$. \end{lemm} \begin{proof} For each small interval $[t,t+dt]$, given $N_{k+1}(t+dt)$ {we look} at the expected number of vertices in $V_1,\dots, V_{k+1}$ that have not yet merged at time $t+dt$ that merge with another vertex between time $t$ and $t+dt$. As in the definition of our coalescent, we distinguish two cases: either there is a large merge at time $t$ or there is a small merge between time $t$ and $t+dt$. The second case is easy to count as the merging occurs by pair at rate given by $\rho$. For the first case we look for every $V_r$ not merged at the value of $\Theta_{t,r}$ and check if there is no other $V_{r'}$ not yet merged that {has} the same value. This, together with the fact that $N_{k+1}(s)\ge N_{k+1}(t)$ for all $s\in [t,t+dt]$ {for all $t\le t+dt$ such that $a-2\delta \le t \le t+dt \le a-\delta$}, gives us the next formula: for every $a-2\delta0} N_{k+1}(t)\theta_j(t)\left(1-(1-\theta_j(t))^{N_{k+1}(t)-1}\right) +2\binom{N_{k+1}(t)}{2}\rho(t,t+dt)\right]. \end{align*} Using the fact that for all $x\in (0,1),y>0$ with $xy>1$ we have $(1-x)^y\leq 1/e$, this implies that \begin{align*} &\E^k[N_{k+1}(t+dt)-N_{k+1}(t)]\\ &\geq \E^k\left[\sum_{j\geq 1\,:\, \theta_j(t)>1/(N_{k+1}(t)-1)} \theta_j(t) \left(1-\frac{1}{e} \right){N_{k+1}(t)} \right]+2\E^k\left[\binom{N_{k+1}(t)}{2}\right]\rho(t,t+dt), \end{align*} Then, by integrating between $a-2\delta$ and $a-\delta$, and using $0\leq N_{k+1}\leq k+1$ for the left hand side, we get: \begin{multline*} k+1 \geq \sum_{\substack{ t\in (a-2\delta,a-\delta), j\geq 1}} \theta_j(t) \left(1- \frac{1}{e}\right)\E^k\left [{N_{k+1}(t)}\1_{\theta_j(t)>1/\left(N_{k+1}(t)-1\right)} \right]\\ +\int_{a-2\delta}^{a-\delta} 2\E^k \left [\binom{N_{k+1}(t)}{2}\right]\rho(\rmd t). \end{multline*} Next, by fixing $\e>0$ and considering only the case where $N_{k+1}(t)>\e k$, we obtain \begin{multline*} k+1 \geq \sum_{\substack{ t\in (a-2\delta,a-\delta), j\geq 1}} \theta_j(t) \left(1-\frac{1}{e}\right){\e k}\probak{N_{k+1}(t)>\e k} \1_{\theta_j(t)>1/(\e k-1)}\\ +\int_{a-2\delta}^{a-\delta}(\e k-1)^2 \probak{N_{k+1}(t)>\e k}\rho(\rmd t). \end{multline*} Then, using that for all $t\geq a-2\delta$, we have $N_{k+1}(t)\geq N_{k+1}(a-2\delta)$, we get \begin{multline*} k+1 \geq \probak{N_{k+1}(a-2\delta)>\e k}\\ \times\left (\left(1-\frac{1}{e}\right){\e k} \sum_{\substack{ t\in (a-2\delta,a-\delta), j\geq 1 \\\theta_j(t)>1/(\e k-1) }} \theta_j(t) +(\e k-1)^2\rho([a-2\delta,a-\delta]) \right). \end{multline*} Using the second part of~\ref{hyp:tightGP}, $k$ is negligible compared to the inside of the huge parenthesis above, so \[ \probak{N_{k+1}(a-2\delta)>\e k}\cv[k] 0. \] Since $\e$ is arbitrary and since $N_{k+1}\leq k+1$ this implies the desired result, and so concludes the section. \end{proof} \section{GP convergence} \label{sec:proofGP} \subsection{Coalescent associated with the genealogies of random vertices in \texorpdfstring{$\T^n$}{T\^n}}\label{sous-section coalescent discret} In this section, we introduce a discrete growth-coalescent process, which describes how the genealogies of random vertices in $\T^n$ merge. In the next section, we will prove a scaling limit for this coalescent toward the similar continuous coalescent introduced in \cref{sec: coalescent continu}. At the end of this section, we explain why this implies the scaling limit of the distance matrix of random vertices in $\T^n$, and so Theorem~\ref{th:GP}. Beforehand, let us formally split high degrees and small degrees. \begin{lemm}\label{lemme epsilon n} There exists a sequence $(\e_n)_{n\ge0}$ decreasing to $0$ slowly enough so that for all $n\ge 0$, there exists a bijection~$I_n : \{ t,\ \theta_1(t)>\e_n \text{ and } \e_n \e_n \text{ and } \e_n < i/n < 1-\e_n\}$ such that for all $\e>0$, \begin{equation}\label{eq:cv atomes L2} \sum_{j \ge 1, t\in (\e, 1-\e)} \left( \frac{d^n_{I_n(t), j}}{D^n_{I_n(t)}} - \theta_j(t) \right)^2 \1_{\theta_j(t)>\e_n } \cv[n] 0 \end{equation} and such that for all $t\ge0$ such that $\theta_1(t)>0$, \begin{equation}\label{eq: cv I sur n vers t} I_n(t)/n \cv[n] t. \end{equation} \end{lemm} \begin{proof} We construct the sequence $\e_n$ as follows. For all $k \ge 1$, let $\eta_k \in (2^{-k-1},2^{-k})$ such that $\eta_k \not \in \{\theta_j(t); \ t \in (0,1), j\ge 1\}$. By~\ref{hyp:atomes} and~\ref{hyp:splitatomes}, there exists $n_k\ge 1$ such that for all $n\ge n_k$, there exists a bijection~$I_{k,n}: \{t \in (\eta_k,1-\eta_k), \ \theta_1(t)> \eta_k\} \to \{i, \ d^n_{i,1}/D^n_i > \eta_k \text{ and } \eta_k < i/n < 1-\eta_k \}$ such that for all $t \in (\eta_k, 1-\eta_k)$, for all $j\ge 1$ such that $\theta_j(t)>\eta_k$, \[ \left|\frac{d^n_{I_{k,n}(t),j}}{D^n_{I_{k,n}(t)}} - \theta_j(t) \right| \le \eta_k \quad \text{and} \quad \left|\frac{I_{k,n}(t)}{n} -t \right| \le \eta_k. \] Then it suffices to take $\e_n \coloneqq \min\{\eta_k; \ k\ge 1 \text{ and } n_k \le n\}$ and define $I_n$ as the $I_{k,n}$ where $k$ is the largest integer such that $n_k \le n$. Then~\eqref{eq: cv I sur n vers t} holds clearly. For~\eqref{eq:cv atomes L2}, note that for all $\e>0$, \begin{align*} \sum_{j \ge 1, t\in (\e, 1-\e)} \left( \frac{d^n_{I_n(t), j}}{D^n_{I_n(t)}} - \theta_j(t) \right)^2 \1_{\theta_j(t)>\e_n } &\le \sum_{j \ge 1, t\in (\e, 1-\e)} \e_n^2 \1_{\theta_j(t)>\e_n }\\ &\le \sum_{j \ge 1, t\in (\e, 1-\e)} \e_n^2\wedge \theta_j(t)^2 \cv[n] 0, \end{align*} where the convergence holds by dominated convergence thanks to~\eqref{eq:pas de coalescence avec proba non nulle}. \end{proof} Then, let us prove a direct consequence of~\ref{hyp:splitatomes} which will be very convenient: \begin{lemm}\label{lemme profil tend vers l'infini} We have for all $\e \in (0,1)$, \[ \min_{ \e n \le i \le (1-\e)n} D^n_i \cv[n] \infty. \] \end{lemm} \begin{proof} Let $\e\in (0,1)$. Let us prove that for all $n$ large enough for all $\e n < i < (1-\e) n$, we have $D^n_i \ge 1/ \e$. For all $i \le \h^n -1$, if $d^n_{i,1} \le \e D^n_i$, then, since $d^n_{i,1}\ge 1$, we deduce that $D^n_i \ge 1/\e$. Besides, note that for all $n$ large enough, if $d^n_{i,1} > \e D^n_i$ for some $\e n < i <(1-\e) n$, then by~\ref{hyp:splitatomes} we have $d^n_{i+1, 1} \le \e D^n_{i+1}$. But we also know that $D^n_i d^n_{i+1,1} \ge D^n_{i+1}$ since there are exactly $D^n_i$ vertices of $\T^n$ at height $i+1$, so that \[ D_{i+1}^n \le d^n_{i+1,1} D^n_i \le \e D^n_{i+1} D^n_i. \] Finally, we also get $D^n_i \ge 1/ \e$ by diving both sides by $D_{i+1}^n$ (that can be done since by~\ref{hyp:naissance}, $\liminf \h^n/n \ge 1$ and so $D_{i+1}^n\ge 1$ for all $n$ large enough and for all $i< (1-\e) n $). \end{proof} Now, {in the rest of this subsection, let us fix $k \in \N$}. To prove the convergence of the rescaled matrix of distances between $k$ vertices $V^n_1,\dots, V^n_k$ taken uniformly at random, we introduce the associated growth-coalescent process and prove that it satisfies a scaling limit towards the coalescent process defined in Section~\ref{sec: coalescent continu}. In order to get some additional independence, we add some vertices at each height to the ancestors of $V^n_1,\dots, V^n_k$ in order to have a constant number of vertices at each height {(this will facilitate the computation of the coalescence probabilities)}. For all $i> \max_{r \in [k]} \h(V^n_r)$: \begin{itemize} \item When $D^n_{i-1} \ge k$, let $(V^n_{i,j})_{1\leq j \leq k}$ be distinct vertices in $\{ (i,j); \, j \le D^n_i \}$ taken uniformly at random. \item When $D^n_{i-1} < k$, we set $V^n_{i,r} = (i, 1)$ for all $r \in [k]$. \end{itemize} Then, we define $V^n_{i,1}, \dots, V^n_{i,k}$ for $i \le \max_{r \in [k]} \h(V^n_r)$ by downward induction on $i$. Given $(V^n_{i+1,j})_{1\leq j \leq k}$ define $(V^n_{i,j})_{1\leq j \leq k}$ as follows: \begin{description} \item[Position of $V^n_{1},\dots, V^n_k$] For all $r \in [k]$, if $\h(V^n_r) = i$, then we set $V^n_{i,r}= V^n_r$. \item[Position of the ancestors] For all $p \ge 1$, for all $1 \le r_1 < \dots 0$ for every $n$ large enough. \end{description} We look at the fathers of $V^n_{i+1,1}, \dots,V^n_{i+1,k}$ in $\T^n$. If $V^n_{i+1,q}$ and $V^n_{i+1,r}$ for some $q \e_n D^n_i$ and $\Theta_{i,r}^n=0$ otherwise. We set for all $i,j \ge 1$, \[ S^{n,k}_{i,j} = \left\{ r \in [k], \; \Theta^n_{i,r}=j\right\}. \] We are now in position to define the growth-coalescent process. Let \[ \left(\left(\Part^{n,k}_r(i)\right)_{r \in [k]}\right)_{i\ge 0} \] be the process defined as follows. We set for all $i >\max_{r \in [k]} \h(V^n_r)$, and for all $r \in [k]$, the initial values $ \Part^{n,k}_r(i) = \emptyset$. Then, for all $i\ge 0$, \begin{description} \item[{Birth}] If $i=\h(V^n_r)$ for some $r \in [k]$, then $\Part_r^{n,k}(i)=\{r\}$; \item[{Small merge}] If $X^n_{q,r,i}=1$ for some $1 \le q0$, we have \begin{equation}\label{eq p n i petit} \max_{\e n \le i \le (1-\e)n} p_{n,i} \cv[n] 0 \end{equation} since $\rho$ has no atoms. Another consequence of~\eqref{eq:cv rho epsilon n} is the following lemma: \begin{lemm}\label{lemme Bernoulli Poisson} For all $k\ge 2$, for any integer $n$ large enough, for $i\ge 0$, let $B^n_{i}$ be independent Bernoulli random variables of parameter $p_{n,i}$. Then we have the {vague} convergence of measures on $(0,1)$ in distribution \[ \sum_{1\le i 0$, \[ d_{\TV}\left( \left(X^n_{i,k}\right)_{\e n < i < (1-\e)n},\; \left(B^{n }_{i,k}\right)_{\e n < i < (1-\e)n} \right) \cv[n] 0, \] where $d_{\TV}$ denotes the total variation distance. \end{lemm} \begin{proof} Let $\e>0$. By definition of $X^n_{i,k}$, it is enough to show that there exists a coupling between $X^n_{i,k}$ and $B^n_{i,k}$ such that \begin{equation}\label{eq:somme des probas X different de B} \sum_{\e n \le i \le (1-\e) n } \P\left(X^n_{i,k} \neq B^n_{i,k}\right) \cv[n] 0. \end{equation} Note that by Lemma~\ref{lemme profil tend vers l'infini}, for $n$ large enough, for all $\e n < i < (1- \e) n$, we have $D^n_i \ge k$, so that the vertices $V^n_{i+1,1}, \dots, V^n_{i+1,k}$ are distinct. First of all, we see that \begin{equation}\label{eq:espX} \E \left[X^n_{i,k}\right] = \sum_{j\ge 1} \frac{\binom{d^n_{i,j}}{2} \binom{D^n_i -2}{k-2}}{\binom{D^n_i}{k}}\; \1_{d^n_{i,j} \le \e_n D^n_i}\\ = \binom{k}{2} p_{n,i}. \end{equation} Besides, we have \begin{align} \E\left[(X^n_{i,k})^2 \right] &= \E\left[X^n_{i,k}\right] \label{espcarre1} \\ &\quad + 6 \E\left[ \sum_{1 \le r_1 < r_2 < r_3 \le k}\ \1_{\substack{V^n_{i+1,r_1}, V^n_{i+1,r_2}, V^n_{i+1,r_3}\\ \text{coalesce in a vertex }\\ \text{of degree at most }\e_n D^n_i}} \right] \label{espcarre2} \\ &\quad + \E \left[\sum_{\substack{1\le r_1 < s_1 \le k\\ 1 \le r_2 < s_2 \le k}} \1_{\{r_1,s_1\}\cap \{r_2,s_2\} = \emptyset}\quad \1_{\mkern-20mu\substack{V^n_{i+1,r_1}, V^n_{i+1,s_1} \\ \text{coalesce in a vertex of}\\ \text{degree at most } \e_n D^n_i \\ \text{and }V^n_{i+1,r_2}, V^n_{i+1,s_2}\\ \text{ coalesce in a vertex of}\\ \text{degree at most } \e_n D^n_i}} \right].\label{espcarre3} \end{align} The first term~\eqref{espcarre1} is $\binom{k}{2} p_{n,i}$ by~\eqref{eq:espX}. The second term~\eqref{espcarre2} is \begin{multline}\label{eq: majoration espcarre2} \sum_{j\ge 1} \frac{\binom{d^n_{i,j}}{3} \binom{D^n_i-3}{k-3}}{\binom{D^n_i}{k}}\;\1_{d^n_{i,j} \le \e_n D^n_i} \\ = \1_{D^n_i\ge 3}\sum_{j\ge 1} \frac{d^n_{i,j}-2}{3} \frac{k-2}{D^n_i-2} \frac{\binom{d^n_{i,j}}{2} \binom{D^n_i-2}{k-2}}{\binom{D^n_i}{k}}\; \1_{d^n_{i,j} \le \e_n D^n_i} \le \e_n \frac{k-2}{3} \binom{k}{2} p_{n,i}, \end{multline} while the third term~\eqref{espcarre3} is \begingroup \allowdisplaybreaks \begin{align}\label{eq: majoration espcarre3} &\sum_{\substack{j_1,j_2\ge 1 \\ j_1 \neq j_2}} \1_{d^n_{i,j_1}, d^n_{i,j_2} \le \e_n D^n_i}\frac{\binom{d^n_{i,j_1}}{2}\binom{d^n_{i,j_2}}{2}\binom{D^n_i-4}{k-4}}{\binom{D^n_i}{k}} + 6 \sum_{j\ge 1} \frac{\binom{d^n_{i,j}}{4} \binom{D^n_i - 4}{k-4}}{\binom{D^n_i}{k}} \1_{d^n_{i,j} \le \e_n D^n_i}\\ &=\1_{D^n_i\ge 4} \binom{k}{2}\binom{k-2}{2} \sum_{\substack{j_1,j_2\ge 1 \\ j_1 \neq j_2}} \frac{\binom{d^n_{i,j_1}}{2}}{\binom{D^n_i}{2}} \frac{\binom{d^n_{i,j_2}}{2}}{\binom{D^n_i-2}{2}} \1_{d^n_{i,j_1}, d^n_{i,j_2} \le \e_n D^n_i}\notag\\ &\quad+ 6 \1_{D^n_i\ge 4} \binom{k}{4} \sum_{j\ge 1} \frac{\binom{d^n_{i,j}}{4}}{\binom{D^n_i}{4}} \1_{d^n_{i,j} \le \e_n D^n_i}\notag \\ &\le\1_{D^n_i\ge 4} \frac{D^n_i(D^n_{i}-1)}{(D^n_i-2)(D^n_i-3)} \binom{k}{2}\binom{k-2}{2}%\\ %&\qquad \sum_{\substack{j_1,j_2\ge 1\notag\\ j_1 \neq j_2}} \frac{\binom{d^n_{i,j_1}}{2}}{\binom{D^n_i}{2}} \frac{\binom{d^n_{i,j_2}}{2}}{\binom{D^n_i}{2}} \1_{d^n_{i,j_1}, d^n_{i,j_2} \le \e_n D^n_i}\notag\\ &\quad+ 6 \1_{D^n_i\ge 4} \binom{k}{4} \sum_{j\ge 1} \frac{\binom{d^n_{i,j}}{2}^2}{\binom{D^n_i}{2}^2} \1_{d^n_{i,j} \le \e_n D^n_i} \notag\\ &\le\1_{D^n_i\ge 4}\left(\frac{D^n_i(D^n_{i}-1)}{(D^n_i-2)(D^n_i-3)} \binom{k}{2}\binom{k-2}{2} + 6 \binom{k}{4}\right)\left(\sum_{j\ge 1} \frac{\binom{d^n_{i,j}}{2}}{\binom{D^n_i}{2}} \1_{d^n_{i,j} \le \e_n D^n_i} \right)^2 \notag\\ &\le 12\binom{k}{2}\binom{k-2}{2} (p_{n,i})^2. \notag \end{align} \endgroup For all $n$ large enough, let $B^n_{i,k}$ be a Bernoulli random variable of parameter $\binom{k}{2} p_{n,i}$ such that $B^n_{i,k}\ge \1_{X^n_{i,k}\ge 1}$. One can indeed perform such a coupling given that\linebreak $\P(X^n_{i,k}\ge 1) \le \E[X^n_{i,k}]=\binom{k}{2} p_{n,i} \le 1$ for $n$ large enough. As a result, \begin{multline*} \sum_{\e n \le i \le (1-\e) n } \P\left(X^n_{i,k} \neq B^n_{i,k}\right)\\ \begin{aligned} &\le \sum_{\e n \le i \le (1-\e) n } \left(\P\left(X^n_{i,k} \neq \1_{X^n_{i,k}\ge 1}\right) + \P\left(B^n_{i,k} \neq \1_{X^n_{i,k}\ge 1}\right) \right)\\ &\le \sum_{\e n \le i \le (1-\e) n } \left( \E\left[ \left| X^n_{i,k} - \1_{X^n_{i,k}\ge 1} \right|^2 \right] +\E\left[X^n_{i,k} \right] -\P\left(X^n_{i,k}\ge1\right) \right) \\ &= \sum_{\e n \le i \le (1-\e) n } \left( \E\left[(X^n_{i,k})^2\right] -\E\left[X^n_{i,k} \right] \right) \\ &\le \sum_{\e n \le i \le (1-\e) n } \left( \e_n \frac{k-2}{3} \binom{k}{2} p_{n,i} + 12 \binom{k}{2}\binom{k-2}{2} (p_{n,i})^2 \right) \text{ by~\eqref{eq: majoration espcarre2} and~\eqref{eq: majoration espcarre3},} \\ &\cv[n] 0 \text{ by~\eqref{eq:cv rho epsilon n} and~\eqref{eq p n i petit}}, \end{aligned} \end{multline*} hence~\eqref{eq:somme des probas X different de B} {follows}. \end{proof} Combining Lemmas~\ref{lemme Bernoulli Poisson} and~\ref{lemme distance en variation totale aux Bernoulli} we obtain the scaling limit of the point processes induced by the $X^n_{q,r,i}$'s which are defined in~\eqref{eq indicatrice petite coalescence}. \begin{coro}\label{cor: cv petites coalescences} We have the {vague} convergence of measures on $(0,1)$ in distribution \[ \left(\sum_{i\ge 0} X^n_{q,r,i} \delta_{i/n} \right)_{1 \le q0$ which are defined in Section~\ref{sec: coalescent continu}. Recall also that $S^\Theta(t,j)\cap [k]= \{ r \in [k], \, \Theta_{t,r} = j\} $. By~\ref{hyp:splitatomes}, we may also assume that $\e_n$ decreases slowly enough and that there exists a sequence $(\eta_n)_{n \ge 0}$ such that $n \eta_n \to \infty$ and for all $n \ge 0$, for all $(i,j),(i',j')$ with $\e_n n < i,i' < (1-\e_n) n$ such that $d^n_{i,j} > \e_n D^n_i$ and $d^n_{i',j'} > \e_n D^n_{i'}$, then we have either $i=i'$ or $| i-i' | > \eta_n n$. \begin{lemm}\label{lem: cv grandes coalescences} For all $\e'>0$, we have the {vague} convergence of measures on $(0,1)\times (\N\cup \{0\})^k$ in distribution \[ \sum_{1 < i \le n} \1_{d^n_{i,1}>\e' D^n_i} \delta_{\left(i/n, \left(\Theta^n_{i,r}\right)_{r \in [k]}\right)} \cvloi[n] \sum_{t\ge 0} \1_{\theta_1(t)>\e'}\delta_{\left(t,(\Theta_{t,r})_{r \in [k]}\right)}. \] \end{lemm} \begin{proof} For all $n,i\ge 0$, $j\ge 1$, let $N^n_{i,j,k}$ be the number of vertices among $V^n_{i+1,1}, \dots, V^n_{i+1,k}$ with father $(i,j)$. Note that by Lemma~\ref{lemme profil tend vers l'infini}, for $n$ large enough, for all $\e n < i < (1- \e) n $, we have $D^n_i \ge k$ so that the vertices $V^n_{i+1,1}, \dots, V^n_{i+1,k}$ are distinct. Conditionally on the $N^n_{i,j,k}$'s for $\e n < i < (1- \e) n $ and $j\ge 1$ such that $d^n_{i,j}>\e_n D^n_i$, the $N^n_{i,j,k}$ vertices which coalesce at $(i,j)$ are chosen uniformly at random among $V^n_{i+1,1}, \dots, V^n_{i+1,k}$. More precisely, the sets $S^{n,k}_{i,j}\coloneqq \{ r \in [k], \ \Theta^n_{i,r} = j\}$ for $j \in \{j \ge 1, \ d^n_{i,j} > \e_n D^n_i \}\cup \{0\}$ form a random partition of $[k]$ with prescribed sizes $N^n_{i,j,k}$ which is chosen uniformly at random independently for all $i\ge 1$ and conditionally on the $N^n_{i,j,k}$'s. Moreover, due to the fact that the $\Theta_{t,r}$'s for $r\in [k]$ are i.i.d., the sets $S^\Theta(t,j) \cap [k]= \{r \in [k], \ \Theta_{t,r}=j \}$ for $j\ge 1$ similarly form a random partition of $[k]$ which is chosen uniformly at random among partitions of $[k]$ with prescribed sizes $N_{t,j,k}^\Theta:=\#\{r\in [k], \ \Theta_{t,r}= j \}$'s for $j\ge 1$, conditionally on the $N_{t,j,k}^\Theta$'s. Recall from Lemma~\ref{lemme epsilon n} the definition of $I_n$. The statement of the lemma is thus implied by the following convergence for all $t\ge 0$ such that $\theta_1(t)>0$: \begin{equation}\label{eq:convergence N_IJ} \left(N^n_{I_n(t), j,k} \right)_{j\ge 1} \cvloi[n] \left(\# \left(S^\Theta(t,j) \cap [k]\right)\right)_{j\ge 1}. \end{equation} Let $t\ge 0$ such that $\theta_1(t)>0$. Let $(m_j)_{j\ge 1}$ be a family of non-negative integers such that $\sum_{j\ge 1} m_j \le k$. Then, for $n$ large enough, \begin{multline*} \P\left( \forall j\ge 1, \ N^n_{I_n(t),j,k} = m_j \right) \\ \begin{aligned} &= \left({\binom{D^n_{I_n(t)}-\sum_{j\ge 1} d^n_{I_n(t), j}\1_{\theta_j(t)>\e_n}}{k-\sum_{j\ge 1} m_j}\prod_{j\ge 1} \binom{d^n_{I_n(t),j}}{m_j}}\right){\binom{D^n_{I_n(t)}}{k}}^{-1} \\ &\cv[n] \left( \frac{\theta_0(t)^{k-\sum_{j\ge 1} m_j}}{(k-\sum_{j\ge 1} m_j)!} \prod_{j\ge 1} \frac{\theta_j(t)^{m_j}}{m_j!} \right) {k!}=\P\left( \forall j \ge 1, \; \# \left(S^\Theta(t,j) \cap [k]\right)= m_j \right), \end{aligned} \end{multline*} by the convergence~\eqref{eq:cv atomes L2}. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop: cv coalescent Skorokhod}] Since the processes take their values in a finite set and since the number of jumps of $((\Part^k_r(t))_{r \in [k]})_{t \in [0,1]}$ is finite {thanks to Lemma~\ref{lemme coalescent continu bien defini}}, it suffices to prove the convergence of the times of jumps and of the jumps. Actually, by definition of the two processes, it suffices to prove the joint convergence of the heights of the uniform vertices~\eqref{eq cv hauteurs unif}, of the small coalescences on $(0,1)$ \begin{equation}\label{eq cv petites coalescences} \left(\sum_{ 1\le i < n}X^n_{q,r,i} \delta_{i/n}\right)_{1 \le q0,\quad \sum_{1\le i < n} \1_{d^n_{i,1} \ge \e' D^n_i} \delta_{\left(i/n, \left(\Theta^n_{i,r}\right)_{r\in [k]}\right)}\cvloi[n] \sum_{t \in (0,1)} \1_{\theta_1(t)>\e'} \delta_{\left(t,(\Theta_{t,r})_{r \in [k]}\right)}. \end{equation} The fact that we can restrict our attention to the degrees $d^n_{i,1} \ge \e' D^n_i$ for $\e'$ arbitrarily small instead of $d^n_{i,1} \ge \e_n D^n_i$ comes from the fact that for all $00$, by a union bound, the probability of having a small coalescence at a height $\eta n \le i \le (1-\eta) n$ such that $d^n_{i,1} \ge \e' D^n_i$ is upperbounded by \begin{multline*} \sum_{\eta n \le i\le (1-\eta) n} \1_{d^n_{i,1} > \e' D^n_i} p_{n,i}\\ \le \left(\max_{\eta n \le i \le (1-\eta)n} p_{n,i}\right) \sum_{\eta n \le i\le (1-\eta) n} \frac{1}{(\e')^2} {\left(\frac{d^n_{i,1}}{{D^n_i}}\right)^2} \1_{d^n_{i,1} > \e' D^n_i} \cv[n] 0, \end{multline*} where the convergence comes from \eqref{eq:pas de coalescence avec proba non nulle},~\eqref{eq:cv atomes L2} and~\eqref{eq p n i petit}. Finally, \eqref{eq cv hauteurs unif} comes from \ref{hyp:naissance} and holds jointly with~\eqref{eq cv petites coalescences} and~\eqref{eq cv grandes coalescences} since the conditional law of \[ \left((\Theta^n_{i,r})_{r \in [k], i\ge 0},\;\bigl(X^n_{q,r,i}\bigr)_{1 \le qH,j\leq D_{i-1}^n})$ is independent from $(\fath^n(i,j))_{1\leq i\leq H,j\leq D_{i-1}^n}$. \end{itemize} For $k\in \N$, we call the set $\trail^n_{k}\coloneqq\{X_{i,j}^n\}_{i\leq \h^n, j\leq k'}$ the $k$-trail (associated to $(X_{i,j}^n)_{i,j}$). Note that in $\trail^n_{k}$ there are exactly $\inf(k,D^n_{i-1})$ vertices of height $i$. \begin{lemm}\label{lem:trailconstruct} There exist some random variables $(X^n_{i,j})_{i\leq \h^n,j\in \N}$ satisfying~\ref{cons:height}, \ref{cons:neq}, \ref{cons:fath}, \ref{cons:inde}. \end{lemm} \begin{proof} We construct the sequence by induction on $j$ and downward induction on $i$. For $j=1$ it suffices to take a uniform random vertex at height $\h^n$ and its ancestors. Assume that for every $1\leq i \leq \h^n$ and $j\leq k$, the vertex $X^n_{i,j}$ is defined. Then we let $X_{\h^n,k+1}^n$ be: \begin{itemize} \item if possible, a uniform random vertex of $\T^n$ of height $\h^n$ not in $\{X_{\h^n,j}^n\}_{1\leq j \leq k}$; \item else, any vertex of height $\h^n$ in $\T^n$. \end{itemize} Then for every $i<\h^n$ given $(X_{h,j}^n)_{h\leq \h^n, 1\leq j\leq k}$ and $X_{i+1,k+1}^n$ we define $X_{i,k+1}^n$ as: \begin{itemize} \item $\fath^n(X_{i+1,k+1}^n)$ if it is not in $\{X_{i,j}^n\}_{1\leq j \leq k}$; \item else, if possible, any vertex in $\T^n$ not in $\{X_{i,j}^n\}_{1\leq j \leq k}$ of height $i$; \item else, if not possible, any vertex in $\T^n$ of height $i$. \end{itemize} A quick induction then shows that this sequence is well defined and satisfies the desired properties. \end{proof} The aim of this section is to prove that $k$-trails approximate well $\T^n$. \begin{prop}\label{pro:chain} For any choice of $(X_{i,j}^n)$ we have \[ \lim_{\e \to 0} \lim_{k\to \infty} \limsup_{n\to \infty} \proba{d_H(\trail_k^n,\T^n)>\e n}=0. \] \end{prop} First, we prove that $k$-trails and $k^2$-trails are close. For every $n\in \N$, we write $\|D^n\|_\infty \coloneqq\max_{0\leq i \leq \h^n} D_i^n$ and let $\Gamma_{<\beta n}$ be the set of vertices of $\T^n$ which are at height smaller than $\beta n$. \begin{lemm}\label{lem:chain} For every $0<\alpha<\beta<1$, for every $n\in \N$, $k\leq \|D^n\|_\infty$ large enough, writing $L=\lceil n/(\log\log k)^2\rceil$, with probability at least $1-1/k$, we have \[ \forall x\in \trail_{k^2}^n,\quad \alpha n+9L<\h(x)<\beta n \implies d^n\left(x,\trail_{k}^n \cap \Gamma_{<\beta n}\right)\leq 9 L. \] \end{lemm} \begin{proof} First note that if $d^n(X^n_{i,j},\trail^n_k \cap \Gamma_{\beta n})>9L$ for some $\alpha n +9L 8L$, so by~\ref{cons:fath}, \[ \#\left\{\alpha n+8L8L\right\}\geq L. \] Thus, by Markov's inequality it is enough to prove that for every $\alpha n+8L8L}\leq \frac{1}{k} \frac{L}{nk^2} . \end{equation} Fix such $i,j$. To this end, we can first compute explicitly for every $1\leq a\leq 4L$, \begin{equation}\label{eq:partialprobakfuse} \proba{\fath^n_{a+1}\left(X^n_{i,j}\right) \notin \left\{X^n_{i-a-1,j}\right\}_{1\leq j \leq k} \,\middle|\,\fath^n_{a}\left(X^n_{i,j}\right) \notin \left\{X^n_{i-a,j}\right\}_{1\leq j \leq k}}. \end{equation} Indeed, by~\ref{cons:inde}, given the event $\fath^n_{a}(X^n_{i,j}) \notin \{X^n_{i-a,j}\}_{1\leq j \leq k}$, the fathers of $\fath^n_{a}(X^n_{i,j})$ and of the $X^n_{i-a,j}$'s for ${1\leq j \leq k}$ have the same law as the (unconditioned) father{s} of any $k+1$ different vertices of height exactly $i-a$. In other words, writing $i'=i-a-1$, if $D_{i'}^n>k$ (note that by~\ref{cons:neq} when $D^n_{i-1-a}\leq k$, a.s. $\fath^n_{a+1}(X^n_{i,j})\in \{X^n_{i-a-1,j}\}_{1\leq j \leq k}$) then~\eqref{eq:partialprobakfuse} equals, dividing according to $({i',\ell})$ the value of $\fath^n_{a+1}(X^n_{i,j})$, \[ \sum_{\ell\ge 1} \frac{d_{i',\ell}^n}{D_{i'}^n} \prod_{1\leq b\leq k}\left (1-\frac{d_{i',\ell}^n-1}{D_{i'}^n-b}\right), \] which we may upper-bound, using that $b\geq 1$ and then that $(1-x)^k\leq 1-\min(1,kx)/2$ for all $x \in [0,1]$, by \[ \sum_{\ell\ge 1} \frac{d_{i',\ell}^n}{D_{i'}^n} \left(1-\frac{d_{i',\ell}^n-1}{D_{i'}^n-1}\right)^k \leq \sum_{\ell\ge 1} \frac{d_{i',\ell}^n}{D_{i'}^n}\left (1-\min\left (1,k\frac{d_{i',\ell}^n-1}{D_{i'}^n-1} \right)/2\right). \] The right-hand side is then equal to $1-\tau_{i'}^n(k)/2$ since $\sum_{\ell\ge1} d_{i',\ell}^n=D_{i'}^n$ and by definition of $\tau$ (see~\eqref{eq def tau} in the introduction), which we may again upper-bound by $e^{-\tau_{i'}^n(k)/2}$ since for every $x>0$, we have $(1-x)\leq e^{-x}$. Thus, \begin{multline*} \proba{d^n\left(X^n_{i,j},\trail^n_k\cap \Gamma_{<\beta n}\right)>8L}\\ \begin{aligned} &\leq \proba{\fath^n_{8L}\left(X^n_{i,j}\right)\notin \left\{X^n_{i-8L,j}\right\}_{1\leq j \leq k}}\\ & \le \prod_{a=0}^{8L-1} \proba{\fath^n_{a+1}\left(X^n_{i,j}\right) \notin \{X^n_{i-a-1,j}\}_{1\leq j \leq k} \,\middle|\,\fath^n_{a}\left(X^n_{i,j}\right) \notin \{X^n_{i-a,j}\}_{1\leq j \leq k}}\\ & \leq \prod_{i-8L2/\e$, and such that for every $n\in \N$ large enough, Lemma~\ref{lem:chain} holds with $\alpha=\e/2$, $\beta=1-\e$, and every $K\leq k \leq \|D^n\|_\infty$. Let $N=N(n,K)$ be the smallest integer such that $K^{2^N}\geq \|D^n\|_\infty$. { Note already that by putting $k=\|D^n\|_\infty$ in~\ref{hyp:tightGHP} we get \[ \log(\|D^n\|_\infty) \leq \tau^n_{i,i+n/(\log \log \|D^n\|_\infty)} \leq 1+n/(\log \log\|D^n\|_\infty)=o(n) \] as $n \to \infty$, so $N=o(n)$.} By Lemma~\ref{lem:chain}, for every $n$ large enough with probability at least $1-\sum_{i=0}^{N-1} 1/K^{2^i}\geq 1-2/K$: \begin{multline}\label{eq:middlechain} \forall 0\leq i \leq N-1, \quad \forall x\in \trail_{K^{2^{i+1}}}^n,\\ \e n< \h(x)<(1-\e) n \implies d^n\left(x,\trail_{K^{2^i}}^n \cap \Gamma_{<(1-\e) n}\right)\\ \le 9 \left\lceil n/(\log \log K^{2^i})^2 \right\rceil. \end{multline} Then writing $\Gamma_{\leq \e n}$ for the set of vertices with height at most $\e n$, \eqref{eq:middlechain} implies \begin{multline*} \forall 0\leq i \leq N-1,\\ d_H\left(\left(\Gamma_{\leq \e n}\cup \trail_{K^{2^i}}^n\right)\cap \Gamma_{<(1-\e) n},\, \left(\Gamma_{\leq \e n}\cup \trail_{K^{2^{i+1}}}^n\right)\cap \Gamma_{<(1-\e) n} \right)\\ \leq 9 \left\lceil n/\left(\log \log K^{2^i}\right)^2 \right\rceil, \end{multline*} which by the triangle inequality implies for every $n,K$ large enough, \begin{multline*} d_H \left(\left(\Gamma_{\leq \e n}\cup \trail_K^n\right)\cap \Gamma_{<(1-\e) n}, \, \left(\Gamma_{\leq \e n}\cup \trail_{K^{2^N}}^n\right)\cap \Gamma_{<(1-\e) n} \right)\\ \begin{aligned} &\leq \sum_{i=0}^{N-1} 9 \left\lceil n/\left(\log \log K^{2^i}\right)^2 \right\rceil \\ &\leq N+9n \sum_{i=1}^\infty \frac{1}{(i\log(2)+\log \log K)^2}\\ &\le N+\frac{40 n}{\log \log K}. \end{aligned} \end{multline*} Then note that by~\ref{cons:neq} and since $K^{2^N}\geq \|D^n\|_\infty$, we know that $\trail_{K^{2^N}}^n$ contains all vertices of $\T^n$. Moreover, $(0,1)\in \trail_K^n$ so every $x\in \Gamma_{\leq \e n}$ is at distance at most $\e n$ of $\trail_K^n$. Finally, if $x$ is a vertex of $\T^n$ such that $\h (x) \ge (1-\e)n$ then by considering the highest ancestor $x'$ of $x$ belonging to $\Gamma_{<(1-\e)n}$, we obtain that $d(x,\Gamma_{\le \e n} \cup \trail^n_K) \le 1+ n\e +d(x',\Gamma_{\le \e n} \cup\trail^n_K)$. Hence, the last line implies \[ d_H \left(\trail_K^n, \T^n \right) \leq 1+ 2\e n + N+\frac{40n}{\log \log K}. \] Since $\e$ can be chosen arbitrarily small, $K$ arbitrarily large, and since $N=o(n)$ this concludes the proof of Proposition~\ref{pro:chain}. \end{proof} \subsection{Approximating \texorpdfstring{$k$}{k}-trails from genealogies of random vertices} By Proposition~\ref{B2}, to extend the GP convergence in Theorem~\ref{th:GP} to the GHP convergence in Theorem~\ref{th:GHP} it suffices to prove the following strong leaf tightness criterion: \[ \forall \delta>0,\quad \lim_{k\to \infty} \limsup_{n\to \infty} \mathbb{P}\bigl(d_H(\T^n,\{V^n_1,V^n_2,\dots,V^n_k\})>\delta n \bigr) = 0. \] where $V^n_1,V^n_2,\dots$ are i.i.d. uniform random vertices in $\T^n$. To that end, by Proposition~\ref{pro:chain}, it is enough to prove that \[ \forall \delta>0, \forall K\in \N, \quad \lim_{k\to \infty} \limsup_{n\to \infty} \proba{\max_{x\in \trail_K^n} d^n(x,\{V^n_1,V^n_2,\dots,V^n_k\})>\delta n} = 0. \] Or even, by noting that if there exists an $x\in \trail_K^n$ such that $d^n(x,\{V^n_1,V^n_2,\dots,V^n_k\})>\delta n$ then either: \begin{itemize} \item $\h(x)<(\delta/2)n$ and $\inf_{1\leq i \leq k} \h(V^n_i)\ge (\delta/2) n$. For every $\delta>0$, the probability of this event goes to $0$ as $k\to \infty,n\to \infty$ by~\ref{hyp:naissance} and since $\nu$ has support $[0,1]$. \item Or $\h(x)\ge(\delta/2)n$, so that by~\ref{cons:fath} there are at least $\lfloor (\delta/4) n\rfloor$ different vertices $y\in \trail_k^n$ such that $d^n(y,\{V^n_1,V^n_2,\dots,V^n_k\})\ge \lfloor (\delta/4) n \rfloor$. \end{itemize} As a result, \begin{align*} &\limsup_{k\to \infty} \limsup_{n\to \infty} \ \proba{\max_{x\in \trail_K^n} d^n(x,\{V^n_1,V^n_2,\dots,V^n_k\})>\delta n}\\ &\le \limsup_{k\to \infty} \limsup_{n\to \infty} \mathbb{P}\bigl( \# \left\{y \in \trail_K^n,\: d^n(y,\{V^n_1,V^n_2,\dots,V^n_k\})\ge\lfloor (\delta/4)n \rfloor \right\} \ge \lfloor (\delta/4)n \rfloor\bigr) \\ &\le \limsup_{k\to \infty} \limsup_{n\to \infty} \frac{1}{\lfloor (\delta/4)n \rfloor} \E\bigl[\# \{y \in \trail_K^n, \: d^n(y,\{V^n_1,V^n_2,\dots,V^n_k\})\ge\lfloor (\delta/4)n \rfloor \} \bigr] \\ &\le \limsup_{k\to \infty} \limsup_{n\to \infty} \frac{K \h^n}{\lfloor (\delta/4)n \rfloor} \max_{1\leq a\leq \h^n,1\leq b\leq K} \proba{d^n(X^n_{a,b},\{V^n_1,V^n_2,\dots,V^n_k\})\ge \lfloor (\delta/4) n\rfloor}, \end{align*} where in the third line we applied Markov's inequality and in the last line we used the fact that $\#\trail_K^n \leq K \h^n$. But since $\h^n \sim n$, it is enough to show that \begin{multline*} \forall \delta>0, \forall K>0, \\ \lim_{k\to \infty} \limsup_{n\to \infty} \max_{1\leq a\leq \h^n,1\leq b\leq K} \proba{d^n(X^n_{a,b},\{V^n_1,V^n_2,\dots,V^n_k\})>{5}\delta n} = 0, \end{multline*} or even by considering the different possible values of $X^n_{a,b}$ it suffices to prove the following result: \begin{prop} \label{pro:MiniTightGHP} Under the setting of Theorem~\ref{th:GHP}, for every $\delta>0$, \[ \lim_{k\to \infty} \limsup_{n\to \infty} \max_{1\leq a\leq \h^n,1\leq b\leq D^n_{a-1}} \mathbb{P}\bigl( d^n((a,b),\{V^n_1,V^n_2,\dots,V^n_k\})>{5}\delta n\bigr) = 0. \] \end{prop} \begin{proof} We follow directly the same approach as in Section~\ref{sec:continuum_leaf-tight} which proves an analogous result for the limit tree. Again by~\ref{hyp:naissance} and since $\supp(\nu)=[0,1]$ we may restrict {ourselves} to the case where $a>2\delta n$. And as in Section~\ref{sec:continuum_leaf-tight}, since we may consider a larger but still bounded number of random vertices $V_1^n, V_2^n,\dots, V_k^n$ and since $\nu$ has support $[0,1]$, we can assume them to have height between $a-\delta n$ and $a$. In other words, it suffices to check that for every $\delta>0$, \begin{align*} \limsup_{n\to \infty} \max_{\substack{2\delta n< a\leq \h^n\\1\leq b\leq D^n_{a-1}}} \proba{d^n((a,b),\{V^n_1,\dots,V^n_k\})> {5}\delta n\,\middle|\, \begin{aligned} &\forall 1\leq i \leq k,\\ &\;a-\delta n<\h(V^n_i) {5}\delta n\bigr) \le \frac{\E^k\left[N_k(a-2\delta n)\right]}{k}. \] The proof of Proposition~\ref{pro:MiniTightGHP} is thus complete as long as we prove the following result. \end{proof} \begin{lemm}\label{lemm4.5} For every $\delta>0$, \[ \lim_{k \to \infty} \limsup_{n \to \infty} \max_{2\delta n < a \le \h^n}\frac{\E^k[N_k(a-2\delta n)]}{k}= 0. \] \end{lemm} \begin{proof} The proof is essentially the same as for Lemma~\ref{lem:end_lim_leaf-tight} with slightly different computations. Let $\delta>0$. Let $k\ge 1$, $n\ge 1$ and $2\delta n < a \le \h^n$. Recall the definition of the sequence $(\e_n)$ from Subsection~\ref{sous-section coalescent discret}. For all $a- 2\delta n \le i \le a - \delta n$, we lowerbound \begin{multline*} \E^k \left[N_k(i+1) - N_k(i) \mid N_k(i+1)\right]\\ \begin{aligned} &\ge \sum_{j=1}^\infty \1_{d^n_{i,j}>\e_n D^n_i} N_k(i+1)\frac{d^n_{i,j}}{D^n_i} \left(1 - \prod_{r=1}^{N_k(i+1)-1} \left(1- \frac{d^n_{i,j}-1}{D^n_i-r}\right)\right) \\ &\quad + 2\binom{N_k(i+1)}{2} \sum_{j=1}^\infty \1_{d^n_{i,j} \le \e_n D^n_i} \frac{d^n_{i,j} (d^n_{i,j} -1)}{D^n_i(D^n_i-1)}\\ &\ge \sum_{j=1}^\infty \1_{d^n_{i,j}>\e_n D^n_i} N_k(i+1)\frac{d^n_{i,j}}{D^n_i} \left(1 - \left(1- \frac{d^n_{i,j}-1}{D^n_i}\right)^{N_k(i+1)-1}\right)\\ &\quad + 2\binom{N_k(i+1)}{2} \sum_{j=1}^\infty \1_{d^n_{i,j} \le \e_n D^n_i} \frac{d^n_{i,j} (d^n_{i,j} -1)}{D^n_i(D^n_i-1)}\\ &\ge \sum_{j=1}^\infty \1_{d^n_{i,j}>\e_n D^n_i \vee \left(1+ D^n_i/\left(N_k(i+1)-1\right)\right)} N_k(i+1) \frac{d^n_{i,j}}{D^n_i} \left(1-\frac{1}{e}\right) \\ &\quad + 2\binom{N_k(i+1)}{2} \sum_{j=1}^\infty \1_{d^n_{i,j} \le \e_n D^n_i} \frac{d^n_{i,j} (d^n_{i,j} -1)}{D^n_i(D^n_i-1)}, \end{aligned} \end{multline*} where in the last inequality we used that for all $x\in (0,1),y>0$ such that $xy>1$ we have $(1-x)^y\leq 1/e$. Next, fix $\eta>0$. Considering only the case where $N_k(i+1)\ge \eta k$ and taking the expectation we get \begin{multline*} \E^k \left[N_k(i+1) - N_k(i) \right] \ge \sum_{j=1}^\infty \1_{d^n_{i,j}>\e_n D^n_i \vee \left(1+ D^n_i/(\eta k-1)\right)} \E^k[N_k(i+1)] \frac{d^n_{i,j}}{D^n_i} \left(1-\frac{1}{e}\right) \\+ \E^k\left[2\binom{N_k(i+1)}{2}\right] \sum_{j=1}^\infty \1_{d^n_{i,j} \le \e_n D^n_i} \frac{d^n_{i,j} (d^n_{i,j} -1)}{D^n_i(D^n_i-1)}. \end{multline*} We then sum over $a-2\delta n \le i \le a - \delta n$ and use the fact that $N_k \le k$, which gives \begin{multline*} k\ge \sum_{a-2 \delta n \le i \le a-\delta n}\sum_{j=1}^\infty \1_{d^n_{i,j}>\e_n D^n_i \vee \left(1+ D^n_i/(\eta k-1)\right)} \E^k[N_k(i+1)] \frac{d^n_{i,j}}{D^n_i} \left(1-\frac{1}{e}\right) \\ + \sum_{a-2 \delta n \le i \le a-\delta n}\E^k\left[2\binom{N_k(i+1)}{2}\right] \sum_{j=1}^\infty \1_{d^n_{i,j} \le \e_n D^n_i} \frac{d^n_{i,j} (d^n_{i,j} -1)}{D^n_i(D^n_i-1)}. \end{multline*} But since for all $a-2 \delta n \le i \le a- \delta n$ we have $N_k(i) \ge N_k(a-2\delta n)$, using Markov's inequality we obtain that \begin{multline}\label{eq majoration nombre de gens qui ne coalescent pas} k\ge \P^k(N_k(a- 2 \delta n)\ge \eta k) (\eta k-1)\\ \times\!\!\!\!\!\!\sum_{a-2 \delta n \le i \le a-\delta n} \left(\sum_{j=1}^\infty \1_{d^n_{i,j}>\e_n D^n_i \vee \left(1+ D^n_i/(\eta k-1)\right)} \frac{d^n_{i,j}}{D^n_i} \left(1-\frac{1}{e}\right)\right.\\ +\left. (\eta k-1)\sum_{j=1}^\infty \1_{d^n_{i,j} \le \e_n D^n_i} \frac{d^n_{i,j} (d^n_{i,j} -1)}{D^n_i(D^n_i-1)} \right). \end{multline} Finally, by~\eqref{eq:cv rho epsilon n}, by~\ref{hyp:atomes} and $\h^n\sim n$, we deduce that \begin{multline*} \liminf_{n\to \infty} \min_{2 \delta n < a \le \h^n} \sum_{a-2 \delta n \le i \le a-\delta n}\\ \left(\sum_{j=1}^\infty \1_{d^n_{i,j}>\e_n D^n_i \vee \left(1+ D^n_i/(\eta k-1)\right)} \frac{d^n_{i,j}}{D^n_i} \left(1-\frac{1}{e}\right) %\right. \\ %\left. + (\eta k-1)\sum_{j=1}^\infty \1_{d^n_{i,j} \le \e_n D^n_i} \frac{d^n_{i,j} (d^n_{i,j} -1)}{D^n_i(D^n_i-1)} \right)\\ \ge \min_{1 \le \ell \le \lceil 2/\delta \rceil} \left(\left(1-\frac{1}{e} \right) \sum_{(\ell-1) \delta/2 < t < \ell \delta/2} \;\sum_{j=1}^\infty \1_{\theta_j(t)>1/(\eta k-1)} \theta_j(t)\right.\\ + (\eta k-1) \rho([(\ell-1)\delta/2, \ell \delta/2]) \Biggr)\cv[k] \infty, \end{multline*} where the last convergence stems from~\ref{hyp:tightGP}. Taking~\eqref{eq majoration nombre de gens qui ne coalescent pas} into account, we have thus proven that for all $\eta>0$, \[ \lim_{k\to \infty} \limsup_{n \to \infty} \max_{2 \delta n Z_{i,n}$ or $i \ge n$. We mainly follow the notation of~\cite{BS15} except that here $n$ is the scale of the height (i.e. the time scale for the GWVE process). Let $(\ell_n)_{n\ge 1}$ be a sequence of positive real numbers tending to $\infty$ which will be the scale of the profile (i.e. the space scale). To simplify the notation, we write $\xi_{i,n} = \xi_{i,n}(1)$ for all $i \in [n]$. As in~\cite{BS15}, we define the rescaled quantities for all integers $n\ge 1,i \in [n]$ and for all $t \in [0,1]$, \begin{gather*} \bxi_{i,n} = \frac{1}{\ell_n} (\xi_{i,n} -1), \quad \alpha_{i,n} = \E\left[\frac{\bxi_{i,n}}{1+\bxi_{i,n}^2}\right],\\ \beta_{i,n} = \E\left[\frac{\bxi_{i,n}^2}{1+\bxi_{i,n}^2} \right] \quad \text{and} \quad X_n(t)= \frac{1}{\ell_n} Z_{\lfloor nt \rfloor,n}. \end{gather*} Intuitively, $\alpha_{i,n}$ is a proxy for the drift and $\beta_{i,n}$ plays the role of a ``variance'' of $\bxi_{i,n}$. We then set for all $n\ge 1, t\in [0,1]$, $x>0$, \begin{gather*} \alpha_n(t) = \ell_n \sum_{i={1}}^{\lfloor nt \rfloor} \alpha_{i,n}, \quad \beta_n(t) = \frac{1}{2} \ell_n \sum_{i={1}}^{\lfloor nt \rfloor }\beta_{i,n} \\ \intertext{and} \mu_n([x,\infty)\times(0,t]) = \ell_n \sum_{i={1}}^{\lfloor nt \rfloor } \P(\bxi_{i,n} \ge x). \end{gather*} Next, we use a simpler version of~\cite[Assumption~A]{BS15} in our framework. We assume that there exist a càdlàg function of finite variation $\alpha: [0,1]\to \R$, an increasing càdlàg function $\beta:[0,1] \to \R$ and a positive measure $\mu$ on $(0,\infty)\times (0,1)$ such that \begin{description} \item[\textcolor{link}{\optionaldesc{$(\mathsf{A1})$}{hyp:A1}}] For all $t \in [0,1],x> 0$ such that $\mu(\{x\} \times (0,t])=0$, \begin{multline*} \alpha_n(t)\cv[n] \alpha(t),\quad \lVert \alpha_n \rVert (t) \cv[n] \lVert \alpha \rVert (t), \quad \beta_n(t) \cv[n] \beta(t)\\ \text{and}\quad \mu_n([x,\infty)\times(0,t]) \cv[n] \mu([x,\infty)\times(0,t]), \end{multline*} where $\lVert \cdot \rVert$ is the total variation. \item[\textcolor{link}{\optionaldesc{$(\mathsf{A'2})$}{hyp:A2}}] The functions $\alpha$, $\beta$ are continuous and the measure $\mu((0,\infty) \times \rmd t)$ has no atoms. \end{description} Note that~\ref{hyp:A1} implies the weak convergence of the restriction of $\mu_n$ to $[x, \infty) \times (0,t]$ towards the restriction of $\mu$ to $[x, \infty) \times (0,t]$. Our assumption~\ref{hyp:A2} makes~\cite[Assumption~(A2)]{BS15} void since the functions $\alpha, \beta$ and $t\mapsto \mu([x,\infty)\times (0,t])$ are continuous for all $x>0$ by~\ref{hyp:A2}. We further assume the following condition coming from~\cite[Proposition~2.2]{BS15}: \begin{description} \item[\textcolor{link}{\optionaldesc{$(\mathsf{A3})$}{hyp:nobottleneck}}] For every $C>0$, \[ \liminf_{n\to \infty} \left(\inf_{1 \le i \le n } \E\left[\xi_{i,n} \1_{\xi_{i,n \le C \ell_n}}\right]\right)>0. \] \end{description} Then, by~\cite[Theorem~2.1]{BS15}, we know that there exists a càdlàg process $(X(t))_{t\in [0,1]}$ taking its values in $[0,\infty]$ such that \begin{equation}\label{eq cv GWVE} (X_n(t))_{t \in [0,1]}\cvloi[n](X(t))_{t\in [0,1]} \end{equation} for the Skorokhod J1 topology on the space $\mathbb{D}([0,1], [0,\infty])$, where $[0,\infty]$ is equipped with the distance $d(x,y) = | e^{-x}-e^{-y}|$, and $\infty$ is an absorbing state of the process $(X(t))_{t\in [0,1]}$. As in the first point of~\cite[Theorem~2.1]{BS15}, we define the {(continuous)} function $\widetilde{\beta}$ by setting for all $t \in [0,1]$, \[ \widetilde{\beta}(t) = \beta(t) - \frac{1}{2} \int_{(0,\infty) \times (0,t]} \frac{x^2}{1+x^2} \mu(\rmd x,\rmd t). \] On the event ``$X(1) \in (0,\infty)$'', we define the measures on $(0,1)$ \[ \nu(\rmd t) = \frac{X(t)\rmd t}{\int_0^1 X(s) \rmd s} \quad\text{and}\quad \rho(\rmd t) = \frac{2\widetilde{\beta}(\rmd t)}{X(t)}, \] where $\widetilde{\beta}(\rmd t)$ is the Lebesgue--Stieltjes measure associated with $\widetilde{\beta}$. We also let\linebreak $(\Delta_+X(t))_{t \in (0,1)}$ be the collection of the positive jumps of $X$ in $(0,1)$ and for all $t\in (0,1)$, we set $\theta_1(t) = \Delta_+X(t)/X(t)$ and $\theta_j(t)=0$ for all $j\ge 2$. We set $\Theta = ((\theta_j(t))_{j\ge 1})_{t\in (0,1)}$. By Skorokhod's representation theorem, we may assume that~\eqref{eq cv GWVE} holds almost surely. Then the consequence of Theorems~\ref{th:GP} and~\ref{th:GHP} is the following: \begin{coro}\label{corollaire CWVE} Suppose that $\alpha, \beta, \mu$ satisfy the conditions~\ref{hyp:A1}, \ref{hyp:A2} and \ref{hyp:nobottleneck}. Assume that $\P(\forall t \in (0,1], \ 00$. \begin{itemize} \item If for all $00$, then on the event ``\,$\forall t \in (0,1], \: X(t) \in (0,\infty)$'', the tree $\T_n/n$ converges in distribution to the random metric space $\T(\nu, \rho, \Theta)$ for the GP topology. \item If for all $\delta>0$, for all $00$ which is not an atom of $\mu(\rmd x \times (0,1))$ and for all non-negative continuous function $\varphi$, the convergences of the non-decreasing functions $t \mapsto \lVert \alpha_n \rVert(t), t\mapsto \beta_n(t)$ and $t\mapsto \int_{[x,\infty) \times (0,t]} \varphi(y) \mu_n(\rmd y, \rmd s)$ hold uniformly on $[0,1]$ since the limits are continuous. Before going further, we introduce some more notation. For all $i \in [n]$ and $\e>0$, let $\alpha_{i,n, \e} \coloneqq \E[\1_{\bxi_{i,n} \le \e} \bxi_{i,n}/(1+ \bxi_{i,n}^2) ]$. Note that~\ref{hyp:A1} and~\ref{hyp:A2} imply that \begin{equation}\label{eq alpha i n epsilon tend vers zero} \lim_{n\to \infty} \ell_n \sup_{i \in [n]} \left| \alpha_{i,n,\e} \right| =0 \end{equation} since the convergences of $t\mapsto \lVert \alpha_n \rVert(t)$ and $t\mapsto \int_{[\e,\infty) \times (0,t]} {x}/{(1+x^2)}\mu_n (\rmd x, \rmd s)$ hold uniformly on $[0,1]$. \begin{rema}\label{remarque pas de grands degrés à la meme hauteur} Note that the assumption that $\mu((0,\infty)\times \rmd t)$ has no atoms rules out the case where two high degree vertices are at the same height. If one removes this assumption, then the $\theta_j(t)$'s could not be described using only the process $X$. {Similarly, the continuity of $\alpha$ ensures, through~\eqref{eq alpha i n epsilon tend vers zero}, that the small degree vertices do not contribute to the jumps of $X$ so that the $\theta_j(t)$'s indeed correspond to the positive jumps of $X$.} {We leave the question of obtaining the scaling limit in the case where $\mu((0,\infty)\times \rmd t)$ has atoms or when $\alpha$ is not continuous as an open problem.} \end{rema} \begin{rema} One can present the above assumptions in the equivalent form. One can replace~\ref{hyp:A1} and~\ref{hyp:A2} by the assumptions that~\ref{hyp:A1} holds, that $\widetilde{\beta}$ is continuous, that $\mu((0,\infty)\times \rmd t)$ has no atoms, that~\cite[Assumption~(A2)]{BS15} holds, i.e. for every $t \in [0,1]$ such that $\Delta \alpha(t) \neq 0$ or $\Delta\beta(t) \neq 0$, \[ {\ell_n} \alpha_{\lfloor n t \rfloor, n} \cv[n] \Delta \alpha (t) \quad \text{and} \quad {\ell_n} \beta_{\lfloor n t \rfloor, n}\cv[n] \Delta \beta (t) \enskip \] and that \[ \lim_{\e \to 0} \limsup_{n\to \infty} \ell_n \sup_{i \in [n]} \left| \alpha_{i,n,\e} \right| =0. \] The above convergence can be seen as a ``near-criticality'' condition. Clearly, the above assumptions are implied by~\ref{hyp:A1} and~\ref{hyp:A2}. But they are actually equivalent. Indeed, from the definition of $\widetilde{\beta}$ and the fact that $\mu((0,\infty)\times \rmd t)$ has no atoms, one can see that $\beta$ is continuous. Moreover, the above two displays and the fact that $\mu((0,\infty)\times \rmd t)$ has no atoms, together with~\ref{hyp:A1} imply the continuity of $\alpha$. One could also add a time-change $\gamma_n : [0,1]\to [0,1]$ converging uniformly to $x\mapsto x$ in the definitions of $\alpha_n, \beta_n$ and $\mu_n$ and in the above assumption on jumps: the assumption~\ref{hyp:A2} would still be satisfied. \end{rema} \begin{rema} The assumption~\eqref{eq hypothese GWVE tight GHP} corresponds to the condition~\ref{hyp:tightGHP} and involves the behavior of the process at every scale, not only the one of the scaling limit, that is why we believe it has to be checked separately in the presence of large degrees. \end{rema} Before starting the proof, let us introduce the following martingale with respect to the natural filtration $(\clF^n_i)_{0 \le i \le n}$ associated to the process $((\xi_{i,n}(j))_{j\ge 1})_{i\in [n]}$. For all $\e>0$ and $\delta>0$, for all $0 \le k\le n$, we define \begin{equation}\label{eq martingale} M^{n,\e, \delta}_k \coloneqq \sum_{i=1}^k \sum_{j=1}^{Z_{i-1,n}} \left(\frac{\bxi_{i,n}(j)}{1+ \bxi_{i,n}(j)^2} - \frac{{\alpha}_{i,n,\e}}{\P(\bxi_{i,n} \le \e)} \right) \1_{\bxi_{i,n}(j) \le \e \text{ and } \delta \ell_n \le Z_{i-1,n}\le \ell_n/\delta}. \end{equation} \begin{lemm}\label{lemme martingale} For all $n\ge 1$ and $\e,\delta>0$, the process $(M^{n,\e,\delta}_k)_{0 \le k \le n}$ is a martingale with respect to the filtration $(\clF^n_k)_{0\le k \le n}$. Moreover, for all $0\le s0$ and $\e'\in (0,1)$, \begin{align*} \lim_{\e \to 0} \limsup_{n\to \infty} \P \left(\sup_{\e'n \le i \le (1-\e')n} \left| \sum_{j=1}^{Z_{i-1,n}} \left(\frac{\bxi_{i,n}(j)}{1+ \bxi_{i,n}(j)^2} - \frac{{\alpha}_{i,n,\e}}{\P(\bxi_{i,n} \le \e)} \right) \1_{\bxi_{i,n}(j) \le \e }\right|\right. \ge \eta \\ \text{and }\forall t\in (0,1], \ X(t)\in (0,\infty)\Biggr) =0. \end{align*} \end{lemm} \begin{proof} The fact that $(M^{n,\e,\delta}_k)_{0\le k \le n}$ is a martingale stems from the definition of $\alpha_{i,n,\e}$. Let $0\le s \e} \right] \right). \end{multline*} Moreover, by~\ref{hyp:A1}, \begin{multline*} \limsup_{n\to \infty} \frac{\ell_n}{\delta}\sum_{i=\lfloor ns \rfloor+1}^{\lfloor nt \rfloor} \left( \beta_{i,n} - \E\left[\frac{\bxi_{i,n}^2}{1+ \bxi_{i,n}^2} \1_{\bxi_{i,n}> \e} \right] \right) \\\le \frac{2}{\delta} \left(\beta(t)-\beta(s) - \frac{1}{2}\int_{(\e, \infty) \times [s,t]} \frac{x^2}{1+x^2} \mu(\rmd x, \rmd t) \right). \end{multline*} We conclude the proof of the first point of Lemma~\ref{lemme martingale} by letting $\e\to 0$. For the second statement of the lemma, let $\e'\in (0,1)$. Thanks to~\eqref{eq cv GWVE}, we have \begin{multline*} \lim_{\delta \to 0} \P \bigl(\forall i \in [\e'n, (1-\e')n], \, \delta \ell_n \le Z_{i-1,n} \le \ell_n/ \delta \text{ and } \forall t \in (0,1], \, X(t)\in (0,\infty)\bigr) \\=\P\left(\forall t \in (0,1], \, X(t) \in (0,\infty) \right). \end{multline*} So, it is enough to show that for all $\delta>0, \eta>0$, \begin{multline*} \lim_{\e \to 0} \limsup_{n\to \infty}\\ \P \left(\sup_{i \in [n]} \left| \sum_{j=1}^{Z_{i-1,n}} \left(\frac{\bxi_{i,n}(j)}{1+ \bxi_{i,n}(j)^2} - \frac{{\alpha}_{i,n,\e}}{\P(\bxi_{i,n} \le \e)} \right) \1_{\bxi_{i,n}(j) \le \e \text{ and } \delta\ell_n \le Z_{i-1, n} \le \ell_n/\delta} \right|\ge \eta \right) =0. \end{multline*} Let $0=t_00$, {for all $\e>0$,} if $\delta_n \to 0$, \begin{multline*} \P\left( \begin{multlined} \exists i_1,i_2 \in [n],\;\exists j_1, j_2 \le A \ell_n,\; (i_1,j_1) \neq (i_2, j_2),\\ | i_1 -i_2 | \le n\delta_n \quad\text{and}\quad \xi_{i_1,n}(j_1), \xi_{i_2,n}(j_2)\ge \e \ell_n \end{multlined} \right) \\ \begin{aligned} &\le A^2 \ell_n^2\sum_{i_1=1}^n \P(\xi_{i_1,n} \ge \e \ell_n) \sum_{i_1 - n \delta_n \le i_2 \le i_1 + n \delta_n}\P(\xi_{i_2,n} \ge \e \ell_n)\\ &\le A^2 \mu_n([\e,\infty)\times(0,1]) \max_{1 \le i_1 \le n} \ell_n \sum_{\substack{i_1 - n \delta_n \le i_2 \le i_1 + n \delta_n\\ {i_2 \in [n]}}}\P(\xi_{i_2,n} \ge \e \ell_n), \end{aligned} \end{multline*} and therefore, {for all $\e>0$,} \begin{equation}\label{eq pas deux gros degres a la meme hauteur} \P\left( \begin{multlined} \exists i_1,i_2 \in [n],\;\exists j_1, j_2 \le A \ell_n,\; (i_1,j_1) \neq (i_2, j_2), \\ | i_1 -i_2 | \le n\delta_n \quad\text{and}\quad \xi_{i_1,n}(j_1),\; \xi_{i_2,n}(j_2)\ge \e \ell_n \end{multlined} \right) \cv[n] 0, \end{equation} since $\mu((0,\infty)\times \rmd t)$ has no atoms and since $\mu_n \to \mu$ by~\ref{hyp:A1}. {Moreover, on the event ``$\forall t \in (0,1], \ X(t) \in (0,\infty)$'', since $X$ is càdlàg, for all $\e>0$, there exists a random variable $Y>0$ such that $\forall t \in [\e,1-\e], \ X(t) \in (Y,1/Y)$. So, by~\eqref{eq cv GWVE}, which holds almost surely, we get that with high probability, for all $ t \in [\e,1-\e]$, we have $ \frac{1}{\ell_n} Z_{\lfloor n t\rfloor, n} \in (Y,1/Y)$. Combined with~\eqref{eq pas deux gros degres a la meme hauteur},} this gives directly~\ref{hyp:splitatomes} with high probability. \end{step} \begin{step}[{Proof of~\ref{hyp:atomes}}] Note that by the Poisson paradigm, thanks to the convergence of $\mu_n$ to $\mu$, we have the vague convergence in distribution of the random measure on $(0,1)\times (0,\infty) \times (0,\infty)$ \begin{equation}\label{eq cv PPP degres} \sum_{i \in [n]} \sum_{j\ge 1} \delta_{\left(j/\ell_n, \xi_{i,n}(j)/\ell_n,i/n\right)} \cvloi \sum_{t\ge 0} \1_{(Y(t), \Xi(t)) \neq \partial} \delta_{(Y(t), \Xi(t),t)}, \end{equation} where $(Y(t), \Xi(t),t)_{t\ge 0}$ is a Poisson point process of intensity $\rmd y \mu(\rmd x,\rmd t)$ and where $\partial$ is a cemetery point. After applying Prohorov's theorem and Skorokhod's representation theorem, we may assume that the above convergence holds jointly with~\eqref{eq cv GWVE} along some subsequence which is again denoted using the index $n$. Let $\delta>0$ which is not an atom of $\mu(\rmd x\times (0,1))$ and let $k \in \N$. Let $t_k\in (0,1)$ be the $k^{\rm th}$ time such that $(Y(t_k), \Xi(t_k))\neq \partial$, $Y(t_k) \le X(t_k)$ and that $\Xi(t_k)\ge \delta$. Such a time is well-defined (or does not exist) thanks to the fact that $\mu([\delta,\infty) \times (0,1))<\infty$. By~\eqref{eq cv GWVE}, by~\eqref{eq cv PPP degres} and since $\mu([\delta,\infty) \times (0,1))<\infty$, if we let $(I_{n,k},J_{n,k})$ be the $k^{\rm th}$ pair (in the lexicographic order) such that $J_{n,k} \le Z_{I_{n,k}-1,n}$ and $\xi_{I_{n,k}, n}(J_{n,k}) \ge \delta \ell_n$, then we have the almost sure convergence \begin{equation}\label{eq cv atomes PPP} \left(I_{n,k}/n,\;J_{n,k}/\ell_n,\; \xi_{I_{n,k}, n}(J_{n,k})/\ell_n\right) \cvps \left(t_k, Y({t_k}),\; \Xi(t_k)\right). \end{equation} For all $i \in [n]$ and $\e>0$, let $ \widetilde{\alpha}_{i,n,\e} \coloneqq {\alpha_{i,n,\e}}/{\P(\bxi_{i,n} \le \e)}$. Note that for all $\e\in (0,\delta)$, \begin{align} \frac{Z_{I_{n,k},n}-Z_{I_{n,k}-1, n}}{\ell_n} = &\sum_{j=1}^{Z_{I_{n,k}-1, n}} \bxi_{I_{n,k}, n}(j) \1_{\bxi_{I_{n,k},n}(j) >\e}\label{premiere ligne increment Z} \\ &+ \sum_{j=1}^{Z_{I_{n,k}-1, n}} \left(\bxi_{I_{n,k},n}(j) -\widetilde{\alpha}_{I_{n,k},n,\e} \right)\1_{\bxi_{I_{n,k},n}(j) \le \e}\label{deuxieme ligne increment Z} \\ &+ \widetilde{\alpha}_{I_{n,k}, n, \e} \# \left\{1 \le j\le Z_{I_{n,k}-1, n}; \ \bxi_{I_{n,k},n}(j) \le \e\right\}.\label{troisieme ligne increment Z} \end{align} By~\eqref{eq cv PPP degres}, by~\eqref{eq cv atomes PPP} and since $\mu((0,\infty)\times \rmd t)$ has no atoms, almost surely, the sum on the first line~\eqref{premiere ligne increment Z} has only one non-zero term when $n$ is large enough: \[ \sum_{j=1}^{Z_{I_{n,k}-1, n}} \bxi_{I_{n,k}, n}(j) \1_{\bxi_{I_{n,k},n}(j) >\e} = \bxi_{I_{n,k}, n}(J_{n,k}) \cvps \Xi(t_k). \] Furthermore, by Lemma~\ref{lemme martingale}, for all $\eta>0$, \begin{multline}\label{eq cv xi sur un plus xi carre petit saut} \lim_{\e\to 0} \limsup_{n\to \infty} \P \left( \left|\sum_{j=1}^{Z_{I_{n,k}-1, n}} \left(\frac{\bxi_{I_{n,k},n}(j)}{1+\bxi_{I_{n,k},n}(j)^2} -\widetilde{\alpha}_{I_{n,k},n,\e} \right)\1_{\bxi_{I_{n,k},n}(j) \le \e}\right|\right. \ge \eta\\ \text{ and } \forall t \in (0,1], \; X(t) \in (0,\infty) \Biggr) =0. \end{multline} But note that for all $\delta>0$, on the event that $Z_{I_{n,k}-1,n} \le \ell_n / \delta$, we have \begin{align*} \left| \sum_{j=1}^{Z_{I_{n,k}-1, n}} \frac{\bxi_{I_{n,k},n}(j)^3}{1+\bxi_{I_{n,k},n}(j)^2} \1_{\bxi_{I_{n,k},n}(j) \le \e} \right| &\le \e \sum_{j=1}^{Z_{I_{n,k}-1, n}} \frac{\bxi_{I_{n,k},n}(j)^2}{1+\bxi_{I_{n,k},n}(j)^2} \1_{\bxi_{I_{n,k},n}(j) \le \e} \\ &\le \e \sum_{i=1}^n \sum_{j=1}^{\lfloor \ell_n / \delta \rfloor} \frac{\bxi_{i,n}(j)^2}{1+ \bxi_{i,n}(j)^2}, \end{align*} and that by~\ref{hyp:A1}, the expectation of the right-hand side converges towards $\e\beta(1)/\delta$, which goes to zero as $\e \to 0$. Thus, for all $\eta>0$, \begin{multline*} \lim_{\e\to 0}\limsup_{n\to \infty} \P \left(\left| \sum_{j=1}^{Z_{I_{n,k}-1, n}} \frac{\bxi_{I_{n,k},n}(j)^3}{1+\bxi_{I_{n,k},n}(j)^2} \1_{\bxi_{I_{n,k},n}(j) \le \e} \right|\right.\ge \eta\\ \text{ and }\forall t \in (0,1], \ X(t) \in (0,\infty) \Biggr)=0. \end{multline*} By taking~\eqref{eq cv xi sur un plus xi carre petit saut} into account, using that $x= x/(1+x^2)+ x^3/(1+x^2)$, we obtain that on the event that $\forall t \in (0,1], \ X(t) \in (0,\infty)$, the second sum~\eqref{deuxieme ligne increment Z} converges in probability as $n\to \infty$ and then $\e\to 0$ towards $0$. In other words, for all $\eta>0$, \begin{multline*} \lim_{\e\to 0} \limsup_{n\to \infty} \P\left( \left|\sum_{j=1}^{Z_{I_{n,k}-1, n}} \left(\bxi_{I_{n,k},n}(j) -\widetilde{\alpha}_{I_{n,k},n,\e} \right)\1_{\bxi_{I_{n,k},n}(j) \le \e}\right|\right. \ge \eta \\\text{ and } \forall t \in (0,1], \ X(t) \in (0,\infty) \Biggr) =0. \end{multline*} For~\eqref{troisieme ligne increment Z}, notice that on the event that $\forall t \in (0,1], \ X(t) \in (0,\infty)$, by~\eqref{eq cv GWVE} we have almost surely \[ \widetilde{\alpha}_{I_{n,k}, n, \e} \# \left\{1 \le j\le Z_{I_{n,k}-1, n}; \; \bxi_{I_{n,k},n}(j) \le \e\right\}= \widetilde{\alpha}_{I_{n,k}, n, \e} O(\ell_n). \] But, by~\eqref{eq alpha i n epsilon tend vers zero}, we know that $ \ell_n\alpha_{I_{n,k}, n,\e} \to 0$ a.s.\@ as $n\to \infty$. Moreover, by~\ref{hyp:A1}, \[ \sup_{i\in [n]} \left(1-\P(\bxi_{i,n} \le \e) \right) = \sup_{i\in [n]} \P(\bxi_{i,n} >\e) \le \sum_{i=1}^n \P(\bxi_{i,n} >\e) = O\left(\frac{1}{\ell_n} \right). \] \pagebreak{} \noindent Therefore, almost surely, $\ell_n \widetilde{\alpha}_{I_{n,k}, n, \e} \to 0$ as $n\to \infty$. This proves that~\eqref{troisieme ligne increment Z} goes to zero almost surely as $n\to \infty$ and then $\e \to 0$ on the event that $\forall t \in (0,1], \ X(t) \in (0,\infty)$. As a result, for all $\eta>0$, \begin{multline*} \lim_{\e \to 0} \limsup_{n\to \infty} \P \left(\left| \frac{Z_{I_{n,k}, n} - Z_{I_{n,k}-1, n}}{\ell_n} - \bxi_{I_{n,k},n}(J_{n,k}) \right |\right. \ge \eta \\ \text{ and } \forall t \in (0,1], \ X(t) \in (0,\infty)\Biggr) =0. \end{multline*} Since the above probability does not depend on $\e$, we deduce that on the event that $\forall t \in (0,1], \ X(t) \in (0,\infty)$, \begin{equation}\label{eq cv proba grand degre} \left| \frac{Z_{I_{n,k}, n} - Z_{I_{n,k}-1, n}}{\ell_n} - \bxi_{I_{n,k},n}(J_{n,k}) \right | \cvproba 0. \end{equation} So, by taking~\eqref{eq cv GWVE} into account, the high degrees correspond to positive jumps of~$X$. Conversely, let $t'_k\in (0,1)$ be the $k^{\rm th}$ positive jump of $X$ of size larger than $\delta$. Let $I'_{n,k}$ be the $k^{\rm th}$ time $i \in [n]$ such that $(Z_{I'_{n,k},n} - Z_{I'_{n,k}-1,n})/\ell_n >\delta$. By~\eqref{eq cv GWVE}, on the event that $X$ has no jump of size exactly $\delta$, we have the convergence \[ \left(\frac{I'_{n,k}}{n},\; \frac{Z_{I'_{n,k}, n}- Z_{I'_{n,k}-1, n}}{\ell_n}\right) \cvps \left(t'_k, \Delta_+ X(t'_k) \right). \] Then, we can reason as above: for all $\e>0$, one can write \begin{multline*} \frac{Z_{I'_{n,k},n}-Z_{I'_{n,k}-1, n}}{\ell_n} = \sum_{j=1}^{Z_{I'_{n,k}-1, n}} \bxi_{I'_{n,k}, n}(j) \1_{\bxi_{I'_{n,k},n}(j) >\e}\\ \begin{aligned} &+ \sum_{j=1}^{Z_{I'_{n,k}-1, n}} \left(\bxi_{I'_{n,k},n}(j) -\widetilde{\alpha}_{I'_{n,k},n,\e} \right)\1_{\bxi_{I'_{n,k},n}(j) \le \e} \\ &+ \widetilde{\alpha}_{I'_{n,k}, n, \e} \# \left\{1 \le j\le Z_{I'_{n,k}-1, n}; \ \bxi_{I'_{n,k},n}(j) \le \e\right\}. \end{aligned} \end{multline*} For the same reasons as above the second and third line converge to zero in probability as $n\to \infty$ and then $\e \to 0$ on the event that $\forall t \in (0,1], \ X(t) \in (0,\infty)$. Moreover, by~\eqref{eq pas deux gros degres a la meme hauteur} we deduce that on the event that $\forall t \in (0,1], \ X(t) \in (0,\infty)$ and that $X$ has no jump of size exactly $\delta$, almost surely, for all $n$ large enough, the first sum has only one non-zero term: \[ \sum_{j=1}^{Z_{I'_{n,k}-1, n}} \bxi_{I'_{n,k}, n}(j) \1_{\bxi_{I'_{n,k},n}(j) >\e} = \bxi_{I'_{n,k}, n}(J'_{n,k}) \] for some $J'_{n,k} \le Z_{I'_{n,k}-1, n}$. This proves that all the jumps of $X$ correspond to high degrees, and that for all $\delta>0$ which is not an atom of $\mu(\rmd x\times (0,1))$, almost surely $X$ does not have a jump of size $\delta$. In particular, by taking~\eqref{eq cv atomes PPP} and~\eqref{eq cv proba grand degre} into account, we deduce that~\ref{hyp:atomes} holds in probability on the event that $\forall t \in (0,1], \linebreak X(t) \in (0,\infty)$. \end{step} \begin{step}[{Proof of the assumption~\eqref{eq:pas de coalescence avec proba non nulle} for $\Theta$}] In other words, we will prove that for all $00$. On the event {that $\forall t \in [a-\delta,b+\delta], \ \delta< X_n(t) < 1/\delta$,} we have \begin{align*} \sum_{i={\lfloor an \rfloor}}^{{\lfloor bn \rfloor}} \sum_{j=1}^{Z_{i-1, n}} \left(\frac{\xi_{i,n}(j){-1}}{Z_{i,n}}\right)^2 \1_{\xi_{i,n}(j)/ Z_{i,n} > \e} &\le\sum_{i=\lfloor an\rfloor}^{\lfloor bn \rfloor} \sum_{j=1}^{\ell_n/\delta} \left(\frac{\xi_{i,n}(j){-1}}{Z_{i,n}}\right)^2 \1_{\xi_{i,n}(j)/ Z_{i,n} > \e} \\ &\le \frac{1}{\delta^2} \sum_{i=1}^n \sum_{j=1}^{\ell_n/\delta} \left(\bxi_{i,n}(j)^2 \wedge 1\right)\1_{\bxi_{i,n}(j) > \e \delta}. \end{align*} Furthermore, using~\ref{hyp:A1} which entails that $\mu_n$ converges {vaguely} towards $\mu$, we obtain that \begin{multline*} {\limsup_{n\to \infty}} \ \E \left[\sum_{i=1}^n \sum_{j=1}^{\ell_n/\delta} \left(\bxi_{i,n}(j)^2 \wedge 1\right)\1_{\bxi_{i,n}(j) > \e \delta}\right]\\ \begin{aligned} &=\limsup_{n\to \infty}\int_{(0,\infty) \times (0,1]} \frac{1}{\delta} (x^2 \wedge 1) \1_{x> \e \delta} \mu_n(\rmd x,\rmd t) \\ &\le \frac{1}{\delta} \int_{(0,\infty) \times(0,1]} (x^2 \wedge 1) \1_{x\ge \e \delta} \mu(\rmd x,\rmd t). \end{aligned} \end{multline*} Moreover, by~\eqref{eq cv GWVE} and~\eqref{eq cv proba grand degre}, when $\mu(\{\varepsilon\} \times(0,1))=0$, \begin{multline*} \sum_{i=\lfloor an \rfloor}^{\lfloor bn \rfloor} \sum_{j=1}^{Z_{i-1, n}} \left(\frac{\xi_{i,n}(j){-1}}{Z_{i,n}}\right)^2 \1_{\substack{\xi_{i,n}(j)/ \ell_n \ge \e \text{ and}\\ \forall t \in [a-\delta,b+\delta], \\ \delta < X_n(t)< 1/\delta}}\\ \cvproba \!\! \sum_{t \in [a,b]} \left(\frac{\Delta_+X(t)}{X(t)} \right)^2 \1_{\substack{\Delta_+ X(t) > \e \\\text{ and } \forall t \in [a-\delta,b+\delta], \\ \delta0$ such that $\mu(\{\varepsilon\} \times(0,1))=0$, \begin{multline*} \E\left[\1_{\forall t \in [a{-\delta},b{+\delta}], \ \delta 0$, {with $\delta$ small enough that $00$ which is not an atom of $\mu(\rmd x \times (0,1))$. As a consequence, by~\eqref{eq cv GWVE},\pagebreak{} since $\widetilde{\beta}_\e$ converges vaguely on $((0,1),\clB((0,1)))$ towards $\widetilde{\beta}$ as $\e \to 0$ and since $\widetilde{\beta}(\rmd t)$ has no atoms, \begin{multline*} \sum_{i=1}^n f(i/n) Z_{i-1,n} \frac{\widetilde{\beta}_{i,n,\e}}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2}\\ = \int_0^1 f\left(t\right) \frac{Z_{\lfloor nt \rfloor-1,n}}{\ell_n} \frac{2\widetilde{\beta}_{n,\e}(\rmd t)}{2\binom{Z_{\lfloor nt \rfloor{-1},n}}{2}/\ell_n^2} \xrightarrow[n\to \infty, \e \to 0]{\text{a.s.}}\int_0^1 f(t) \frac{2\widetilde{\beta} (\rmd t)}{X(t)}, \end{multline*} in the sense that the limit as $\e \to 0$ of the limsup as $n\to \infty$ of the difference is a.s. zero. Thus, we deduce that, on the event ``$\forall t \in (0,1], \ X(t) \in (0, \infty)$'', \begin{equation}\label{eq cv petits sauts1} \sum_{i=1}^n f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{\bxi_{i,n}(j)^2}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2} \1_{\bxi_{i,n}(j)<\e} \xrightarrow[n\to \infty, \e \to 0]{\P} \int_0^1 f(t) \frac{2\widetilde{\beta} (\rmd t)}{X(t)}, \end{equation} in the sense that for all $\eta>0$, the limsup as $n\to \infty$ of the probability that the difference is larger in absolute value than $\eta$ goes to zero as $\e \to 0$. But we want the above convergence with a $Z_{i,n}$ instead of $Z_{i-1,n}$. To that aim, let us write \begin{equation}\label{eq def rho n epsilon} \widetilde{\rho}_{n,\e}(\rmd t)\coloneqq \sum_{i=1}^n \delta_{i/n}(\rmd t) \sum_{j=1}^{Z_{i-1,n}} \frac{\bxi_{i,n}(j)^2}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2} \1_{\bxi_{i,n}(j)<\e}. \end{equation} Then~\eqref{eq cv petits sauts1} implies the vague convergence in probability on $(0,1)$ of $\widetilde{\rho}_{n,\e}(\rmd t)$ towards $2 \widetilde{\beta}(\rmd t)/ X(t)$ as $n\to \infty$ and then $\e \to 0$. Let $ g_n(t) \coloneqq f(t) {\binom{Z_{\lfloor nt\rfloor-1, n}}{2}}/{\binom{Z_{\lfloor n t\rfloor, n}}{2}} $. On the event ``$\forall t \in (0,1], \ X(t) \in (0, \infty)$'', the sequence of random functions $g_n$ with compact support $[a,b]$ converges a.s. for the $J_1$ topology of Skorokhod towards $g(t) \coloneqq f(t) {X(t-)^2}/{X(t)^2}$. So, by Lemma~\ref{lemme technique cv skorokhod}, \begin{equation}\label{eq cv integrale gn rho n epsilon} \begin{aligned} \int_0^1 g_n(t) \widetilde{\rho}_{n,\e}(\rmd t) \xrightarrow[n\to \infty, \e \to 0]{\P} \int_0^1 g(t) \frac{2 \widetilde{\beta}(\rmd t)}{X(t)} &= \int_0^1 f(t) \frac{X(t-)^2}{X(t)^2}\frac{2 \widetilde{\beta}(\rmd t)}{X(t)}\\ &= \int_0^1 f(t) \frac{2 \widetilde{\beta}(\rmd t)}{X(t)}, \end{aligned} \end{equation} where in the last equality, we use the facts that $\widetilde{\beta}$ has no atoms and the fact that for every $t$ except for a random countable subset we have $X(t-)=X(t)$. In other words, \begin{equation}\label{eq cv petits sauts} \sum_{i=1}^n f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{\bxi_{i,n}(j)^2}{2\binom{Z_{i,n}}{2}/\ell_n^2} \1_{\bxi_{i,n}(j)<\e} \xrightarrow[n\to \infty, \e \to 0]{\P} \int_0^1 f(t) \frac{2\widetilde{\beta} (\rmd t)}{X(t)}. \end{equation} Let us then show that~\eqref{ligne 3 petits sauts} is negligible, i.e. that \begin{equation}\label{eq reste tend vers zero} \sum_{i =1}^{n} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)/\ell_n}{2\binom{Z_{i,n}}{2}/\ell_n^2}\1_{\bxi_{i,n}(j)<\e} \xrightarrow[n\to \infty, \e \to 0]{\P} 0. \end{equation} In order to show~\eqref{eq reste tend vers zero}, let us first show that \begin{equation}\label{eq reste tend vers zero2} \sum_{i =1}^{n} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)/\ell_n}{2\binom{Z_{i-1,n}}{2}/\ell_n^2}\1_{\bxi_{i,n}(j)<\e} \cvproba[n] 0. \end{equation} Using that $(1+1/\ell_n^2) \bxi_{i,n}(j) \le (1+\bxi_{i,n}(j)^2) \bxi_{i,n}(j)$ and that for all $x\le \e$, we have $x\le x/(1+x^2)+ \e x^2/(1+x^2)$, we bound \begin{multline}\label{encadrement somme petits sauts} \frac{1+1/\ell_n^2}{\ell_n}\sum_{i =1}^{n} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)/\left(1+\bxi_{i,n}(j)^2\right)}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2}\;\1_{\bxi_{i,n}(j)<\e} \\ \begin{aligned} &\le \sum_{i =1}^{n} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)/\ell_n}{2\binom{Z_{i-1,n}}{2}/\ell_n^2}\;\1_{\bxi_{i,n}(j)<\e} \\ &\le \frac{1}{\ell_n}\sum_{i =1}^{n} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)/\left(1+\bxi_{i,n}(j)^2\right)}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2}\;\1_{\bxi_{i,n}(j)<\e} \\ &\quad + \frac{\e}{\ell_n}\sum_{i =1}^{n} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)^2/\left(1+\bxi_{i,n}(j)^2\right)}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2}\;\1_{\bxi_{i,n}(j)<\e}. \end{aligned} \end{multline} By~\eqref{eq cv petits sauts1} and since $\ell_n \to \infty$, we know that the above last term on the right goes to zero as $n\to \infty$ in probability. Moreover, \begin{multline*} \left|\E\left[\frac{1}{\ell_n}\sum_{i =1}^{n} \1_{\delta \ell_n \le Z_{i-1,n}\le \ell_n/\delta} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)/\left(1+\bxi_{i,n}(j)^2\right)}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2} \right]\right| \\ \begin{aligned} &=\left| \E\left[\frac{1}{\ell_n}\sum_{i =1}^{n} { \1_{\delta \ell_n \le Z_{i-1,n}\le \ell_n/\delta} }f(i/n) {Z_{i-1,n}} \frac{ \alpha_{i,n} }{2\binom{Z_{i{-1},n}}{2}/\ell_n^2}\right] \right|\\ & \le \E\left[\frac{1}{\ell_n}\sum_{i =1}^{n} { \1_{\delta \ell_n \le Z_{i-1,n}\le \ell_n/\delta} }f(i/n) {Z_{i-1,n}} \frac{ | \alpha_{i,n} | }{2\binom{Z_{i{-1},n}}{2}/\ell_n^2}\right] \\ &\le\frac{1}{\ell_n} \max(f) \frac{1}{\delta} \frac{\lVert \alpha_n \rVert(1)}{\delta^2} \cv[n] 0, \end{aligned} \end{multline*} where the last inequality comes from the fact that $f$ has compact support $[a,b]$ where the convergence stems from~\ref{hyp:A1} and from the fact that $\ell_n \to \infty$. Furthermore, we bound from above the variance as follows: \begingroup \allowdisplaybreaks \begin{align*} &\E\left[\left(\frac{1}{\ell_n}\sum_{i=1}^n f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{\bxi_{i,n}(j)/(1+ \bxi_{i,n}(j)^2) - \alpha_{i,n}}{2 \binom{Z_{i-1,n}}{2}/ \ell_n^2} \1_{\delta \ell_n \le Z_{i-1,n} \le \ell_n/ \delta}\right)^2 \right]\\ &\quad\le \frac{1}{\ell_n^2} \E\left[\sum_{i=1}^n f(i/n)^2 \sum_{j=1}^{Z_{i-1,n}} \frac{\left(\bxi_{i,n}(j)/\left(1+ \bxi_{i,n}(j)^2\right) - \alpha_{i,n}\right)^2}{4 \binom{Z_{i-1,n}}{2}^2/ \ell_n^4} \1_{\delta \ell_n \le Z_{i-1,n} \le \ell_n/ \delta} \right]\\ &\quad\le \frac{1}{\ell_n^2} \max(f)^2\E\left[\sum_{i=1}^n \sum_{j=1}^{Z_{i-1,n}} \frac{\bxi_{i,n}(j)^2/\left(1+ \bxi_{i,n}(j)^2\right)^2}{4 \binom{Z_{i-1,n}}{2}^2/ \ell_n^4} \1_{\delta \ell_n \le Z_{i-1,n} \le \ell_n/ \delta} \right]\\ &\quad\le \frac{1}{\ell_n^2} \max(f)^2\E\left[\sum_{i=1}^n \sum_{j=1}^{Z_{i-1,n}} \frac{\bxi_{i,n}(j)^2/\left(1+ \bxi_{i,n}(j)^2\right)}{4 \binom{Z_{i-1,n}}{2}^2/ \ell_n^4} \1_{\delta \ell_n \le Z_{i-1,n} \le \ell_n/ \delta} \right]\\ &\quad\le \frac{1}{\ell_n^2} \max(f)^2 \E\left[\sum_{i=1}^n \frac{\ell_n}{\delta} \frac{\beta_{i,n}}{\delta^4}\right]\\ &\quad=O\left(\frac{1}{\ell_n^2} \right). \end{align*} \endgroup Therefore, by letting $\delta\to 0$, on the event ``$\forall t \in (0,1], \ X(t) \in (0,\infty)$'', \[ \frac{1}{\ell_n}\sum_{i =1}^{n} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)/(1+\bxi_{i,n}(j)^2)}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2} \cvproba 0. \] Besides, by~\eqref{eq cv GWVE} and~\eqref{eq cv proba grand degre}, we know that for all $\e>0$ which is not the size of a positive jump of $X$: \begin{multline*} \sum_{i =1}^{n} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)/\left(1+\bxi_{i,n}(j)^2\right)}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2} \1_{\bxi_{i,n}(j)\ge\e}\\ \cvproba \sum_{t \in (0,1)} f(t) \frac{\Delta_+X(t)/\left(1+\Delta_+X(t)^2\right)}{X(t)^2} \1_{\Delta_+ X(t) \ge \e}, \end{multline*} so that \[ \frac{1}{\ell_n}\sum_{i =1}^{n} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)/\left(1+\bxi_{i,n}(j)^2\right)}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2} \1_{\bxi_{i,n}(j)\ge\e} \cvproba 0. \] Thus, \[ \frac{1}{\ell_n}\sum_{i =1}^{n} f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{ \bxi_{i,n}(j)/\left(1+\bxi_{i,n}(j)^2\right)}{2\binom{Z_{i{-1},n}}{2}/\ell_n^2} \1_{\bxi_{i,n}(j) < \e} \cvproba 0. \] Thanks to~\eqref{encadrement somme petits sauts}, we deduce~\eqref{eq reste tend vers zero2}. Next, we need to check that~\eqref{eq reste tend vers zero2} implies~\eqref{eq reste tend vers zero}. Note that since $| \bxi_{i,n}(j)/\ell_n | \le \bxi_{i,n}(j)^2$, we have \begin{multline*} \left|\sum_{i=1}^n f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{\bxi_{i,n}(j)}{\ell_n} \1_{\bxi_{i,n}(j)<\e} \left(\frac{1}{2 \binom{Z_{i-1,n}}{2}/\ell_n^2}- \frac{1}{2 \binom{Z_{i,n}}{2}/\ell_n^2}\right) \right|\\ \begin{aligned} &\le \sum_{i=1}^n f(i/n) \sum_{j=1}^{Z_{i-1,n}} \bxi_{i,n}(j)^2 \1_{\bxi_{i,n}(j)<\e} \left| \frac{1}{2 \binom{Z_{i-1,n}}{2}/\ell_n^2}- \frac{1}{2 \binom{Z_{i,n}}{2}/\ell_n^2}\right|\\ &= \int_0^1 r_n(t) \widetilde{\rho}_{n,\e}(\rmd t), \end{aligned} \end{multline*} where we recall that $\widetilde{\rho}_{n,\e}$ is defined in~\eqref{eq def rho n epsilon} and we set for all $t \in [0,1]$, \[ r_n(t)\coloneqq f(t) \left| 1- \frac{\binom{Z_{\lfloor nt \rfloor -1, n}}{2}} {\binom{Z_{\lfloor nt \rfloor, n}}{2}}\right|. \] Then, by the same reasoning as in~\eqref{eq cv integrale gn rho n epsilon}, we obtain the convergence \[ \int_0^1 r_n(t) \widetilde{\rho}_{n,\e} (\rmd t) \xrightarrow[n\to \infty, \e \to 0]{\P} 0. \] Hence, \[ \left|\sum_{i=1}^n f(i/n) \sum_{j=1}^{Z_{i-1,n}} \frac{\bxi_{i,n}(j)}{\ell_n} \1_{\bxi_{i,n}(j)<\e} \left(\frac{1}{2 \binom{Z_{i-1,n}}{2}/\ell_n^2}- \frac{1}{2 \binom{Z_{i,n}}{2}/\ell_n^2}\right) \right| \xrightarrow[n\to \infty, \e \to 0]{\P} 0. \] Thus, we have shown~\eqref{eq reste tend vers zero}. Combining~\eqref{eq coalescence grands sauts}, \eqref{eq cv petits sauts}, \eqref{eq reste tend vers zero}, we conclude that on the event ``$\forall t \in (0,1], \ X(t) \in (0,\infty)$'', \[ \sum_{i =1}^{n} f(i/n)\sum_{j=1}^{Z_{i-1,n}} \frac{\binom{\xi_{i,n}(j)}{2}}{\binom{Z_{i,n}}{2}} \cvproba[n]\sum_{t\in (0,1)} f(t) \frac{\Delta_+ X(t)^2}{X(t)^2} + \int_0^1 f(t) \frac{2 \widetilde{\beta}(\rmd t)}{X(t)}. \] This proves~\ref{hyp:coalescence}. \end{step} \begin{step}[{Checking~\ref{hyp:tightGP} and~\ref{hyp:tightGHP}}] The assumption that for all $00$ entails readily~\ref{hyp:tightGP}. Indeed, when $\int_{(0,\infty)\times[a,b]} x \mu(\rmd x,\rmd t) = \infty$, almost surely, for all $\delta>0$, we have $ \sum_{a\le t \le b} \Xi(t) \1_{Y(t) \le \delta}=\infty $, where $(\Xi(t), Y(t))_{t\ge 0}$ is the Poisson point process introduced in~\eqref{eq cv PPP degres}, so that on the event ``$\forall t \in (0,1], \ X(t) \in (0,\infty)$'', a.s. \[ \sum_{a \le t \le b} \Xi(t) \1_{Y(t) \le X(t)} = \infty. \] Thus, on the event ``$\forall t \in (0,1], \ X(t) \in (0,\infty)$'', a.s. \[ \sum_{a \le t \le b} \frac{\Delta_+ X(t)}{X(t)} = \infty. \] Finally, under the assumption~\eqref{eq hypothese GWVE tight GHP}, for all $\delta\in (0,1)$, the condition~\ref{hyp:tightGHP} is satisfied with probability at least $1-\delta$.\qedhere \end{step}\let\qed\relax \end{proof} \appendix \section{Background on the GP, GH and GHP topologies}\label{sec:background} \subsection{The Gromov--Prokhorov (GP) topology} \label{GPdef} A measured metric space is a triple $(X,d,\mu)$ such that $(X,d)$ is a Polish space and $\mu$ is a Borel probability measure on $X$. Two such spaces $(X,d,\mu)$, $(X',d',\mu')$ are called GP-isometry-equivalent if and only if there exists an isometry $f:\supp(\mu)\to \supp(\mu')$ such that if $f_\star \mu$ is the image of $\mu$ by $f$ then $f_\star \mu=\mu'$. Let $\mathbb{K}_{\GP}$ be the set of GP-equivalent classes of measured metric spaces. Given a measured metric space $(X,d,\mu)$, we write $[X,d,\mu]$ for the GP-isometry-equivalence class of $(X,d,\mu)$ and frequently use the notation $X$ for either $(X,d,\mu)$ or $[X,d,\mu]$. We now recall the definition of the Prokhorov distance. Consider a metric space $(X,d)$. For every $A\subset X$ and $\e>0$ let $A^\e\coloneqq \{x\in X, d(x,A)<\e\}$ be the open $\varepsilon$-neighborhood of $A$. Then given two (Borel) probability measures $\mu$, $\nu$ on $X$, the Prokhorov distance between $\mu$ and $\nu$ is defined by \begin{multline*} d_P(\mu, \nu)\\ \coloneqq \inf\left\{ \e>0\colon \mu(A)\leq \nu (A^\e)+\e\;\text{ and }\;\nu(A)\leq \mu(A^\e)+\e, \text{ for all Borel set }A\subset X \right\}. \end{multline*} The Gromov--Prokhorov (for short GP) distance is an extension of the Prokhorov's distance: for every $[X,d,\mu],[X',d',\mu']\in \K_{\GP}$ the Gromov--Prokhorov distance between $X$ and $X'$ is defined by \[ d_{\GP}([X,d,\mu],[X',d',\mu'])\coloneqq\inf_{S,\phi,\phi'} d_P(\phi_\star \mu, \phi'_\star\mu'), \] where the infimum is taken over all metric spaces $S$ and isometric embeddings $\phi :X\to S$, $\phi' :X'\to S$. $d_{\GP}$ is indeed a distance on $\K_{\GP}$ and $(\K_{\GP},d_{\GP})$ is a Polish space (see e.g.~\cite{ADH13}). We use another convenient characterization of the GP topology using the convergence of distance matrices: for every measured metric space $(X,d^X,\mu^X)$ let $(x_i^X)_{i\in \N}$ be a sequence of i.i.d. random variables of common distribution $\mu^X$ and let $M^X\coloneqq(d^X(x_i^X,x_j^X))_{i,j\in \N}$. We have the following result from~\cite{Loh13}, \begin{lemm}\label{equivGP} Let $(X^n)_{n\in \N} \in \K_{\GP}^\N$ and let $X\in \K_{\GP}$ then $X^n\to^{\GP}X$ as $n\to \infty$ if and only if $M^{X^n}$ converges in distribution toward $M^X$ for the product topology. \end{lemm} \subsection{The Gromov--Hausdorff (GH) topology} \label{GH} Let $\K_{\GH}$ be the set of isometry-equivalent classes of compact metric spaces. For every metric space $(X,d)$, we write $[X,d]$ for the isometry-equivalence class of $(X,d)$, and frequently write $X$ for either $(X,d)$ or $[X,d]$. For every metric space $(X,d)$, the Hausdorff distance between $A,B\subset X$ is given~by \[ d_H(A,B)\coloneqq \inf\{\e>0, A\subset B^\e, B\subset A^\e \}. \] The Gromov--Hausdorff distance between $[X,d]$,$[X',d']\in \K_{\GH}$ is given by \[ d_{\GH}([X,d],[X',d'])\coloneqq\inf_{S,\phi,\phi'} \left (d_H(\phi(X), \phi'(X')) \right), \] where the infimum is taken over all metric spaces $S$ and isometric embeddings $\phi :X\to S$, $\phi' :X'\to S$. $d_{\GH}$ is indeed a distance on $\K_{\GH}$ and $(\K_{\GH},d_{\GH})$ is a Polish space (see e.g.~\cite{ADH13}). \subsection{The Gromov--Hausdorff--Prokhorov (GHP) topology} Two measured metric spaces $(X,d,\mu)$, $(X',d',\mu')$ are called GHP-isometry-equiva\-lent if and only if there exists an isometry $f:X\to X'$ such that if $f_\star \mu$ is the image of $\mu$ by $f$ then $f_\star \mu=\mu'$. Let $\K_{\GHP}\subset\K_{\GP}$ be the set of isometry-equivalence classes of compact measured metric spaces. The Gromov--Hausdorff--Prokhorov distance between $[X,d,\mu]$,$[X',d',\mu']\in \K_{\GHP}$ is given by \[ d_{\GHP}([X,d,\mu],[X',d',\mu'])\coloneqq\inf_{S,\phi,\phi'} \bigl(d_P(\phi_\star \mu, \phi'_\star\mu')+d_H(\phi(X), \phi'(X')) \bigr), \] where the infimum is taken over all metric spaces $S$ and isometric embeddings $\phi :X\to S$, $\phi' :X'\to S$. $d_{\GHP}$ is indeed a distance on $\K_{\GHP}$ and $(\K_{\GHP},d_{\GHP})$ is a Polish space (see~\cite{ADH13}). \section{Leaf-tightness criteria}\label{sec:leaf} In this section, $\bfX=((X^n,d^n,p^n))_{n\in \N}$ denotes a sequence of random compact measured metric spaces GHP-measurable. For every $n\in \N$, let $(x_i^n)_{i\in \N}$ be a sequence of i.i.d.\@ random variables of common distribution $p^n$, then let $M^n\coloneqq(d^n(x_i^n,x_j^n))_{i,j\in \N}$. We say that $\bfX$ is weak-leaf-tight if and only if \begin{equation}\label{eq:weak-leaf-tight} \forall \delta>0, \lim_{k\to \infty} \limsup_{n\to \infty} \proba{d^n(x^{n}_{k+1},\{x^n_1,x^n_2,\dots,x^n_k\})>\delta} = 0. \end{equation} We say that $\bfX$ is strong-leaf-tight if and only if \begin{equation}\label{eq:strong-leaf-tight} \forall \delta>0, \lim_{k\to \infty} \limsup_{n\to \infty} \proba{d_H(X^n,\{x^n_1,x^n_2,\dots,x^n_k\})>\delta } = 0. \end{equation} Those criteria were first introduced by Aldous~\cite{Ald91,Ald93}. \begin{prop}\label{B.1} If $(M^n)_{n\in \N}$ converges weakly toward a random matrix $M$, $\bfX$ is weak-leaf-tight, and for every $n\in \N$, $(X^n,d^n)$ is a random tree equipped with random edge lengths, then $\bfX$ converges weakly for the GP topology toward a random measured $\R$-tree $(X,d,p)$. Furthermore, if $(x_i)_{i\in \N}$ are i.i.d. random variables of law $p$ then $M^X\coloneqq(d(x_i,x_j))_{i,j\in \N}=^{(d)} M$. \end{prop} \begin{proof} The result is directly adapted from Aldous~\cite[Theorem~3]{Ald93} using modern formalism. \end{proof} \begin{prop}[{\cite[Proposition~B.1]{BBKKbis23+}}]\label{B2} If $(M^n)_{n\in \N}$ converges weak\-ly toward a random matrix $M$, and $\bfX$ is strong-leaf-tight, then $\bfX$ converges weakly for the GHP topology toward a random compact measured metric space $(X,d,p)$. Furthermore, if $(x_i)_{i\in \N}$ are i.i.d. random variables of law $p$ then $M^X\coloneqq(d(x_i,x_j))_{i,j\in \N}=^{(d)} M$. In addition, a.s. $p$ has full support. \end{prop} \bibliography{20260304_blancRenaudie} \end{document}