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Thomas Duquesne

Publications and source records attributed to Thomas Duquesne.

At least 19 recordsLinked to original sources
arXiv2026

The continuous Derrida-Retaux branching process in the Brownian CRT

The Derrida-Retaux continuous branching process, which is conjectured to be the scaling limit of the corresponding discrete model and of many other critical hierarchical renormalization models, is a process of cells evolving inhomogeneously on the time interval $[0, 1)$ via linear growth of each cell and independent splitting of cells at a specific rate. In this article, we show that the Derrida-Retaux continuous branching process is, on the one hand, encoded by a family of processes converging to a Brownian motion and on the other hand, fully obtained from the continuum Brownian tree encoded by the limiting Brownian path by time reversal and through length erasure of this tree. This direct representation explains the specific laws governing the significant quantities featuring the model (total mass, number of cells, law of large numbers) and provides a better understanding of its dynamics.

math.PR↗
arXiv2026

Mass erasure on measured $\mathbb{R}$-trees, applications to L\'evy forests

Let $h>0$. For a complete and separable $\mathbb{R}$-tree $(T,d)$ equipped with a root $\rho$ and a finite Borel measure $\mu$, we define the $h$-mass-erased tree by removing from $T$ all fringe subtrees of mass less than $h$ and we equip it with a suitable measure such that the erasure operators $(\mathcal{E}_h)_{h\ge 0}$ form a semigroup that is continuous for the Gromov-weak topology. Then, we say that a sequence $\boldsymbol{\mu}_n=(T_n,d_n,\rho_n,\mu_n)$, $n\in\mathbb{N}$, converges in the sense of mass erasure if $(\mathcal{E}_h\boldsymbol{\mu}_n)_{n\in\mathbb{N}}$ converges Gromov-weakly for all $h\ge 0$. This notion of convergence is strictly weaker than Gromov-weak convergence and we establish criteria to relate the two notions. We define a distance function that metrizes convergence in the sense of mass erasure. By extending the notion of measured $\mathbb{R}$-trees to allow mass on the boundary (the far ends of infinite geodesics), we obtain a complete metric space. Next, we identify random trees of finite type (that is, discrete trees with edge lengths) satisfying the regenerative branching property as a specific class of measured (sub)critical GW forests. We then show that this class of trees is preserved by mass erasure and we compute the law of these mass-erased GW forests explicitly. Finally, we establish a limit theorem for these measured (sub)critical GW forests to converge to standard measured L\'evy forests, i.e. those whose total mass has the same distribution as the total population of a continuous-state branching process. This includes cases with bounded variation by crucially using the convergence in the sense of mass erasure and it extends the cases studied previously.

math.PR↗
arXiv2026

Scaling limit of the range of tree-valued branching random walks in random environmen

We study a branching random walk (BRW) taking its values in a random tree $\bT$ (seen as a family tree) with an infinite line of ancestors that is a variant of a supercritical Galton--Watson (GW) tree with offspring distribution $\nu$. The transition probabilities of the BRW are those of a critical biased random walk on $\bT$: namely, the probability to move from $x$ to one of its $k_x$ children is $1/(\mathtt{m}_\nu+k_x)$ and the probability to move from $x$ to the direct parent of $x$ is $\mathtt{m}_\nu/(\mathtt{m}_\nu+k_x)$. Here $\ttm_\nu$ stands for the mean of $\nu$. The BRW is indexed by a critical GW tree conditioned to have $n$ {vertices} and whose offspring distribution is in the domain of attraction of an $\alpha$-stable law with $\alpha \ino (1, 2]$. We denote by $\cR_n$ the range of the BRW, i.e., ~the set of all sites in $\bT$ visited by the BRW. Under a moment assumption for $\nu$, we prove that if we view $\cR_n$ as a random subtree of $\bT$ equipped with its graph distance $d_{\mathtt{gr}}$ and with its occupation measure $\ttm^{_{(n)}}_{{\mathtt{occ}}}$ then there exists a scaling sequence $s_n \! \to \! \infty$ such that conditionally given the environment $\bT$, the measured metric space $(\cR_n, s_n^{-1}d_{\mathtt{gr}} , \frac{_1}{^n}\ttm^{_{(n)}}_{{\mathtt{occ}}} )$ weakly converges in the Gromov--Hausdorff--Prokhorov sense to a random measured compact real tree introduced by Curien, Le Gall \& Miermont in \cite{CuLGMi13} called the Brownian cactus with $\alpha$-stable branching mechanism. This work extends in random environment the result from D., K., Lin \& Torri \cite{DuKhLiTo22} which deals with the case where $\bT$ is a regular tree.

math.PR↗
arXiv2025

Hipster random walks, random series-parallel graph and random homogeneous systems

We study a class of random homogeneous systems. Our main result says that under suitable general assumptions, these systems converge weakly, upon a suitable normalization, to the probability distribution with density $\frac34 \, (1-x^2) \, {\bf 1}_{\{ x\in (-1, \, 1)\} }$. Two special cases are of particular interest: for the effective resistance of the critical random series-parallel graph, our result gives an affirmative answer to a conjecture of Hambly and Jordan (Adv. Appl. Probab. 2004) and further conjectures of Addario-Berry et al. (Probab.Theory Related Fields 2020) and Derrida whereas for the hipster random walk, we recover a previous result of Addario-Berry et al.~(Probab. Theory Related Fields 2020).

math.PR↗
arXiv2022

Scaling limits of tree-valued branching random walks

We consider a branching random walk (BRW) taking its values in the $\mathtt{b}$-ary rooted tree $\mathbb W_{ \mathtt{b}}$ (i.e. the set of finite words written in the alphabet $\{ 1, \ldots, \mathtt{b} \}$, with $\mathtt{b}\! \geq \! 2$). The BRW is indexed by a critical Galton--Watson tree conditioned to have $n$ vertices; its offspring distribution is aperiodic and is in the domain of attraction of a $γ$-stable law, $γ\in (1, 2]$. The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on $\mathbb W_{ \mathtt{b}}$ (reflection at the root of $\mathbb W_{ \mathtt{b}}$ and otherwise: probability $1/2$ to move closer to the root of $\mathbb W_{ \mathtt{b}}$ and probability $1/(2\mathtt{b})$ to move away from it to one of the $\mathtt{b}$ sites above). We denote by $\mathcal R_{\mathtt{b}} (n)$ the range of the BRW in $\mathbb W_{ \mathtt{b}}$ which is the set of all sites in $\mathbb W_{\mathtt{b}}$ visited by the BRW. We first prove a law of large numbers for $\# \mathcal R_{\mathtt{b}} (n)$ and we also prove that if we equip $\mathcal R_{\mathtt{b}} (n)$ (which is a random subtree of $\mathbb W_{\mathtt{b}}$) with its graph-distance $d_{\mathtt{gr}}$, then there exists a scaling sequence $(a_n)_{n\in \mathbb N}$ satisfying $a_n \! \rightarrow \! \infty$ such that the metric space $(\mathcal R_{\mathtt{b}} (n), a_n^{-1}d_{\mathtt{gr}})$, equipped with its normalised empirical measure, converges to the reflected Brownian cactus with $γ$-stable branching mechanism: namely, a random compact real tree that is a variant of the Brownian cactus introduced by N. Curien, J-F. Le Gall and G. Miermont.

math.PR↗
arXiv2020

Limits of multiplicative inhomogeneous random graphs and Lévy trees: The continuum graphs

Motivated by limits of critical inhomogeneous random graphs, we construct a family of sequences of measured metric spaces that we call continuous multiplicative graphs, that are expected to be the universal limit of graphs related to the multiplicative coalescent (the Erdős--Rényi random graph, more generally the so-called rank-one inhomogeneous random graphs of various types, and the configuration model). At the discrete level, the construction relies on a new point of view on (discrete) inhomogeneous random graphs that involves an embedding into a Galton--Watson forest. The new representation allows us to demonstrate that a processus that was already present in the pionnering work of Aldous [Ann. Probab., vol.~25, pp.~812--854, 1997] and Aldous and Limic [Electron. J. Probab., vol.~3, pp.~1--59, 1998] about the multiplicative coalescent actually also (essentially) encodes the limiting metric: The discrete embedding of random graphs into a Galton--Watson forest is paralleled by an embedding of the encoding process into a Lévy process which is crucial in proving the very existence of the local time functionals on which the metric is based; it also yields a transparent approach to compactness and fractal dimensions of the continuous objects. In a companion paper, we show that the continuous Lévy graphs are indeed the scaling limit of inhomogeneous random graphs.

math.PR↗
arXiv2020

Limits of multiplicative inhomogeneous random graphs and Lévy trees: Limit theorems

We consider a natural model of inhomogeneous random graphs that extends the classical Erd\H os-Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous [AOP 1997]. In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic [EJP 1998] have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov-Hausdorff-Prokhorov convergence to a class of random compact measured metric spaces that have been introduced in a companion paper. Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general "critical" regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton-Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via regime-specific proofs, for instance: the case of Erd\H os-Rényi random graphs obtained by Addario-Berry, Goldschmidt and B. [PTRF 2012], the asymptotic homogeneous case as studied by Bhamidi, Sen and Wang [PTRF 2017], or the power-law case as considered by Bhamidi, Sen and van der Hofstad [PTRF 2018].

math.PR↗
arXiv2015

Decomposition of Levy trees along their diameter

We study the diameter of L{é}vy trees that are random compact metric spaces obtained as the scaling limits of Galton-Watson trees. L{é}vy trees have been introduced by Le Gall and Le Jan (1998) and they generalise Aldous' Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of L{é}vy trees and we prove that it is realized by a unique pair of points. We prove that the law of L{é}vy trees conditioned to have a fixed diameter r $\in$ (0, $\infty$) is obtained by glueing at their respective roots two independent size-biased L{é}vy trees conditioned to have height r/2 and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. As an application of this decomposition of L{é}vy trees according to their diameter, we characterize the joint law of the height and the diameter of stable L{é}vy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalised stable trees, which generalises the identity due to Szekeres (1983) in the Brownian case.

math.PR↗
arXiv2014

Exact packing measure of the range of $ψ$-Super Brownian motions

We consider super processes whose spatial motion is the $d$-dimensional Brownian motion and whose branching mechanism $ψ$ is critical or subcritical; such processes are called $ψ$-super Brownian motions. If $d\!>\!2\bgamma/(\bgamma\!-\!1)$, where $\bgamma\!\in\!(1,2]$ is the lower index of $ψ$ at $\infty$, then the total range of the $ψ$-super Brownian motion has an exact packing measure whose gauge function is $g(r)\! =\! (\log\log1/r) / φ^{-1} ( (1/r\log\log 1/r)^{2})$, where $φ\! =\! ψ^\prime\! \circ \! ψ^{\!-1}$. More precisely, we show that the occupation measure of the $ψ$-super Brownian motion is the $g$-packing measure restricted to its total range, up to a deterministic multiplicative constant only depending on $d$ and $ψ$. This generalizes the main result of \cite{Duq09} that treats the quadratic branching case. For a wide class of $ψ$, the constant $2\bgamma/(\bgamma\!-\!1)$ is shown to be equal to the packing dimension of the total range.

math.PR↗
arXiv2014

Scaling limits and influence of the seed graph in preferential attachment trees

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barabási-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel \& Rácz concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric structure of nodes of large degrees in a plane version of Barabási-Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov-Hausdorff sense towards a random compact metric space which we call the Brownian looptree. The latter is constructed as a quotient space of Aldous' Brownian Continuum Random Tree and is shown to have almost sure Hausdorff dimension $2$.

math.PR↗
arXiv2014

On the Eve property for CSBP

We consider the population model associated to continuous state branching processes and we are interested in the so-called Eve property that asserts the existence of an ancestor with an overwhelming progeny at large times, and more generally, in the possible behaviours of the frequencies among the population at large times. In this paper, we classify all the possible behaviours according to the branching mechanism of the continuous state branching process.

math.PR↗
arXiv2012

Hereditary tree growth and Levy forests

We introduce the notion of a hereditary property for rooted real trees and we also consider reduction of trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a special case of hereditary reduction. We only consider the metric structure of trees, and our framework is the space $\bT$ of pointed isometry classes of locally compact rooted real trees equipped with the Gromov-Hausdorff distance. Some of the main results of the paper are a general tightness criterion in $\bT$ and limit theorems for growing families of trees. We apply these results to Galton-Watson trees with exponentially distributed edge lengths. This class is preserved by hereditary reduction. Then we consider families of such Galton-Watson trees that are consistent under hereditary reduction and that we call growth processes. We prove that the associated families of offspring distributions are completely characterised by the branching mechanism of a continuous-state branching process. We also prove that such growth processes converge to Levy forests. As a by-product of this convergence, we obtain a characterisation of the laws of Levy forests in terms of leaf-length erasure and we obtain invariance principles for discrete Galton-Watson trees, including the super-critical cases.

math.PR↗
arXiv2011

Exceptionally small balls in stable trees

The $γ$-stable trees are random measured compact metric spaces that appear as the scaling limit of Galton-Watson trees whose offspring distribution lies in a $γ$-stable domain, $γ\in (1, 2]$. They form a specific class of Lévy trees (introduced by Le Gall and Le Jan in1998) and the Brownian case $γ= 2$ corresponds to Aldous Continuum Random Tree (CRT). In this paper, we study fine properties of the mass measure, that is the natural measure on $γ$-stable trees. We first discuss the minimum of the mass measure of balls with radius $r$ and we show that this quantity is of order $r^{\fracγ{γ-1}} (\log1/r)^{-\frac{1}{γ-1}}$. We think that no similar result holds true for the maximum of the mass measure of balls with radius $r$, except in the Brownian case: when $γ= 2$, we prove that this quantity is of order $r^2 \log 1/r$. In addition, we compute the exact constant for the lower local density of the mass measure (and the upper one for the CRT), which continues previous results.

math.PR↗
arXiv2010

The exact packing measure of Lévy trees

We study fine properties of Lévy trees that are random compact metric spaces introduced by Le Gall and Le Jan in 1998 as the genealogy of continuous state branching processes. Lévy trees are the scaling limits of Galton-Watson trees and they generalize Aldous's continuum random tree which corresponds to the Brownian case. In this paper we prove that Lévy trees have always an exact packing measure: We explicitely compute the packing gauge function and we prove that the corresponding packing measure coincides with the mass measure up to a multiplicative constant.

math.PR↗
arXiv2010

Packing and Hausdorff measures of stable trees

In this paper we discuss Hausdorff and packing measures of random continuous trees called stable trees. Stable trees form a specific class of Lévy trees (introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum random tree (1991) which corresponds to the Brownian case. We provide results for the whole stable trees and for their level sets that are the sets of points situated at a given distance from the root. We first show that there is no exact packing measure for levels sets. We also prove that non-Brownian stable trees and their level sets have no exact Hausdorff measure with regularly varying gauge function, which continues previous results from a joint work with J-F Le Gall (2006).

math.PR↗
arXiv2008

An Elementary Proof of Hawkes's Conjecture on Galton-Watson Trees

In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge function for the boundary of supercritical Galton-Watson trees under a certain assumption on the tail at the infinity of the total mass of the branching measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as other other precise results on fractal properties of the boundary of Galton-Watson trees. The goal of this paper is to provide an elementary proof of Hawkes's conjecture under a less restrictive assumption than in T. Watanabe's paper, by use of size-biased Galton-Watson trees introduced by Lyons, Pemantle and Peres in 1995.

math.PR↗
arXiv2008

The Packing Measure of the Range of Super-Brownian Motion

We prove that the total range of Super-Brownian motion with quadratic branching mechanism has an exact packing measure with respect to the gauge function $g(r)=r^4 (\log \log1/r)^{-3}$ in super-critical dimensions $d\geq 5$. More precisely, we prove that the total occupation measure of Super-Brownian motion is equal to the $g$-packing measure restricted to its range, up to a deterministic multiplicative constant that only depends on space dimension $d$.

math.PR↗