arXiv subjects

Delphin Sénizergues

Publications and source records attributed to Delphin Sénizergues.

12 recordsLinked to original sources

arXiv2026

Total disconnectedness and percolation for the supports of super-tree random measures

Super-tree random measures (STRMs) were introduced by Allouba, Durrett, Hawkes and Perkins as a simple stochastic model which emulates a superprocess at a fixed time. A STRM $ν$ arises as the a.s. limit of a sequence of empirical measures for a discrete time particle system which undergoes independent supercritical branching and independent random displacement (spatial motion) of children from their parents. We study the connectedness properties of the closed support of a STRM ($\mathrm{supp}(ν)$) for a particular choice of random displacement. Our main results are distinct sufficient conditions for the a.s. total disconnectedness (TD) of $\mathrm{supp}(ν)$, and for percolation on $\mathrm{supp}(ν)$ which will imply a.s. existence of a non-trivial connected component in $\mathrm{supp}(ν)$. We illustrate a close connection between a subclass of these STRM's and super-Brownian motion (SBM). For these particular STRM's the above results give a.s. TD of the support in three and higher dimensions and the existence of a non-trivial connected component in two dimensions, with the three-dimensional case being critical. The latter two-dimensional result assumes that $p_c(\mathbb{Z}^2)$, the critical probability for site percolation on $\mathbb{Z}^2$, is less than $1-e^{-1}$. (There is strong numerical evidence supporting this condition although the known rigorous bounds fall just short.) This gives evidence that the same connectedness properties should hold for SBM. The latter remains an interesting open problem in dimensions $2$ and $3$ ever since it was first posed by Don Dawson over $30$ years ago.

math.PR↗

arXiv2025

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

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

math.PR↗

arXiv2025

The longest increasing subsequence of Brownian separable permutons

We establish a scaling limit result for the length $\operatorname{LIS}(σ_n)$ of the longest increasing subsequence of a permutation $σ_n$ of size $n$ sampled from the Brownian separable permuton $\boldsymbolμ_p$ of parameter $p\in(0,1)$, which is the universal limit of pattern-avoiding permutations. Specifically, we prove that \[\frac{\operatorname{LIS}(σ_n)}{n^α}\;\underset{n\to\infty}{\overset{\mathrm{a.s.}}{\longrightarrow}}\; X,\] where $α=α(p)$ is the unique solution in the interval $(1/2,1)$ to the equation \[\frac{1}{4^{\frac{1}{2α}}\sqrtπ}\,\frac{Γ\big(\tfrac{1}{2}-\tfrac{1}{2α}\big)}{Γ\big(1-\tfrac{1}{2α}\big)}=\frac{p}{p-1},\] and $X=X(p)$ is a non-deterministic and a.s. positive and finite random variable, which is a measurable function of the Brownian separable permuton. Notably, the exponent $α(p)$ is an increasing continuous function of $p$ with $α(0^+)=1/2$, $α(1^-)=1$ and $α(1/2)\approx0.815226$, which corresponds to the permuton limit of uniform separable permutations. We prove analogous results for the size of the largest clique of a graph sampled from the Brownian cographon of parameter $p\in(0,1)$.

math.PR↗

arXiv2025

The height of the infection tree

We are interested in the geometry of the ``infection tree'' in a stochastic SIR (Susceptible-Infectious-Recovered) model, starting with a single infectious individual. This tree is constructed by drawing an edge between two individuals when one infects the other. We focus on the regime where the infectious period before recovery follows an exponential distribution with rate $1$, and infections occur at a rate $λ_{n} \sim \fracλ{n}$ where $n$ is the initial number of healthy individuals with $λ>1$. We show that provided that the infection does not quickly die out, the height of the infection tree is asymptotically $κ(λ) \log n$ as $n \rightarrow \infty$, where $κ(λ)$ is a continuous function in $λ$ that undergoes a second-order phase transition at $λ_{c}\simeq 1.8038$. Our main tools include a connection with the model of uniform attachment trees with freezing and the application of martingale techniques to control profiles of random trees.

math.PR↗

arXiv2024

Maximum Agreement Subtrees and Hölder homeomorphisms between Brownian trees

We prove that the size of the largest common subtree between two uniform, independent, leaf-labelled random binary trees of size $n$ is typically less than $n^{1/2-\varepsilon}$ for some $\varepsilon>0$. Our proof relies on the coupling between discrete random trees and the Brownian tree and on a recursive decomposition of the Brownian tree due to Aldous. Along the way, we also show that almost surely, there is no $(1-\varepsilon)$-Hölder homeomorphism between two independent copies of the Brownian tree.

math.PR↗

arXiv2022

Height of weighted recursive trees with sub-polynomially growing total weight

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen at random with probability proportional to its weight. In the case where the total weight of the tree at step $n$ grows polynomially in $n$, we obtained in (Pain-Sénizergues 2022) an asymptotic expansion for the height of the tree, which falls into the university class of the maximum of branching random walks. In this paper, we consider the case of a total weight growing sub-polynomially in $n$ and obtain asymptotics for the height of the tree in several regimes, showing that universality is broken and exhibiting new behaviors.

math.PR↗

arXiv2022

Decorated stable trees

We define decorated $α$-stable trees which are informally obtained from an $α$-stable tree by blowing up its branchpoints into random metric spaces. This generalizes the $α$-stable looptrees of Curien and Kortchemski, where those metric spaces are just deterministic circles. We provide different constructions for these objects, which allows us to understand some of their geometric properties, including compactness, Hausdorff dimension and self-similarity in distribution. We prove an invariance principle which states that under some conditions, analogous discrete objects, random decorated discrete trees, converge in the scaling limit to decorated $α$-stable trees. We mention a few examples where those objects appear in the context of random trees and planar maps, and we expect them to naturally arise in many more cases.

math.PR↗

arXiv2021

Correction terms for the height of weighted recursive trees

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of weights, the first order for the height of such trees has been recently established in [Electron. J. Probab. 26 (2021), Paper No. 80] by one of the authors. In this paper, we obtain the second and third orders in the asymptotic expansion of the height of weighted recursive trees, under similar assumptions. Our methods are inspired from those used to prove similar results for branching random walks. Our results also apply to a related model of growing trees, called the preferential attachment tree with additive fitnesses.

math.PR↗

arXiv2020

Stable graphs: distributions and line-breaking construction

For $α\in (1,2]$, the $α$-stable graph arises as the universal scaling limit of critical random graphs with i.i.d. degrees having a given $α$-dependent power-law tail behavior. It consists of a sequence of compact measured metric spaces (the limiting connected components), each of which is tree-like, in the sense that it consists of an $\mathbb R$-tree with finitely many vertex-identifications (which create cycles). Indeed, given their masses and numbers of vertex-identifications, these components are independent and may be constructed from a spanning $\mathbb R$-tree, which is a biased version of the $α$-stable tree, with a certain number of leaves glued along their paths to the root. In this paper we investigate the geometric properties of such a component with given mass and number of vertex-identifications. We (1) obtain the distribution of its kernel and more generally of its discrete finite-dimensional marginals; we will observe that these distributions are related to the distributions of some configuration models (2) determine the distribution of the $α$-stable graph as a collection of $α$-stable trees glued onto its kernel and (3) present a line-breaking construction, in the same spirit as Aldous' line-breaking construction of the Brownian continuum random tree.

math.PR↗

arXiv2020

Geometry of weighted recursive and affine preferential attachment trees

We study two models of growing recursive trees. For both models, initially the tree only contains one vertex $u_1$ and at each time $n\geq 2$ a new vertex $u_n$ is added to the tree and its parent is chosen randomly according to some rule. In the \emph{weighted recursive tree}, we choose the parent $u_k$ of $u_n$ among $\{u_1,u_2,\dots, u_{n-1}\}$ with probability proportional to $w_k$, where $(w_n)_{n\geq1}$ is some deterministic sequence that we fix beforehand. In the \emph{affine preferential attachment tree with fitnesses}, the probability of choosing any $u_k$ is proportional to $a_k+\mathrm{deg}^{+}(u_k)$, where $\mathrm{deg}^{+}(u_k)$ denotes its current number of children, and the sequence of \emph{fitnesses} $(a_n)_{n\geq 1}$ is deterministic and chosen as a parameter of the model. We show that for any sequence $(a_n)_{n\geq 1}$, the corresponding preferential attachment tree has the same distribution as some weighted recursive tree with a \emph{random} sequence of weights (with some explicit distribution). We then prove almost sure scaling limit convergences for some statistics associated with weighted recursive trees as time goes to infinity, such as degree sequence, height, profile and also the weak convergence of some measures carried on the tree. Thanks to the connection between the two models, these results also apply to affine preferential attachment trees.

math.PR↗

arXiv2020

Growing random graphs with a preferential attachment structure

The aim of this paper is to develop a method for proving almost sure convergence in Gromov-Hausodorff-Prokhorov topology for a class of models of growing random graphs that generalises Rémy's algorithm for binary trees. We describe the obtained limits using some iterative gluing construction that generalises the famous line-breaking construction of Aldous' Brownian tree. In order to do that, we develop a framework in which a metric space is constructed by gluing smaller metric spaces, called \emph{blocks}, along the structure of a (possibly infinite) discrete tree. Our growing random graphs seen as metric spaces can be understood in this framework, that is, as evolving blocks glued along a growing discrete tree structure. Their scaling limit convergence can then be obtained by separately proving the almost sure convergence of every block and verifying some relative compactness property for the whole structure. For the particular models that we study, the discrete tree structure behind the construction has the distribution of an affine preferential attachment tree or a weighted recursive tree. We strongly rely on results concerning those two models of random trees and their connection, obtained in a companion paper.

math.PR↗

arXiv2019

Random gluing of metric spaces

We construct random metric spaces by gluing together an infinite sequence of pointed metric spaces that we call blocks. At each step, we glue the next block to the structure constructed so far by randomly choosing a point on the structure and then identifying it with the distinguished point of the block. The random object that we study is the completion of the structure we obtain after an infinite number of steps. We introduce a sequence $(w_n)_{n\geq1}$ that we call the weights of the blocks. The probability at each step that the next block is glued onto any of the preceding blocks is proportional to its weight. We suppose that the blocks are i.i.d. copies of the same random metric space, scaled by deterministic factors that we call $(λ_n)_{n\geq 1}$. We work under some conditions on the distribution of the blocks ensuring that they a.s. have dimension $d$, for some $d>0$. The main contribution of this paper is the computation of the Hausdorff dimension of the set $\mathcal{L}$ of points which appear during the completion procedure, which we call the leaves, when $(λ_n)_{n\geq 1}$ and $(w_n)_{n\geq1}$ typically behave like a power of $n$, say $n^{-α}$ for the scaling factors and $n^{-β}$ for the weights. For a large domain of $α$ and $β$, we have $\mathrm{dim_H}(\mathcal{L})=α^{-1}$. However for $β>1$ and $α<1/d$, our results reveal an interesting phenomenon: the dimension has a non-trivial dependence in $α$, $β$ and $d$, namely $\mathrm{dim_H}(\mathcal{L})=α^{-1}(2β-1-2\sqrt{(β-1)(β-dα)})$.}

math.PR↗