arXiv subjects

Louigi Addario-Berry

Publications and source records attributed to Louigi Addario-Berry.

At least 19 recordsLinked to original sources

arXiv2026

A shape theorem for BBM in a periodic environment

We consider the long-time behaviour of binary branching Brownian motion (BBM) where the branching rate depends on a periodic spatial heterogeneity. We prove that almost surely as $t\to\infty$, the heterogeneous BBM at time $t$, normalized by $t$, approaches a deterministic convex shape with respect to Hausdorff distance. Our approach relies on establishing tail bounds on the probability of existence of BBM particles lying in half-spaces, which in particular yields the asymptotic speed of propagation of projections of the BBM in every direction. Our arguments are primarily probabilistic in nature, but additionally exploit the existence of a "front speed" (or minimal speed of a pulsating traveling front solution) for the Fisher-KPP reaction-diffusion equation naturally associated to the BBM.

math.PR↗

arXiv2026

Optimal root recovery for uniform attachment trees and $d$-regular growing trees

We consider root-finding algorithms for random rooted trees grown by uniform attachment. Given an unlabeled copy of the tree and a target accuracy $\varepsilon > 0$, such an algorithm outputs a set of nodes that contains the root with probability at least $1 - \varepsilon$. We focus on the algorithm introduced by Bubeck, Devroye and Lugosi (2017) and proved to be optimal by Crane and Xu (2021). We prove that, for the optimal algorithm, an output set of size $\exp(O(\log^{1/2}(1/\varepsilon)))$ suffices; this bound is sharp and answers a question of Bubeck, Devroye and Lugosi (2017). We prove similar bounds for random regular trees that grow by uniform attachment, strengthening a result of Khim and Loh (2017).

cs.DS↗

arXiv2026

Scaling limits of multitype Bienaymé trees

We consider critical multitype Bienaymé trees that are either irreducible or possess a critical irreducible component with attached subcritical components. These trees are studied under two distinct conditioning frameworks: first, conditioning on the value of a linear combination of the numbers of vertices of given types; and second, conditioning on the precise number of vertices belonging to a selected subset of types. We prove that, under a finite exponential moment condition, the scaling limit as the tree size tends to infinity is given by the Brownian Continuum Random Tree. Additionally, we establish strong nonasymptotic tail bounds for the height of such trees. Our main tools include a flattening operation applied to multitype trees and sharp estimates regarding the structure of monotype trees with a given sequence of degrees.

math.PR↗

arXiv2026

Symmetric Cooperative Motion in Higher Dimensions

We prove a distributional convergence result for a multidimensional version of symmetric cooperative motion which was introduced and studied in one dimension in \cite{HRW, SCM1}. Our approach relies on framing the associated recursive distributional equation as a discretization of the porous medium equation. A major challenge is to analyze the behaviour of finite difference schemes which approximate weak solutions of the porous medium equation with unbounded initial data. In overcoming this difficulty, we perform a detailed analysis of the probability mass function of symmetric cooperative motion, in which we introduce several new comparison arguments for the discrete process. Consequently, along the way, we establish a novel multidimensional convergence result for a finite difference scheme approximating the ZKB/Barenblatt solution of the porous medium equation, which is of independent interest.

math.PR↗

arXiv2026

Diameters and mixing times for giant components of random graphs with given degrees

A sequence $D = \{d_1,...d_n\}$ is a feasible degree sequence if there is a graph on $\{1,...,n\}$ such that $i$ has degree $d_i$. For such a sequence, $G(D)$ is a graph chosen uniformly at random from those with the given degree sequence. We consider sequences $\{D_\ell\}_{\ell \geq 1}$ of feasible degree sequences which have a giant component. We show that with high probability this giant component is unique, and bound its diameter and the mixing time of the random walk on it. We also bound the size and diameter of the other components and show that many of these bounds are tight.

math.CO↗

arXiv2026

What is The Probability That A Random Graph With A Given Degree Sequence is Connected?

An $n$-tuple $D=(d(1),\dots,d(n))$ is a \emph{feasible degree sequence} if there is a graph on $\{1,\dots,n\}$ such that $i$ has degree $d(i)$. Any such graph will have $m=\sum_{i=1}^n d(i)/2$ edges. Letting $G(D)$ be a graph chosen uniformly from those with the given degree sequence, we upper-bound the probability that $G(D)$ is disconnected based on the number of vertices of degree $d$ for small $d$, and develop a powerful tool for proving such bounds. If there are any vertices of degree zero the probability $G$ is disconnected is $1$, so we assume there are no such vertices. Our results then imply that if there are $o(\sqrt{m})$ vertices of degree $1$ and $o(m)$ vertices of degree 2 then with high probability $G$ is connected, while if there are no vertices of degree 1 or 2 then the probability $G$ is disconnected is $O(\frac{n^4}{m^6})$.

math.PR↗

arXiv2026

The statistical threshold for planted matchings and spanning trees

In this paper, we study the problem of detecting the presence of a planted perfect matching or spanning tree in an Erdős--Rényi random graph. More precisely, we study the hypothesis testing problem where the statistician observes a graph on $n$ vertices. Under the null hypothesis, the graph is a realization of an Erdős--Rényi random graph $G(n,q)$, while under the alternative hypothesis, the graph is the union of an Erdős--Rényi random graph and a random perfect matching (or random spanning tree). In order to avoid trivial detection by counting edges, we adjust the alternative hypothesis so that the expected number of edges under both distributions coincides. We prove that in both problems, when $q\gg n^{-1/2}$, no test can perform better than random guessing, while for $q\ll n^{-1/2}$, there exist computationally efficient tests that guess correctly with high probability.

math.ST↗

arXiv2025

On exponentially height-penalized random trees

Given $n \in \mathbb{N}$ and $μ\in \mathbb{R}$, a $\textit{$μ$-height-biased tree of size $n$}$ is a random plane tree $\mathbf{\mathbf{T}}_n$ with $n$ vertices with law given by $\mathbb{P}(\mathbf{T}=t) \propto e^{-μh(t)}$, where $t$ ranges over fixed plane trees with $n$ vertices, and $h(t)$ is the height of $t$. Fix a sequence $(μ_n)_{n \ge 1}$ of real numbers, and for $n \ge 1$ let $\mathbf{T}_n$ be a $μ$-height-biased tree of size $n$. Durhuus and Ünel (2023) described the asymptotic behaviour of $h(\mathbf{T}_n)$ when $μ_n \equiv μ\in \mathbb{R}$ is fixed. In this work, we extend their results to arbitrary sequences of positive parameters depending on $n$. Most notably, we show that such a tree behaves like a height-biased Continuum Random Tree (CRT) when $μ_n$ is of order $1/\sqrt{n}$; that its height is asymptotically $(2π^2n/μ_n)^{1/3}$ when $μ_n$ is of larger order than $1/\sqrt{n}$ and of smaller order than $n$; and that its height converges to a fixed constant when $μ_n$ is of order at least $n$, with some random jumps under specific conditions on $μ_n$. We additionally prove various results on second order behaviours, and large deviation principles for the height, for different regimes of $μ_n$. Finally, we describe new statistics of these trees, covering their widths, their root degrees, and the local structure around their roots.

math.PR↗

arXiv2025

Universal diameter bounds for random graphs with given degrees

Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so $\mathrm{diam}^+(G)=\infty$ if $G$ is not connected. Fix a sequence $(d_1,\ldots,d_n)$ of positive integers, and let $\mathbf{G}$ be a uniformly random connected simple graph with $V(\mathbf{G})=[n]:=\{1,\ldots,n\}$ such that $\mathrm{deg}_{\mathbf{G}}(v)=d_v$ for all $v \in [n]$. We show that, unless a $1-o(1)$ proportion of vertices have degree $2$, then $\mathbb{E}[\mathrm{diam}(\mathbf{G})]=O(\sqrt{n})$. It is not hard to see that this bound is best possible for general degree sequences (and in particular in the case of trees, in which $\sum_{v=1}^n d_v = 2(n-1)$). We also prove that this bound holds without the connectivity constraint. As a key input to the proofs, we show that graphs with minimum degree $3$ are with high probability connected and have logarithmic diameter: if $\min(d_1,\ldots,d_n) \ge 3$ and $\mathbf{G}$ is a uniformly random simple graph with $V(\mathbf{G})=[n]$ such that $\mathrm{deg}_{\mathbf{G}}(v)=d_v$ for all $v \in [n]$, then $\mathrm{diam}^+(\mathbf{G})=$ $O_{\mathbb{P}}(\log n)$; this bound is also best possible.

math.PR↗

arXiv2025

Infinite Schnyder Woods

It is well-known that any finite triangulation possesses a unique maximal Schnyder wood. We introduce Schnyder woods of infinite triangulations, and prove there exists a unique maximal Schnyder wood of any infinite triangulation with finite boundary, and of the uniform infinite half-planar triangulation. Furthermore, the maximal Schnyder wood of the uniform infinite planar triangulation is the limit of maximal Schnyder woods of large finite random triangulations. Several structural properties of infinite Schnyder woods are also described.

math.CO↗

arXiv2025

Kingman's coalescent on a random graph

We introduce a generalization of Kingman's coalescent on $[n]$ that we call the Kingman coalescent on a graph $G = ([n],E)$. Specifically, we generalize a forest valued representation of the coalescent introduced in Addario-Berry and Eslava (2018). The difference between the Kingman coalescent on $G$ and the normal Kingman coalescent on $[n]$ is that two trees $T_1,T_2$ with roots $ρ_1,ρ_2$ can merge if and only if $\{ρ_1,ρ_2\} \in E$. When this process finishes (when there are no trees left that can merge anymore), we are left with a random spanning forest that we call a Kingman forest of $G$. In this article, we study the Kingman coalescent on Erdős-Rényi random graphs, $G_{n,p}$. We derive a relationship between the Kingman coalescent on $G_{n,p}$ and uniform random recursive trees, which provides many answers concerning structural questions about the corresponding Kingman forests. We explore the heights of Kingman forests as well as the sizes of their trees as illustrative examples of how to use the connection. Our main results concern the number of trees, $C_{n,p}$, in a Kingman forest of $G_{n,p}$. For fixed $p \in (0,1)$, we prove that $C_{n,p}$ converges in distribution to an almost surely finite random variable as $n \to \infty$. For $p = p(n)$ such that $p \to 0$ and $np \to \infty$ as $n \to \infty$, we prove that $C_{n,p}$ converges in probability to $\frac{2(1-p)}{p}$.

math.PR↗

arXiv2025

Revisiting scaling limits for critical inhomogeneous random graphs with finite third moments

We consider the rank-1 inhomogeneous random graph in the Brownian regime in the critical window. Aldous studied the weights of the components, and showed that this ordered sequence converges in the $\ell^2$-topology to the ordered excursions of a Brownian motion with parabolic drift when appropriately rescaled (http://doi.org/10.1214/aop/1024404421), as the number of vertices $n$ tends to infinity. We show that, under the finite third moment condition, the same conclusion holds for the ordered component sizes. This in particular proves a result claimed by Bhamidi, Van der Hofstad and Van Leeuwaarden (https://doi.org/10.1214/EJP.v15-817). We also show that, for the large components, the ranking by component weights coincides with the ranking by component sizes with high probability as $n \to \infty$.

math.PR↗

arXiv2025

Discrete snakes with globally centered displacements

We prove a scaling limit for globally centered discrete snakes on size-conditioned critical Bienaymé trees. More specifically, under a global finite variance condition, we prove convergence in the sense of random finite-dimensional distributions of the head of the discrete snake (suitably rescaled) to the head of the Brownian snake driven by a Brownian excursion. When the third moment of the offspring distribution is finite, we further prove uniform functional convergence under a necessary tail condition on the displacements. We also consider displacement distributions with heavier tails, for which we instead obtain convergence to a variant of the hairy snake introduced by Janson and Marckert. We further give two applications of our main result. Firstly, we obtain a scaling limit for the difference between the height process and the Łukasiewicz path of a size-conditioned critical Bienaymé tree. Secondly, we obtain a scaling limit for the difference between the height process of a size-conditioned critical Bienaymé tree and the height process of its associated looptree.

math.PR↗

arXiv2025

Critical trees are neither too short nor too fat

We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaymé trees. Our bounds are optimal at this level of generality. We also obtain precise asymptotics for offspring distributions within the domain of attraction of a Cauchy distribution, under a local regularity assumption. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaymé trees.

math.PR↗

arXiv2024

Scaling limits of random graphs

This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric structure of these random objects in the limit as their size goes to infinity, with distances appropriately rescaled. We start with the simplest setting of random trees, before turning to various examples of random graphs, including the critical Erdős--Rényi random graph.

math.PR↗

arXiv2024

Leaf Stripping on Uniform Attachment Trees

In this note we analyze the performance of a simple root-finding algorithm in uniform attachment trees. The leaf-stripping algorithm recursively removes all leaves of the tree for a carefully chosen number of rounds. We show that, with probability $1 - ε$, the set of remaining vertices contains the root and has a size only depending on $ε$ but not on the size of the tree.

math.PR↗

arXiv2024

Refined Horton-Strahler numbers I: a discrete bijection

The Horton-Strahler number of a rooted tree $T$ is the height of the tallest complete binary tree that can be homeomorphically embedded in $T$. The number of full binary trees with $n$ internal vertices and Horton-Strahler number $s$ is known to be the same as the number of Dyck paths of length $2n$ whose height $h$ satisfies $\lfloor \log_2(1+h)\rfloor=s$. In this paper, we present a new bijective proof of the above result, that in fact strengthens and refines it as follows. We introduce a sequence of trees $(τ_i,i \ge 0)$ which "interpolates" the complete binary trees, in the sense that $τ_{2^h-1}$ is the complete binary tree of height $h$ for all $h \ge 0$, and $τ_{i+1}$ strictly contains $τ_i$ for all $i \ge 0$. Defining $\mathcal{S}(T)$ to be the largest $i$ for which $τ_i$ can be homeomorphically embedded in $T$, we then show that the number of full binary trees $T$ with $n$ internal vertices and with $\mathcal{S}(T)=h$ is the same as the number of Dyck paths of length $2n$ with height $h$. (We call $\mathcal{S}(T)$ the refined Horton-Strahler number of $T$.) Our proof is bijective and relies on a recursive decomposition of binary trees (resp. Dyck paths) into subtrees with strictly smaller refined Horton-Strahler number (resp. subpaths with strictly smaller height). In a subsequent paper, we will show that the bijection has a continuum analogue, which transforms a Brownian continuum random tree into a Brownian excursion and under which (a continuous analogue of) the refined Horton-Strahler number of the tree becomes the height of the excursion.

math.CO↗

arXiv2024

Patricia's Bad Distributions

The height of a random PATRICIA tree built from independent, identically distributed infinite binary strings with arbitrary diffuse probability distribution $μ$ on $\{0,1\}^\mathbb{N}$ is studied. We show that the expected height grows asymptotically sublinearly in the number of leaves for any such $μ$, but can be made to exceed any specific sublinear growth rate by choosing $μ$ appropriately.

math.PR↗