Related papers: The harmonic measure of balls in random trees
We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green…
Consider the Erd\H{o}s-Renyi random graph on n vertices where each edge is present independently with probability c/n, with c>0 fixed. For large n, a typical random graph locally behaves like a Galton-Watson tree with Poisson offspring…
We consider a random walk on a supercritical Galton-Watson tree with leaves, where the transition probabilities of the walk are determined by biases that are randomly assigned to the edges of the tree. The biases are chosen independently on…
This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees ("Otter trees"), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size $n$ is proved to…
The Horton-Strahler number -- also called the register function -- is a combinatorial tool that quantifies the branching complexity of a rooted tree. We study the law of the Horton-Strahler number of stable Galton-Watson trees conditioned…
We study a natural analogue of Ulam's problem for random rooted trees distributed according to a Plancherel-type measure. This probability measure is closely related to the classical Plancherel measure on integer partitions. For a…
We prove that the dimension drop phenomenon holds for the harmonic measure associated to a transient random walk in a random environment (as defined by R. Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on an infinite…
Given a finite typed rooted tree $T$ with $n$ vertices, the {\em empirical subtree measure} is the uniform measure on the $n$ typed subtrees of $T$ formed by taking all descendants of a single vertex. We prove a large deviation principle in…
We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root…
It is well known that if $h$ is a nonnegative harmonic function in the ball of $\RR^{d+1}$ or if $h$ is harmonic in the ball with integrable boundary values, then the radial limit of $h$ exists at almost every point of the boundary. In this…
We consider a uniform spanning tree in a $\delta$-square grid approximation of a planar domain $\Omega$. For given integer $n\ge 2$, we condition the tree on the following $n$-arm event: we pick $n$ branches, emanating from $n$ points…
We consider a specific random graph which serves as a disordered medium for a particle performing biased random walk. Take a two-sided infinite horizontal ladder and pick a random spanning tree with a certain edge weight $c$ for the…
We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed-points and the convergence of…
Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable…
The Horton-Strahler number, also known as the register function, provides a tool for quantifying the branching complexity of a rooted tree. We consider the Horton-Strahler number of critical Galton-Watson trees conditioned to have size $n$…
The Horton-Strahler number of a tree is a measure of its branching complexity; it is also known in the literature as the register function. We show that for critical Galton-Watson trees with finite variance conditioned to be of size $n$,…
In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$.…
We study a new type of random minimum spanning trees. It is built on the complete graph where each vertex is given a weight, which is a positive real number. Then, each edge is given a capacity which is a random variable that only depends…
The metric dimension of a graph $G$ is the minimal size of a subset $R$ of vertices of $G$ that, upon reporting their graph distance from a distingished (source) vertex $v^\star$, enable unique identification of the source vertex $v^\star$…
This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a…