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Jean-François Le Gall

Publications and source records attributed to Jean-François Le Gall.

At least 19 recordsLinked to original sources
arXiv2026

Large deviations for the CRT

We establish large deviation principles both for the CRT in the Gromov-Hausdorff topology and for the CRT weighted by its mass measure in the Gromov-Hausdorff-Prohorov topology. In the first case, the rate function of a tree $\mathcal{T}$ is half the square of the length of $\mathcal{T}$. In the second case, the rate function of a pair $(\mathcal{T},μ)$ is half the integral with respect to the length measure of $\mathcal{T}$ of the inverse of the absolutely continuous part of $μ$.

math.PR↗
arXiv2025

Peeling the Brownian half-plane

We establish a new spatial Markov property of the Brownian half-plane. According to this property, if one removes a hull centered at a boundary point, the remaining space equipped with an intrinsic metric is still a Brownian half-plane, which is independent of the part that has been removed. This is an analog of the well-known peeling procedure for random planar maps. We also investigate several distributional properties of hulls centered at a boundary point, and we provide a new construction of the Brownian half-plane giving information about distances from a half-boundary.

math.PR↗
arXiv2025

The area of spheres in the Brownian plane

We consider the area of spheres centered at the distinguished point in the Brownian plane. As a function of the radius, the resulting process has continuously differentiable sample paths. Furthermore, the pair consisting of the process and its derivative is time-homogeneous Markov and satisfies an explicit stochastic differential equation.

math.PR↗
arXiv2024

Drilling holes in the Brownian disk: The Brownian annulus

We give a new construction of the Brownian annulus based on removing a hull centered at the distinguished point in the free Brownian disk. We use this construction to prove that the Brownian annulus is the scaling limit of Boltzmann triangulations with two boundaries. We also prove that the space obtained by removing hulls centered at the two distinguished points of the Brownian sphere is a Brownian annulus. Our proofs rely on a detailed analysis of the peeling by layers algorithm for Boltzmann triangulations with a boundary.

math.PR↗
arXiv2024

Spatial Markov property in Brownian disks

We derive a new representation of the Brownian disk in terms of a forest of labeled trees, where labels correspond to distances from a subset of the boundary. We then use this representation to obtain a spatial Markov property showing that the complement of a hull centered at a boundary point of a Brownian disk is again a Brownian disk, with a random perimeter, and is independent of the hull conditionally on its perimeter. Our proofs rely in part on a study of the peeling process for triangulations with a boundary, which is of independent interest. The results of the present work will be applied to a continuous version of the peeling process for the Brownian half-plane in a companion paper.

math.PR↗
arXiv2023

The Markov property of local times of Brownian motion indexed by the Brownian tree

We consider the model of Brownian motion indexed by the Brownian tree, which has appeared in a variety of different contexts in probability, statistical physics and combinatorics. For this model, the total occupation measure is known to have a continuously differentiable density. Although the density process indexed by nonnegative reals is not Markov, we prove that the pair consisting of the density and its derivative is a time-homogeneous Markov process. We also establish a similar result for the local times of one-dimensional super-Brownian motion. Our methods rely on the excursion theory for Brownian motion indexed by the Brownian tree.

math.PR↗
arXiv2022

The volume measure of the Brownian sphere is a Hausdorff measure

We prove that the volume measure of the Brownian sphere is equal to a constant multiple of the Hausdorff measure associated with the gauge function $h(r)=r^4\log\log(1/r)$. This shows in particular that the volume measure of the Brownian sphere is determined by its metric structure. As a key ingredient of our proofs, we derive precise estimates on moments of the volume of balls in the Brownian sphere.

math.PR↗
arXiv2021

Geodesic stars in random geometry

A point of a metric space is called a geodesic star with $m$ arms if it is the endpoint of $m$ disjoint geodesics. For every $m\in\{1,2,3,4\}$, we prove that the set of all geodesic stars with $m$ arms in the Brownian sphere has dimension $5-m$. This complements recent results of Miller and Qian, who proved that this dimension is smaller than or equal to $5-m$.

math.PR↗
arXiv2020

Spine representations for non-compact models of random geometry

We provide a unified approach to the three main non-compact models of random geometry, namely the Brownian plane, the infinite-volume Brownian disk, and the Brownian half-plane. This approach allows us to investigate relations between these models, and in particular to prove that complements of hulls in the Brownian plane are infinite-volume Brownian disks.

math.PR↗
arXiv2020

The Brownian disk viewed from a boundary point

We provide a new construction of Brownian disks in terms of forests of continuous random trees equipped with nonnegative labels corresponding to distances from a distinguished point uniformly distributed on the boundary of the disk. This construction shows in particular that distances from the distinguished point evolve along the boundary as a five-dimensional Bessel bridge. As an important ingredient of our proofs, we show that the uniform measure on the boundary, as defined in the earlier work of Bettinelli and Miermont, is the limit of the suitably normalized volume measure on a small tubular neighborhood of the boundary. Our construction also yields a simple proof of the equivalence between the two definitions of the Brownian half-plane.

math.PR↗
arXiv2018

Growth-fragmentation processes in Brownian motion indexed by the Brownian tree

We consider the model of Brownian motion indexed by the Brownian tree. For every $r\geq 0$ and every connected component of the set of points where Brownian motion is greater than $r$, we define the boundary size of this component, and we then show that the collection of these boundary sizes evolves when $r$ varies like a well-identified growth-fragmentation process. We then prove that the same growth-fragmentation process appears when slicing a Brownian disk at height $r$ and considering the perimeters of the resulting connected components.

math.PR↗
arXiv2018

Brownian geometry

We present different continuous models of random geometry that have been introduced and studied in the recent years. In particular, we consider the Brownian map, which is the universal scaling limit of large planar maps in the Gromov-Hausdorff sense, and the Brownian disk, which appears as the scaling limit of planar maps with a boundary. We discuss the construction of these models, and we emphasize the role played by Brownian motion indexed by the Brownian tree.

math.PR↗
arXiv2018

Excursion theory for Brownian motion indexed by the Brownian tree

We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical Itô theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connectedcomponent is itself a continuous tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with branching mechanism $ψ(u)=\sqrt{8/3}\,u^{3/2}$. Furthermore, conditionally on the boundary lengths, the different excursions are independent, and we determine their conditional distribution in terms of an excursion measure $\mathbb{M}_0$ which is the analog of the Itô measure of Brownian excursions. We provide various descriptions of the excursion measure $\mathbb{M}_0$, and we also determine several explicit distributions, such as the joint distribution of the boundary length and the mass of an excursion under $\mathbb{M}_0$. We use the Brownian snake as a convenient tool for defining and analysing the excursions of our tree-indexed Brownian motion.

math.PR↗
arXiv2017

Brownian disks and the Brownian snake

We provide a new construction of the Brownian disks, which have been defined by Bettinelli and Miermont as scaling limits of quadrangulations with a boundary when the boundary size tends to infinity. Our method is very similar to the construction of the Brownian map, but it makes use of the positive excursion measure of the Brownian snake which has been introduced recently. This excursion measure involves a random continuous tree whose vertices are assigned nonnegative labels, which correspond to distances from the boundary in our approach to the Brownian disk. We provide several applications of our construction. In particular, we prove that the uniform measure on the boundary can be obtained as the limit of the suitably normalized volume measure on a small tubular neighborhood of the boundary. We also prove that connected components of the complement of the Brownian net are Brownian disks, as it was suggested in the recent work of Miller and Sheffield. Finally, we show that connected components of the complement of balls centered at the distinguished point of the Brownian map are independent Brownian disks, conditionally on their volumes and perimeters.

math.PR↗
arXiv2017

The harmonic measure of balls in random trees

We study properties of the harmonic measure of balls in typical large discrete trees. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is of order $n$, most of the harmonic measure is supported on a boundary set of size approximately equal to $n^β$, where $β\approx0.78$ is a universal constant. To derive such results, we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree, and we first deal with the case of critical Galton-Watson trees conditioned to have height greater than $n$. An important ingredient of our approach is the analogous continuous model (related to Aldous' continuum random tree), where the dimension of harmonic measure of a level set of the tree is equal to $β$, whereas the dimension of the level set itself is equal to $1$. The constant $β$ is expressed in terms of the asymptotic distribution of the conductance of large critical Galton-Watson trees.

math.PR↗
arXiv2016

Subordination of trees and the Brownian map

We discuss subordination of random compact R-trees. We focus on the case of the Brownian tree, where the subordination function is given by the past maximum process of Brownian motion indexed by the tree. In that particular case, the subordinate tree is identified as a stable Levy tree with index 3/2. As a more precise alternative formulation, we show that the maximum process of the Brownian snake is a time change of the height process coding the Levy tree. We then apply our results to properties of the Brownian map. In particular, we recover, in a more precise form, a recent result of Miller and Sheffield identifying the metric net associated with the Brownian map.

math.PR↗