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Résumé :We consider a branching random walk in dimension 1 in which particles are killed when they go below zero. We prove a conjecture of Aldous on the tail distribution of the total progeny Z. In the subcritical case, we confirm that Z has a power law tail. In the critical case, we show that Z has a tail distribution of order 1/n log^2(n). The tools used in the proof are the study of the tail distribution of the maximum of a killed branching random walk, and Yaglom-type theorems on particles that exceed a high level. This is joint work with Yueyun Hu and Olivier Zindy.
Résumé : This is a joint work with Pietro Caputo (Roma 3) and Djalil Chafaï (Paris Est). Consider an n x n random matrix whose entries are i.i.d. complex random variables in the domain of attraction of an alpha-stable law, with 0< alpha <2. We will explain the connection between this type of random matrices and the Poisson Weighted Infinite Tree introduced by David Aldous in 1992 in the context of probabilistic combinatorial optimization.
Résumé : Consider a Poisson Point Process P in the plane (with Lebesgue intensity measure). A very natural object that can be defined on this set of points is the so called Minimal Spanning Tree (MST). In some sense this tree minimizes the sum of its euclidean edge lengths among all possible spanning trees on P. The MST was studied by David Aldous in various geometrical contexts going from the euclidean case (for example the MST on a PPP on R^d) to the "mean field" case (for example the MST on complete graphs). An important aspect of David Aldous' work on the MST (and more generally on random trees) is to describe what is the typical "geometry" of these random trees.
In this talk, I will describe a joint work with G. Pete and O. Schramm which is exactly in this spirit: we prove that the MST defined on the triangular lattice T (which is a variant of the above euclidean MST in the plane) has a unique scaling limit as the mesh of the triangular lattice goes to zero. I will give some intriguing properties of this limiting planar random tree. Furthermore, in the same way as critical Z^2 and critical site percolation on T are conjectured to be in the same universality class, the euclidean MST and the lattice MST should also be in the same "universality class". In particular, the limiting object we obtain should also describe macroscopic properties of the above Euclidean MST on the plane.
Résumé : We will discuss several aspects of David Aldous' contributions to probability theory, including random trees, and especially the continuum random tree (CRT), Brownian triangulations of the disk, asymptotics for the Erdös-Rényi random graph in the critical window, coalescence theory and random mappings. We will also describe some of the subsequent work, which has been much inspired by Aldous' papers.
Résumé : We will discuss joint work with Nicolas Curien in which we study existence and uniqueness of certain random hyperbolic triangulations of the disk.