Probability Seminar, Statistical Laboratory

# Probability Seminar, Statistical Laboratory

#### Tuesdays 4.30-5.30 pm, MR 12. Organisers:Nathanaël Berestycki, and Mike Tehranchi. Webpage of the probability group

Tea beforehand at 4pm in the common room, and drinks afterwards at the Punter.

### Lent 2010

 Date Speaker Title Notes 12 January Daniel Remenik (Cornell University) Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations (abstract) 19 January Gregory Miermont (Paris-Sud Orsay). Scaling limits of random planar maps with large faces (abstract) 25 January Marc Lelarge (ENS Paris) Diffusion and cascading behavior in random networks (click here for details) Note unusual day and time: 3pm in MR5 (joint with Networks). 2 February Perla Sousi (Cambridge). Collisions of random walks. (abstract) 9 February Clement Mouhot (CNRS and Cambridge). Quantitative chaos propagation estimates for jump processes (abstract). 16 February Geoffrey Grimmett (Cambridge) Embeddings, entanglement, and percolation (abstract). 23 February Tom Kurtz (University of Wisconsin) Applications of a Markov mapping theorem (abstract). 2 March Simon Harris (University of Bath) Cancelled 5 March Wendelin Werner (Orsay and ENS Paris) Random shapes and Sierpinski-type carpets (abstract). Special colloquium, note special details: Friday 2pm in MR2 9 March Saul Jacka (Warwick) The noisy veto-voter model: a Recursive Distributional Equation on [0,1](abstract)

#### 12 January (Daniel Remenik). Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations. Abstract:

We consider a branching-selection system in $\R$ with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N tends to infinity, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integro-differential equation. We also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether c >= a or c < a, where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener-Hopf equations. This is joint work with Rick Durrett.

#### 19 January (Gregory Miermont). Scaling limits of random planar maps with large faces. Abstract:

We discuss asymptotics of large random maps in which the distribution of the degree of a typical face has a polynomial tail. When the number of vertices of the map goes to infinity, the appropriately rescaled distances from a base vertex can be described in terms of a new random process, defined in terms of a field of Brownian bridges over the so-called stable trees. This allows to obtain weak convergence results in the Gromov-Hausdorff sense for these "maps with large faces", viewed as metric spaces by endowing the set of their vertices with the graph distance. The limiting spaces form a one-parameter family of "stable maps", in a way parallel to the fact that the so-called Brownian map is the conjectured scaling limit for families of maps with faces-degrees having exponential tails. This work takes part of its motivation from the study of statistical physics models on random maps. Joint work with J.-F. Le Gall.

#### 2 February (Perla Sousi). Collisions of random walks. Abstract:

Regarding his 1920 paper proving recurrence of random walks in Z^2, Polya wrote that his motivation was to determine whether 2 independent random walks in Z^2 meet infinitely often. Of course, in this case, the problem reduces to the recurrence of a single random walk in Z^2, by taking differences. Perhaps surprisingly, however, there exist graphs G where a single random walk is recurrent, yet G has the “finite collision property”: two independent random walks in G collide only finitely many times almost surely. Some examples were constructed by Krishnapur and Peres (2004), who asked whether critical Galton-Watson trees conditioned on nonextinction also have this property. In this talk I will answer this question as part of a systematic study of the finite collision property. In particular, for two classes of graphs, wedge combs and spherically symmetric trees, we exhibit a phase transition for the finite collision property when growth parameters are varied. I will state the main theorems and give some ideas of the proofs. This is joint work with Martin Barlow and Yuval Peres.

#### 9 February (Clement Mouhot). Quantitative chaos propagation estimates for jump processes. Abstract:

This talk devoted to a joint work in collaboration with Stephane Mischler about the mean-field limit for systems of indistinguables particles undergoing collision processes. As formulated by [Kac, 1956] this limit is based on the chaos propagation, and we (1) prove and quantify this property for Boltzmann collision processes with unbounded collision rates (hard spheres or long-range interactions), (2) prove and quantify this property \emph{uniformly in time}. This yields the first chaos propagation result for the spatially homogeneous Boltzmann equation for true (without cut-off) Maxwell molecules whose Master equation shares similarities with the one of a Lévy process and the first quantitative chaos propagation result for the spatially homogeneous Boltzmann equation for hard spheres (improvement of the convergence result of [Sznitman, 1984]). Moreover our chaos propagation results are the first uniform in time ones for Boltzmann collision processes (to our knowledge), which partly answers the important question raised by Kac of relating the long-time behavior of a particle system with the one of its mean-field limit. Our results are based on a new method which reduces the question of chaos propagation to the one of proving a purely functional estimate on some generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting non-linear equation (stability estimates).

#### 16 February (Geoffrey Grimmett). Embeddings, entanglement, and percolation. Abstract:

Can there exist a monotone embedding of one infinite random word inside another, with bounded gaps? What can be said about the critical point for the existence of an infinite entangled' set of open edges in the percolation model on the cubic lattice? These two questions are connected through a new type of percolation process, called Lipschitz percolation'. It will be shown how to embed higher-dimensional words, and to obtain the best (so far) lower bound for the entanglement critical point.

#### 23 February (Tom Kurtz). Applications of a Markov mapping theorem. Abstract:

Conditions in terms of generators are given under which the image of a solution of a martingale problem with values in one metric space gives a solution of a martingale problem with values in the image space. Applications of the general result will be discussed including Burke's theorem, uniqueness for stochastic partial differential equations, and limit theorems for branching processes.

#### 5 March (Wendelin Werner). Random shapes and Sierpinski-type carpets. Abstract:

I will discuss various constructions of a class of natural conformally invariant random fractals. This talk is based on joint work with Scott Sheffield, and it will be accessible to non-specialists.

#### 9 March (Saul Jacka). The noisy veto-voter model: a Recursive Distributional Equation on [0,1]. Abstract:

We study a particular example of a recursive distributional equation (RDE) on the unit interval. We first transform the problem to amore tractable one, then identify the invariant distributions, the corresponding basins of attraction" and address the issue of endogeny for the associated tree-indexed problem, making use of an extension of a result of Warren. Along the way, we identify some novel martingales associated with Galton-Watson branching processes.