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.

### Michaelmas 2010

 Date Speaker Title Notes 12 October Ronnie Sircar (Princeton) Stochastic Differential Games and Applications to Energy and Consumer Goods Markets (abstract) 19 October Christophe Garban (CNRS, Lyon) Near-critical scaling limits (abstract) 26 October Bernard Derrida (ENS Paris) Statistics at the tip of a branching random walk, and simple models of evolution with selection (abstract) 2 November Arnab Sen (Cambridge) Coalescing systems of non-Brownian particles (abstract). 4 November Fabrice Debbasch (Paris VI) Relativistic stochastic processes (abstract). Note special day and time: 4pm 9 November Stephanie Jacquot (Cambridge) Bulk scaling limit of the Laguerre ensemble (abstract). 16 November No seminar that day. 23 November Ismael Bailleul (Cambridge) Lifetime of relativistic diffusions (abstract). 26 November Stephane Boucheron (Paris VII) Concentration inequalities by the entropy method, variations (details) Joint with statistics 30 November Michael Keane (Wesleyan University) TBA (abstract) Talk at 2 pm.

#### 12 Oct, Ronnie Sircar. Stochastic Differential Games and Applications to Energy and Consumer Goods Markets.

We discuss Cournot and Bertrand models of oligopolies, first in the context of static games and then in dynamic models. The static games, involving firms with different costs, lead to questions of how many competitors actively participate in a Nash equilibrium and how many are sidelined or blockaded from entry. The dynamic games lead to systems of nonlinear partial differential equations for which we discuss asymptotic and numerical approximations. Applications include competition between energy producers in the face of exhaustible resources such as oil (Cournot); and markets for substitutable consumer goods (Bertrand). Joint work with Chris Harris, Sam Howison and Andrew Ledvina.

#### 19 Oct, Christophe Garban. Near-critical scaling limits.

Consider percolation on the triangular grid in the plane. The scaling limit (as n goes to infinity) of a critical rescaled percolation \omega_{p_c}^n on the triangular grid of mesh 1/n is a very rich probabilistic object (it is the analog of the Brownian motion for the Random walk), and is now well understood thanks to the works of Smirnov and Schramm.

The purpose of a joint work with Gabor Pete and Oded Schramm is to visualize'' the phase transition which occurs at p_c, from the perspective of the scaling limit. A first natural attempt in this direction is to consider the scaling limit as n goes to infinity of non-critical rescaled percolations \omega_p^n (with p \neq p_c). Unfortunately, such scaling limits are "degenerate". In order to obtain non-trivial "off-critical" scaling limits, $p$ and $n$ need to be rescaled accordingly.

I will describe in this talk what the natural renormalization is if one wishes to observe the emergence of an infinite cluster, seen from the continuous. The main result in joint work with G. Pete and O. Schramm is that "up to scaling", there is a unique near-critical scaling limit. This near-critical limit is not conformally invariant anymore, but one can nevertheless give a precise description of its "conformal defect".

#### 26 October, Bernard Derrida. Statistics at the tip of a branching random walk, and simple models of evolution with selection.

The statistics at the tip of a branching random walk can be studied using the Fisher-KPP equation. The whole limiting measure, at the tip, can be understood in terms of the way the delay of a traveling wave solution of the F-KPP equation depends on its initial condition. Several analytical properties of the distribution of the distances between the rightmost particles can be predicted. This work was motivated by the study of the statistical properties of genealogies of evolving populations under selection.

#### 2 Nov. Arnab Sen. Coalescing systems of non-Brownian particles

A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the starting set is compact. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We show that Arratia's conclusion is valid for Brownian motions on the Sierpinski gasket and for stable processes on the real line with stable index greater than one. Joint work with Steve Evans and Ben Morris.

#### 4 Nov. Fabrice Debbasch. Relativistic stochastic processes.

Modelling irreversible phenomena in relativistic continuous media is still the object of active research. The simplest irreversible phenomenon is particle diffusion and the simplest models of particle diffusions are stochastic processes. Relativistic stochastic processes first appeared in the physics literature 13 years ago. I will explain how these processes are constructed and how they differ from their traditionnal Galilean counterparts. I will also present the hydrodynamic limit of these processes and discuss what it teaches us on the very notion of relativistic dissipative hydrodynamics. Possible cosmological applications will also be addressed briefly.

#### 9 Nov. Stephanie Jacquot. Bulk scaling limit of the Laguerre ensemble.

Random matrix theory has found many applications in physics, statistics and engineering since its inception. The eigenvalues of random matrices are often of particular interest. The standard technique for studying local eigenvalue behavior of a random matrix distribution involves the following steps. We first choose a family of n x n random matrices which we translate and rescale in order to focus on a particular region of the spectrum, and then we let n \to\infty. When this procedure is performed carefully, the limiting eigenvalue behavior often falls into one of three classes: soft edge, hard edge or bulk. In the world of random matrices, three ensembles are of particular interest: the Hermite, Laguerre and Jacobi \beta-ensembles. In this talk I will present a joint work with Benedek Valkó. We consider the \beta-Laguerre ensemble, a family of distributions generalizing the joint eigenvalue distribution of the Wishart random matrices. We show that the bulk scaling limit of these ensembles exists for all \beta > 0 for a general family of parameters and it is the same as the bulk scaling limit of the corresponding \beta-Hermite ensemble.

#### 16 Nov, Ismael Bailleul. Lifetime of relativistic diffusions.

Relativistic diffusions are models of random motion in spacetime of an object moving with a speed less than the speed of light. These processes are the Lorentzian analogues of Brownian motion in a Riemannian context. In so far as they are defined in purely geometric terms, it is very likely that part (or all?) of the geometry of the ambient spacetime may be recovered from the probablistic behaviour of these processes. In a Riemannian setting, this probabilistic view on geometry is well-illustrated by Weyl and Pleyel formulas for the heat kernel of Brownian motion where local and global informations about the geometry appear.