Stochastic Analysis Seminar

The vacant set is the set of vertices not visited by a random walk on a graph G before a given time T. In the talk, I will discuss properties of this random subset of the graph, the phase transition conjectured in its connectivity properties (in the `thermodynamic limit'

when the graph grows), and the relation of the problem to the random interlacement percolation.  I will then concentrate on the case when G is a large-girth expander or a random regular graph, where the conjectured phase transition (and much more) can be proved.

We define a notion of discrete Ricci curvature for a metric measure space by looking at whether "small balls are closer than their centers are". In a Riemannian manifolds this gives back usual Ricci curvature up to scaling. This definition is very easy to apply in a series of examples such as graphs (eg the discrete cube has positive curvature). We are able to generalize several Riemannian theorems in positive curvature, such as concentration of measure and the log-Sobolev inequality. This definition also allows to prove new theorems both in the Riemannian and discrete case: for example improved bounds on spectral gap of the Laplace-Beltrami operator, and fast convergence results for some Markov Chain Monte Carlo methods

Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they have also significant geometrical applications. Particular cases of planar maps are p-angulations, where each face (meaning each component of the complement of edges) has exactly p adjacent edges. Random planar maps have been used in theoretical physics, where they serve as models of random geometry.Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces.More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all p-angulations with n vertices. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n to the power -1/4. Both in the case p=3 and when p>3 is even, we prove that the resulting random metric spaces converge as n tends to infinity to a universal object called the Brownian map. This convergence holds in the sense of the Gromov-Hausdorff distance between compact metric spaces. In the particular case of triangulations (p=3), this solves an open problem stated by Oded Schramm in his 2006 ICM paper. As a key tool, we use bijections between planar maps and various classes of labeled trees

To identify the Martin boundary for a transient Markov chain with Green's function G(x,y), one has to identify all possible limits Lim G(x,y_n)/G(0,y_n) with y_n "tending to infinity". For homogeneous random walks, these limits are usually obtained from the exact asymptotics of Green's function G(x,y_n). For non-homogeneous random walks, the exact asymptotics af Green's function is an extremely difficult problem. We discuss several examples where Martin boundary can beidentified by using large deviation technique. The minimal Martin boundary is in general not homeomorphic to the "radial"  compactification obtained by Ney and Spitzer for homogeneous random walks in Z^d : convergence of a sequence of points y_n toa point on the Martin boundary does not imply convergence of the sequence y_n/|y_n| on the unit sphere. Such a phenomenon is a consequence of non-linear optimal large deviation trajectories.

We will consider a branching Brownian motion where particles have a drift $-\rho$, binary branch at rate $\beta$ and are killed if they hit the origin. This process is supercritical  if $\beta>\rho^2/2$ and we will discuss the survival probability in the regime as criticality is approached. (Joint work with Elie Aidekon)

We treat the first n terms of general orthogonal series evolving with n as the partial sum process, and proved that under Menshov-Rademacher condition, the partial sum process can be enhanced into a geometric 2-rough process. For Fourier series, the condition can be improved, with an equivalent condition on limit function identified.

I will report joint work with Wolfhard Hansen, Tomasz Jakubowski, Sebastian Sydor and Karol Szczypkowski on perturbations of semigroups and integral kernels, ones which produce comparable semigroups and integral kernels.