MARK KAC SEMINAR

Random walk in random environment models are natural extensions of the classical simple random walk model where the transition kernel is modified according to a random environment. Such models have been studied since the 1970’s and are known to exhibit regimes where the behaviour of the random walk is significantly different from the classical model. In particular, the random walk can be subballistic and behave non-diffusively. Such phenomena are typically explained by the occurrence of “trapping”.

I will start this talk by giving an introduction to random walk in random environment models and by mentioning some key results. My focus will be on the case where the random environment itself is evolving dynamically with time. For such models, the mixing properties of the dynamics play a central role. In order to prove asymptotic properties for the random walk (such as LLN and CLT), often strong (uniform) mixing assumptions have been made (such as cone mixing). Recently there have been several advances where these mixing assumptions have been relaxed. These approaches, however, mostly rely on model specific properties of the dynamics and/or are perturbative.

In this talk I will present a new and general approach to tackle this problem which requires certain mixing assumptions strictly weaker than the strong mixing assumptions present in the literature. This approach goes by studying the invariant laws of the “environment as seen from the walker”-process and is based on joint work with Florian Völlering (Bath University). If time allows it, I will also discuss some more concrete examples, in particular, the random walk on the supercritical contact process.

This talk deals with stochastic geometry in connection with convex and integral geometry. More precisely, we use a point process of the Euclidean space as a basic object to generate random convex sets and convex tessellations. This kind of discrete structures arise notably in computational geometry, which may explain the importance of their average-case analysis.

We concentrate on the asymptotic description of the random polytope defined as the convex hull of a binomial or Poisson point process, and we investigate several associated random functionals including the number of extreme points and volume. In their 1963 seminal work, A. Rényi and R. Sulanke were the first to obtain limiting expectations in the planar case and to point out the completely different behaviors of uniform inputs inside a smooth convex body and inside a convex polytope. Though central limit theorems were obtained by M. Reitzner, I. Bárány and V. Vu between 2004 and 2010, there have been relatively few results concerning the asymptotic variances.

Our main contribution is the construction of a scaling limit of the boundary of the random polytope and the use of this limit process to calculate explicit limiting variances. The method applies to uniform points inside a smooth or polytopal convex body and to Gaussian points as well. We emphasize the case of uniform points inside a convex polytope, where the localization of extreme points near the vertices of the polytope plays a central role. If time allows, we will also briefly address similar asymptotic questions for Poisson–Voronoi tessellations.

The main theme of this talk comes from a joint work with J.E. Yukich.