Modul: MAT971 Stochastische Prozesse

## Spread of infection by random walks - Multi-scale percolation along a Lipschitz surface

Dr. Peter Gracar talk

Speaker invited by: Prof. Dr. Jean Bertoin

**Date:** 07.11.18 **Time:** 17.15 - 18.15 **Room:** ETH HG G 43

A conductance graph on $\mathbb{Z}^d$ is a nearest-neighbor graph where all of the edges have positive weights assigned to them. We first consider a point process of particles on the nearest neighbour graph $(\mathbb{Z}^d,E)$ and show some known results about the spread of infection between particles performing continuous time simple random walks. Next, we extend consider the case of uniformly elliptic random graphs on $\mathbb{Z}^d$ and show that the infection spreads with positive speed also in this more general case. We show this by developing a general multi-scale percolation argument using a two-sided Lipschitz surface that can also be used to answer other questions of this nature.

Joint work with Alexandre Stauffer.