MARK KAC SEMINAR

In the classical model of first passage percolation on Z^d, edges are assigned iid random lengths taking values in [0,∞), which endows the graph Z^d with a random distance. The shape theorem ensures then that the ball of radius n for this random distance asymptotically looks like a ball of radius n for a certain norm on R^d, depending on the distribution of the length of an edge.

In this talk, we will study an extension of this model, by allowing the random length to be infinite (with a not too large probability). This is equivalent to studying first-passage percolation on the random graph given by an infinite cluster of supercritical Bernoulli percolation. We will discuss the existence of a shape theorem in this context, and the continuity properties of the norm giving the asymptotic shape with respect to the law of the length of an edge.

This is a work in collaboration with Olivier Garet and Marie Théret.

Given a finite weighted oriented graph, we study a certain probability measure on the set of spanning rooted oriented forests of the graph. As I will explain, this object has some similarities and connections with the so called random cluster model. I will give an overview of the results we proved and on some related work in progress.

In particular, we prove that the set of roots sampled from this measure is a determinantal process, characterized by a possibly non-symmetric kernel with complex eigenvalues. We further derive several results relating this measure to the Markov process associated with the starting graph, to the spectrum of its generator and to hitting times of subsets of the graph.

This is based on a recent joint work with Alexandre Gaudillière.