Statistics and Stochastic Seminar

We consider a random connection model (RCM) on a general space driven by a Poisson process whose intensity measure is scaled by a parameter $t\ge 0$. An important special case is the stationary marked RCM (in Euclidean space), containing the Boolean model with general compact grains and the so-called weighted RCM as special cases. We say that the infinite clusters are deletion stable if the removal of a Poisson point cannot split a cluster in two or more infinite clusters. We prove that this stability together with a natural irreducibility assumption implies uniqueness of the infinite cluster.  We then show that the infinite clusters of the stationary marked RCM are deletion stable. It follows that an irreducible stationary marked RCM can have at most one infinite cluster which extends and unifies several results in the literature. An important ingredient of our proofs are differentiability and convexity properties of the cluster density which are of interest in their own right.

The talk is based on recent joint work with Mikhail Chebunin. Some of the main ideas come from a seminal paper by Aizenman, Kesten and Newman (1987), treating discrete percolation models.

Zoom meeting link

Password: 499804

This seminar is part of the Probability Victoria seminar series.