Subcritical regimes in some models of continuum percolation

TL;DR

This paper investigates conditions under which certain continuum percolation models do not exhibit percolation, focusing on multiscale Boolean models and stable marriage models involving Poisson processes.

Contribution

It provides sufficient conditions for the absence of percolation in two specific continuum percolation models, extending understanding of their non-percolating regimes.

Findings

Established general conditions for non-percolation.

Analyzed multiscale Boolean percolation model.

Studied stable marriage percolation model.

Abstract

We consider some continuum percolation models. We are mainly interested in giving some sufficient conditions for absence of percolation. We give some general conditions and then focuse on two examples. The first one is a multiscale percolation model based on the Boolean model. It was introduced by Meester and Roy and subsequently studied by Menshikov, Popov and Vachkovskaia. The second one is based on the stable marriage of Poisson and Lebesgue introduced by Hoffman, Holroyd and Peres and whose percolation properties have been studied by Freire, Popov and Vachkovskaia.