Invariance under quasi-isometries of subcritical and supercritical behaviour in the Boolean model of percolation

TL;DR

This paper proves that the subcritical and supercritical phases of the Poisson Boolean percolation model are invariant under mm-quasi-isometries in various metric spaces, including manifolds, Lie groups, and Cayley graphs.

Contribution

It establishes the invariance of percolation phases under quasi-isometries across diverse metric spaces, extending understanding of phase transitions in geometric probability models.

Findings

Invariance of phases under mm-quasi-isometries in Polish metric spaces.

Existence of subcritical phase in Gromov spaces with bounded growth.

Application to percolation in Riemannian manifolds, Lie groups, and Cayley graphs.

Abstract

In this work we study the Poisson Boolean model of percolation in locally compact Polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. In other words, we prove that if the Poisson Boolean model of percolation is subcritical or supercritical (or exhibits phase transition) in a metric space M which is mm-quasi-isometric to a metric space N, then these phases also exist for the Poisson Boolean model of percolation in N. Then we apply these results to understand the phenomenon of phase transition in a large family of metric spaces. Indeed, we study the Poisson Boolean model of percolation in the context of Riemannian manifolds, in a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the existence of a subcritical phase in Gromov spaces with bounded growth at some scale.