Boolean Percolation on Doubling Graphs

TL;DR

This paper investigates conditions under which percolation does not occur in Boolean models on doubling graphs, providing new insights into percolation thresholds and ergodicity in these structures.

Contribution

It introduces new sufficient conditions for non-percolation and ergodicity on doubling graphs, expanding understanding of percolation behavior in these metric spaces.

Findings

Identifies conditions for absence of percolation on certain graphs.

Provides criteria for ergodicity of the Boolean percolation model.

Applies results to three specific families of graphs.

Abstract

We consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.