Bernoulli and self-destructive percolation on non-amenable graphs

TL;DR

This paper investigates properties of infinite percolation clusters on non-amenable graphs, showing that removing sites in infinite clusters affects the critical threshold and demonstrating the emergence of infinite clusters with minimal reinforcement.

Contribution

It introduces a novel approach using mass-transport to analyze percolation thresholds on non-amenable graphs and extends understanding of self-destructive percolation processes.

Findings

Critical percolation threshold can be arbitrarily close to the original as p approaches p_c

Removing sites in infinite clusters impacts percolation properties

Infinite clusters can form with minimal reinforcement

Abstract

In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement of infinite percolation clusters. An approach based on mass-transport is adapted to show that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as p approaches p_c. Closely related is the self-destructive percolation process, introduced by J. van den Berg and R. Brouwer, for which we prove that an infinite cluster emerges for any small reinforcement.