Counterexamples for percolation on unimodular random graphs

TL;DR

This paper constructs specific examples of unimodular random graphs that challenge existing conjectures about percolation thresholds and cluster behavior, demonstrating that these conjectures do not hold in the broader class of unimodular graphs.

Contribution

The paper provides the first known counterexamples of unimodular random graphs that disprove two conjectures extending from transitive graphs to unimodular graphs.

Findings

Counterexample with $p_c=p_u$ in nonamenable graphs

Counterexample with $p_c<1$ and infinite cluster at criticality

Disproof of two conjectures by Benjamini and Schramm in unimodular graphs

Abstract

We construct an example of a bounded degree, nonamenable, unimodular random rooted graph with $p_c=p_u$ for Bernoulli bond percolation, as well as an example of a bounded degree, unimodular random rooted graph with $p_c<1$ but with an infinite cluster at criticality. These examples show that two well-known conjectures of Benjamini and Schramm are false when generalised from transitive graphs to unimodular random rooted graphs.