Critical percolation of virtually free groups and other tree-like graphs

TL;DR

This paper introduces a method to determine the critical probability for percolation on tree-like graphs, including Cayley graphs of virtually free groups, using branching processes and algebraic computations.

Contribution

It develops a novel approach linking percolation thresholds to multi-type Galton--Watson processes on tree-like structures, enabling explicit calculations of critical probabilities.

Findings

Provides a method to compute $p_c$ using the first-moment matrix.

Shows $p_c$ is algebraic when pieces are finite.

Applies the method to Cayley graphs of virtually free groups.

Abstract

This article presents a method for finding the critical probability $p_c$ for the Bernoulli bond percolation on graphs with the so-called tree-like structure. Such a graph can be decomposed into a tree of pieces, each of which has finitely many isomorphism classes. This class of graphs includes the Cayley graphs of amalgamated products, HNN extensions or general groups acting on trees. It also includes all transitive graphs with more than one end. The idea of the method is to find a multi-type Galton--Watson branching process (with a parameter $p$) which has finite expected population size if and only if the expected percolation cluster size is finite. This provides sufficient information about $p_c$. In particular, if the pairwise intersections of pieces are finite, then $p_c$ is the smallest positive $p$ such that $\operatorname {det}(M-1)=0$, where $M$ is the first-moment matrix of the branching process. If the pieces of the tree-like structure are finite, then $p_c$ is an algebraic number and we give an algorithm computing $p_c$ as a root of some algebraic function. We show that any Cayley graph of a virtually free group (i.e., a group acting on a tree with finite vertex stabilizers) with respect to any finite generating set has a tree-like structure with finite pieces. In particular, we show how to compute $p_c$ for the Cayley graph of a free group with respect to any finite generating set.