Growth dichotomy for unimodular random rooted trees

TL;DR

This paper proves that unimodular random rooted trees with bounded degree have a well-defined growth rate above a critical threshold, using a novel method to analyze the return probabilities of lazy random walks on percolation clusters.

Contribution

It introduces the 2-3-method for establishing the existence of growth exponents and extends the Cohen-Grigorchuk co-growth formula to invariant percolation clusters.

Findings

Growth rate exists above the critical threshold $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,",

The growth of the cluster can be expressed via the limiting exponent of the return probability of the lazy random walk.

Abstract

We show that the growth of a unimodular random rooted tree $(T,o)$ of degree bounded by $d$ always exists, assuming its upper growth passes the critical threshold $\sqrt{d-1}$. This complements Timar's work who showed the possible nonexistence of growth below this threshold. The proof goes as follows. By Benjamini-Lyons-Schramm, we can realize $(T,o)$ as the cluster of the root for some invariant percolation on the $d$-regular tree. Then we show that for such a percolation, the limiting exponent with which the lazy random walk returns to the cluster of its starting point always exists. We develop a new method to get this, that we call the 2-3-method, as the usual pointwise ergodic theorems do not seem to work here. We then define and prove the Cohen-Grigorchuk co-growth formula to the invariant percolation setting. This establishes and expresses the growth of the cluster from the limiting exponent, assuming we are above the critical threshold.