Growth dichotomy for unimodular random rooted trees

Abért, Miklós and Fraczyk, Mikolaj and Hayes, Ben (2025) Growth dichotomy for unimodular random rooted trees. ANNALS OF PROBABILITY, 53 (5). pp. 1627-1644. ISSN 0091-1798

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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 √ 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.

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