##### Department of Mathematics,

University of California San Diego

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### Math 211B - Group Actions Seminar

## Prof. Ben Hayes

#### University of Virginia

## Growth dichotomy for unimodular random rooted trees

##### 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 of the square root of 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.

Host: Brandon Seward

### February 15, 2024

### 10:00 AM

APM 7321

Research Areas

Ergodic Theory and Dynamical Systems****************************