An extension of the cogrowth formula to arbitrary subsets of the tree

TL;DR

This paper generalizes the cogrowth formula to compute the exponential decay rates of probabilities for random walks ending in specific subsets of regular and biregular trees, extending previous results to broader cases.

Contribution

It introduces a new formula that relates the decay rates of random walks ending in subsets of trees, generalizing the classical cogrowth formula to arbitrary subsets and biregular trees.

Findings

Derived a formula for exponential decay rates in regular trees.

Extended the formula to biregular trees.

Unified the understanding of random walk probabilities in different tree structures.

Abstract

What is the probability that a random walk in the free group ends in a proper power? Or in a primitive element? We present a formula that computes the exponential decay rate of the probability that a random walk on a regular tree ends in a given subset, in terms of the exponential decay rate of the analogous probability of the non-backtracking random walk. This generalizes the well-known cogrowth formula of Grigorchuk, Cohen and Northshield. We also extend the formula to arbitrary subsets of the biregular tree.