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

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.

16 pages, 1 figure, added a formula for subsets of the biregular tree