This analysis extends the cogrowth formula to arbitrary subsets of trees, revealing decay rates of random walks in these contexts.
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.
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Doron Puder (2026) studied this question.
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