Phase transition for the asymptotic entropy of branching random walks on groups

2026/07/22 by Jeremie Brieussel, Robin Kaiser, Martin Klötzer +1
#math.PR

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Abstract

We consider supercritical branching random walks (BRW) on infinite countable groups G and we prove that the asymptotic entropy of the empirical distributions of the BRW has a phase transition at ρ_* = e^h(μ), where h(μ) is the asymptotic entropy of the underlying random walk on G with step distribution μ. Below this value ρ_*, the asymptotic empirical entropy of BRW equals the logarithm of the exponential growth rate of the population. Above this value, it is constantly equal to the asymptotic entropy of the underlying random walk. In particular, this answers questions from Kaimanovich-Woess [MR4663513, Section 6.3] about the existence and the behavior of the asymptotic entropy.

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