AI 中文总结
研究无限可数群上超临界分支随机游走渐近熵的相变，证明其在\(\rho_* = \e^{h(\mu)}\)处发生相变，低于此值时渐近经验熵与种群指数增长率对数有关，高于此值时恒等于基础随机游走渐近熵，回答了相关问题。
AI 中文摘要
我们考虑无限可数群\(G\)上的超临界分支随机游走（简称BRW），证明了BRW经验分布的渐近熵在\(\rho_* = \e^{h(\mu)}\)处有相变，其中\(h(\mu)\)是\(G\)上具有步分布\(\mu\)的基础随机游走的渐近熵。低于\(\rho_*\)时，BRW的渐近经验熵等于种群指数增长率的对数。高于此值时，它恒等于基础随机游走的渐近熵。特别地，这回答了Kaimanovich-Woess [MR4663513，第6.3节]关于渐近熵的存在性和行为的问题。
英文摘要
We consider supercritical branching random walks (BRW) on 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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