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

We consider supercritical branching random walks (BRW shortly) on infinite countable groups $G$ and we prove that the asymptotic entropy of the empirical distributions of the BRW has a phase transition at $\rho_* = \e^{h(\mu)}$, where $h(\mu)$ is the asymptotic entropy of the underlying random walk on $G$ with step distribution $\mu$.

Below this value $\rho_*$, 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.