Extremes of local times for simple random walks on symmetric trees

We consider local times of the simple random walk on the $b$-ary tree of depth $n$ and study a point process which encodes the location of the vertex with the maximal local time and the properly centered maximum over leaves of each subtree of depth $r_n$ rooted at the $(n-r_n)$ level, where $(r_n)_{n \geq 1}$ satisfies $\lim_{n \to \infty} r_n = \infty$ and $\lim_{n \to \infty} r_n/n \in [0, 1)$. We show that the point process weakly converges to a Cox process with intensity measure $\alpha Z_{\infty} (dx) \otimes e^{-2\sqrt{\log b}~y}dy$, where $\alpha > 0$ is a constant and $Z_{\infty}$ is a random measure on $[0, 1]$ which has the same law as the limit of a critical random multiplicative cascade measure up to a scale factor. As a corollary, we establish convergence in law of the maximum of local times over leaves to a randomly shifted Gumbel distribution.