2025/06/11 by Yoshihiro Abe, Abe, Yoshihiro, Marek Biskup +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #05C81 #60G50 #60G70 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2506.09592
openalex publication_date 2025/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a continuous-time simple random walk on a regular rooted tree of depth n in two settings: either the walk is started from a leaf vertex and run until the tree root is first hit or it is started from the root and run until it has spent a prescribed amount of time there. In both cases we show that the extremal process associated with centered square-root local time on the leaves tends, as n→∞, to a decorated Poisson point process with a random intensity measure. While the intensity measure is specific to the local-time problem at hand, the decorations are exactly those for the tree-indexed Markov chain (a.k.a. Branching Random Walk or Gaussian Free Field) with normal step distribution. The proof demonstrates the latter by way of a Lindeberg-type swap of the decorations of the two processes which itself relies on a well-known isomorphism theorem.