Convergence in law for the capacity of the range of a critical branching random walk

2022/03/07 by Tianyi Bai, Bai, Tianyi, Yueyun Hu +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2203.03188

openalex publication_date 2022/03/07 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

Let R_n be the range of a critical branching random walk with n particles on \mathbb Z^d, which is the set of sites visited by a random walk indexed by a critical Galton--Watson tree conditioned on having exactly n vertices. For d∈\3, 4, 5\, we prove that n^-\fracd-24 \mathttcap^(d)(R_n), the renormalized capacity of R_n, converges in law to the capacity of the support of the integrated super-Brownian excursion. The proof relies on a study of the intersection probabilities between the critical branching random walk and an independent simple random walk on \mathbb Z^d.

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