Capacity of the range of branching random walks in low dimensions

2021/04/24 by Tianyi Bai, Bai, Tianyi, Yueyun Hu +1
Mathematics · Physics and Astronomy · #60J65 #60J80 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · doi:10.48550/arxiv.2104.11898

openalex publication_date 2021/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider a branching random walk (V_u)_{u∈ \mathcal T^IGW} in \mathbb Z^d with the genealogy tree \mathcal T^IGW formed by a sequence of i.i.d. critical Galton-Watson trees. Let R_n be the set of points in \mathbb Z^d visited by (V_u) when the index u explores the first n subtrees in \mathcal T^IGW. Our main result states that for d∈ \3, 4, 5\, the capacity of R_n is almost surely equal to n^{(d-2)/(2)+o(1)} as n → ∞.

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• Capacity of the range of tree-indexed random walk
• Branching capacity of a random walk range
• Range and critical generations of a random walk on Galton-Watson trees
• Branching capacity of a random walk in \mathbb Z⁵
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