2009/12/08 by Ming Fang, Fang, Ming, Ofer Zeitouni +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #60G50 #60J80 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60G50 #msc:60J80
paper · pdf · doi:10.48550/arxiv.0912.1392
arxiv created 2009/12/08 · openalex publication_date 2009/12/08 · arxiv updated 2010/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbbT denote a rooted b-ary tree and let \Sv\_v∈ \mathbbT denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function Λ(⋅). Let mn denote the minimum of the variables Sv over all vertices at the nth generation, denoted by \mathbbDn. Under mild conditions, mn/n converges almost surely to a constant, which for convenience may be taken to be 0. With Sv=max\Sw:\rm w is on the geodesic connecting the root to v\, define Ln=min_v∈ \mathbbDn Sv. We prove that Ln/n1/3 converges almost surely to an explicit constant l0. This answers a question of Hu and Shi.