On the maximal displacement of critical branching random walk in random environment

Wenxin Fu and Wenming Hong

Stochastic Processes and their Applications, 2026, vol. 199, issue C

Abstract: In this article, we consider the maximal displacement of critical branching random walk in random environment. Let Mn be the maximal displacement of a particle in generation n, and Zn be the total population in generation n, M be the rightmost point ever reached by the branching random walk. Under some reasonable conditions, we prove a conditional limit theorem: L(σ−1/2n−3/4Mn|Zn>0)⟹L(AΛ), where random variable AΛ is related to a standard Brownian meander. And there exist some positive constant C1 and C2, such that C1⩽lim infx→∞x−2/3P(M>x)⩽lim supx→∞x−2/3P(M>x)⩽C2. Compared with the constant environment case (Lalley and Shao (2015)), it reveals that, the conditional limit speed for Mn in random environment (i.e., n3/4) is significantly greater than that of constant environment case (i.e., n1/2), and so is the tail probability for M (i.e., x−2/3 vs x−2). Our method is based on the path large deviation for the reduced critical branching random walk in random environment.

Keywords: Branching random walk; Random environment; Maximal displacement; Large deviation principle; Reduced processes (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1016/j.spa.2026.104973

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