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

2025/03/20 by Fu, Wenxin, Hong, Wenming
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.15841

Abstract

In this article, we study the maximal displacement of critical branching random walk in random environment. Let M_n be the maximal displacement of a particle in generation n, and Z_n 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( \dfracM_n√σ n^(3)/(4) |Z_ngt;0) \dcon L(A_Λ), where random variable A_Λ is related to the standard Brownian meander. And there exist some positive constant C₁ and C₂, such that C₁\leqslant\liminf_x→∞x^(2)/(3)¶(Mgt;x) \leqslant \limsup_x→∞ x^(2)/(3)¶(Mgt;x) \leqslant C₂. Compared with the constant environment case (Lalley and Shao (2015)), it revaels that, the conditional limit speed for M_n in random environment (i.e., n^(3)/(4)) is significantly greater than that of constant environment case (i.e., n^(1)/(2)), and so is the tail probability for the 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.

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