Asymptotic behavior for the quenched survival probability of a supercritical branching random walk in random environment with a barrier

TL;DR

This paper studies the asymptotic behavior of the quenched survival probability of a supercritical branching random walk in a random environment with a barrier, revealing precise convergence rates as the barrier parameter approaches zero.

Contribution

It extends previous results to the random environment case, providing explicit asymptotic convergence of the survival probability near the critical barrier.

Findings

 rac{ ext{log} ho_ ext{L}( ext{varepsilon})}{ ext{sqrt} ext{varepsilon}} o ext{constant} < 0 as ext{varepsilon} o 0

The convergence holds in probability, almost surely, and in L^p under certain conditions

The results generalize earlier work from deterministic to random environments.

Abstract

We introduce a random barrier to a supercritical branching random walk in an i.i.d. random environment $\{\mathcal{L}_n\}$ indexed by time $n,$ i.e., in each generation, only the individuals born below the barrier can survive and reproduce. At generation $n$ ($n\in\mathbb{N}$), the barrier is set as $\chi_n+\varepsilon n,$ where $\{\chi_n\}$ is a random walk determined by the random environment. Lv \& Hong (2024) showed that for almost every $\mathcal{L}:=\{\mathcal{L}_n\},$ the quenched survival probability (denoted by $\varrho_{\mathcal{L}}(\varepsilon)$) of the particles system will be 0 (resp., positive) when $\varepsilon\leq 0$ (resp., $\varepsilon>0$). In the present paper, we prove that $\sqrt{\varepsilon}\log\varrho_\mathcal{L}(\varepsilon)$ will converge in Probability/ almost surely/ in $L^p$ to an explicit negative constant (depending on the environment) as $\varepsilon\downarrow 0$ under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case.