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

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.