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 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), the barrier is set as χₙ+ε n, where \χₙ\ is a random walk determined by the random environment. Lv & Hong (2024) showed that for almost every L:=ₙ\, the quenched survival probability (denoted by L(ε)) of the particles system will be 0 (resp., positive) when ε≤ 0 (resp., ε>0). In the present paper, we prove that √εlogL(ε) will converge in Probability/ almost surely/ in Lᵖ to an explicit negative constant (depending on the environment) as ε↓ 0 under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case.