Survival of a long random string among hard Poisson traps

In [AJM26], we gave large-time asymptotic bounds on the annealed survival probability of a moving polymer taking values in ${\mathbb R}^d, d \geq 1$. This polymer is a solution of a stochastic heat equation driven by additive spacetime white noise on $[0,T] \times [0,J]$, in an environment of Poisson traps. For fixed $J$, the annealed survivial probability decays exponentially with rate proportional to $T^{d/(d+2)}$. In this work we examine the large $J$ asymptotics of the annealed survival probability for any fixed time $T>0$. We prove upper and lower bounds for the annealed survival probability in the cases of hard obstacles. Our bounds decay exponentially with rate proportional to $J^{d/(d+2)}$. The exponents also depend on time $T >0$.