Survival of a long random string among hard Poisson traps

TL;DR

This paper investigates the asymptotic behavior of the survival probability of a stochastic heat equation-driven polymer in a Poisson trap environment, focusing on large spatial domain limits and providing bounds for survival rates.

Contribution

It extends previous work by analyzing large spatial domain asymptotics of survival probability for a stochastic heat equation polymer in Poisson traps, deriving bounds dependent on domain size.

Findings

Survival probability decays exponentially with rate proportional to J^{d/(d+2)}.

Bounds depend on the spatial domain size J and time T.

Provides upper and lower bounds for the annealed survival probability.

Abstract

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 $T >0$.