Survival Probability of a Random Walk Among a Poisson System of Moving Traps

TL;DR

This paper analyzes the survival probability of a random walk amid moving traps modeled by a Poisson system, deriving asymptotic decay rates in different dimensions and confirming the Pascal principle's role in maximizing survival chances.

Contribution

It provides new proofs and results on the asymptotic decay of survival probabilities for random walks among moving traps, including explicit constants and the validation of the Pascal principle.

Findings

Annealed survival probability decays as e^{- ext{constant} imes t^{1/2}} in 1D

Decay as e^{- ext{constant} imes t / ext{log} t} in 2D

Decay as e^{- ext{constant} imes t} in higher dimensions (d>=3)

Abstract

We review some old and prove some new results on the survival probability of a random walk among a Poisson system of moving traps on Z^d, which can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential. We show that the annealed survival probability decays asymptotically as e^{-\lambda_1\sqrt{t}} for d=1, as e^{-\lambda_2 t/\log t} for d=2, and as e^{-\lambda_d t} for d>= 3, where \lambda_1 and \lambda_2 can be identified explicitly. In addition, we show that the quenched survival probability decays asymptotically as e^{-\tilde \lambda_d t}, with \tilde \lambda_d>0 for all d>= 1. A key ingredient in bounding the annealed survival probability is what is known in the physics literature as the Pascal principle, which asserts that the annealed survival probability is maximized if the random walk stays at a fixed position. A corollary of independent interest is that the expected cardinality of the range of a continuous time symmetric random walk increases under perturbation by a deterministic path.