A limit theorem for the survival probability of a simple random walk among power-law renewal traps

TL;DR

This paper establishes a limit theorem for the survival probability of a one-dimensional simple random walk among power-law distributed traps, revealing universal and non-universal features in the asymptotic behavior.

Contribution

It proves a convergence law for the logarithm of the survival probability, involving a universal scaling limit and a variational formula with parameters like trap crossing cost.

Findings

Convergence in law of the rescaled logarithm of survival probability

Emergence of a Poisson point process as the universal limit

Identification of the asymptotic crossing cost parameter

Abstract

We consider a one-dimensional simple random walk surviving among a field of static soft traps : each time it meets a trap the walk is killed with probability 1--e --$\beta$ , where $\beta$ is a positive and fixed parameter. The positions of the traps are sampled independently from the walk and according to a renewal process. The increments between consecutive traps, or gaps, are assumed to have a power-law decaying tail with exponent $\gamma$ > 0. We prove convergence in law for the properly rescaled logarithm of the quenched survival probability as time goes to infinity. The normalization exponent is $\gamma$/($\gamma$ + 2), while the limiting law writes as a variational formula with both universal and non-universal features. The latter involves (i) a Poisson point process that emerges as the universal scaling limit of the properly rescaled gaps and (ii) a function of the parameter $\beta$ that we call asymptotic cost of crossing per trap and that may, in principle, depend on the details of the gap distribution. Our proof suggests a confinement strategy of the walk in a single large gap. This model may also be seen as a (1 + 1)-directed polymer among many repulsive interfaces, in which case $\beta$ corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finite-volume free energy. Along the way we prove a stochastic monotonicity property for the hitting time of the killed random walk with respect to the non-killed one, that could be of interest in other contexts, see Proposition 3.5.