A LIMIT THEOREM FOR THE SURVIVAL PROBABILITY OF A SIMPLE RANDOM WALK AMONG POWER-LAW RENEWAL OBSTACLES

• 成果类型: Article
• 署名作者: Poisat, Julien; Simenhaus, Francois
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite PSL; Universite Paris-Dauphine
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1551
• 发表日期: 2020
• 页码: 2030-2068
• 关键词: polymer

摘要

We consider a one-dimensional simple random walk surviving among a field of static soft obstacles: each time it meets an obstacle the walk is killed with probability 1 - e(-beta), where beta is a positive and fixed parameter. The positions of the obstacles are sampled independently from the walk and according to a renewal process. The increments between consecutive obstacles, 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 nonuniversal 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 obstacle 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.