A limit theorem for the survival probability of a simple random walk among powerlaw renewal obstacles
Poisat, Julien; Simenhaus, François (2019), A limit theorem for the survival probability of a simple random walk among powerlaw renewal obstacles, The Annals of Applied Probability
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Article accepté pour publication ou publiéExternal document link
https://hal.archivesouvertes.fr/hal01878052Date
2019Journal name
The Annals of Applied ProbabilityPublisher
Institute of Mathematical Statistics
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Poisat, JulienCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Simenhaus, François
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
We consider a onedimensional 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 −β, where β 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 powerlaw decaying tail with exponent γ > 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 γ/(γ + 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 β 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 β corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finitevolume free energy.Subjects / Keywords
Random walks in random obstacles; polymers in random environments; parabolicAnderson model; survival probability; FKG inequalities; RayKnight theoremsRelated items
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