A limit theorem for the survival probability of a simple random walk among powerlaw renewal traps
Poisat, Julien; Simenhaus, François (2018), A limit theorem for the survival probability of a simple random walk among powerlaw renewal traps. https://basepub.dauphine.fr/handle/123456789/18486
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Document de travail / Working paperExternal document link
https://hal.archivesouvertes.fr/hal01878052Date
2018Series title
Cahier de recherche CEREMADE, Université ParisDauphinePages
36
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Show full item recordAuthor(s)
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 traps : each time it meets a trap the walk is killed with probability 1−e −β , where β 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 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 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 β corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finitevolume free energy. Along the way we prove a stochastic monotonicity property for the hitting time of the killed random walk with respect to the nonkilled one, that could be of interest in other contexts, see Proposition 3.5.Subjects / Keywords
Random walks in random traps; polymers in random environments; parabolicAnderson model; survival probability; FKG inequalities; RayKnight theoremsRelated items
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