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

Poisat, Julien; Simenhaus, François (2018), A limit theorem for the survival probability of a simple random walk among power-law renewal traps. https://basepub.dauphine.fr/handle/123456789/18486

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155109792211805.pdf (515.7Kb)
Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-01878052
Date
2018
Series title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Pages
36
Metadata
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Author(s)
Poisat, Julien
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Simenhaus, François
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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 −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 power-law 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 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 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 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.
Subjects / Keywords
Random walks in random traps; polymers in random environments; parabolicAnderson model; survival probability; FKG inequalities; Ray-Knight theorems

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