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

Poisat, Julien; Simenhaus, François (2020), A limit theorem for the survival probability of a simple random walk among power-law renewal obstacles, Annals of Applied Probability, 30, 5, p. 2030-2068. 10.1214/19-AAP1551

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Type
Article accepté pour publication ou publié
Date
2020
Journal name
Annals of Applied Probability
Volume
30
Number
5
Publisher
Institute of Mathematical Statistics
Pages
2030-2068
Publication identifier
10.1214/19-AAP1551
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 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 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 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 finite-volume free energy.
Subjects / Keywords
Random walks in random obstacles; polymers in random environments; parabolicAnderson model; survival probability; FKG inequalities; Ray-Knight theorems

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