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October 2020 A limit theorem for the survival probability of a simple random walk among power-law renewal obstacles
Julien Poisat, François Simenhaus
Ann. Appl. Probab. 30(5): 2030-2068 (October 2020). DOI: 10.1214/19-AAP1551

Abstract

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.

Citation

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Julien Poisat. François Simenhaus. "A limit theorem for the survival probability of a simple random walk among power-law renewal obstacles." Ann. Appl. Probab. 30 (5) 2030 - 2068, October 2020. https://doi.org/10.1214/19-AAP1551

Information

Received: 1 May 2019; Revised: 1 November 2019; Published: October 2020
First available in Project Euclid: 15 September 2020

MathSciNet: MR4149522
Digital Object Identifier: 10.1214/19-AAP1551

Subjects:
Primary: 60K35, 60K37

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 5 • October 2020
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