A LIMIT THEOREM FOR THE SURVIVAL PROBABILITY OF A SIMPLE RANDOM WALK AMONG POWER-LAW RENEWAL OBSTACLES
Résumé
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 $1-e^{-\gb}$, where $\gb$ 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 $\gga/(\gga+2)$, 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 $\gb$ that we call {\it 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 $\gb$ corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finite-volume free energy.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)
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