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

Julien Poisat (1) , François Simenhaus (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Domaines: Probabilités [math.PR]

Julien Poisat, François Simenhaus. A limit theorem for the survival probability of a simple random walk among power-law renewal obstacles. The Annals of Applied Probability, In press, ⟨10.1214/19-AAP1551⟩. ⟨hal-01878052v2⟩