Abstract:
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 a fixed probability. 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. We prove convergence in law for the properly rescaled logarithm of the quenched survival probability, as time goes to infinity. 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 quantity that we call "asymptotic cost of crossing per trap" and that may, in principle, depends 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 the survival probability corresponds to the partition function and its logarithm to the finite-volume free energy. This is joint work with François Simenhaus (CEREMADE).
Biography:
Julien Poisat is a Maître de conférences at CEREMADE (Université Paris-Dauphine) working in the field of probability theory and statistical mechanics. He obtained his Ph.D. in 2012 under the supervision of Nadine Guillotin-Plantard, at Université Lyon 1. Between 2012 and 2014 he was a Postdoctoral Researcher in Frank den Hollander’s group at Leiden University. He is mainly interested in the study of phase transitions (such as pinning and folding) in homogeneous and disordered polymer models, as well as related models of interacting random walks.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai