Bulk deviations for the simple random walk
In this talk we aim at establishing large-deviation estimates for the probability that the average value over a large box of some local observable of the field of occupation times of the simple random walk exceeds a given positive value. When the rare event occurs, we are in the presence of a certain “high density regime” and the random walk is locally well approximated by random interlacements with a slowly varying intensity. This can be used as a pivotal tool to obtain exact exponential rates for the probability of the deviant behaviour. In fact, the proof of the lower bound relies on the introduction of a near optimal strategy via the so called tilted walks --- originally constructed by Li (2017) --- which can be coupled with random interlacements at mesoscopic scales. Importantly, the lower bound matches at leading order the corresponding upper bound derived by Sznitman (2023), and is given in terms of a certain constrained variational problem. As an application, we look into the question of how costly it is for the simple random walk to disconnect an excessive fraction of points from an enclosing box. The talk is based on the joint work with M. Nitzschner (Hong Kong University of Science and Technology).
Area: IS22 - Universality, Stochastic PDEs and Random Growth (Giuseppe Cannizzaro)
Keywords: large deviation, random walk
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