Titolo: On the cost of covering a fraction of a macroscopic body by a simple random walk.

Abstract: In this talk we aim at establishing large deviation estimates for the probability that a simple random walk on the Euclidean lattice (d>2) covers a substantial fraction of a macroscopic body. It turns out that, when such rare event happens, the random walk is locally well approximated by random interlacements with a specific intensity, which can be used as a pivotal tool to obtain precise exponential rates. Random interlacements have been introduced by Sznitman in 2007 in order to describe the local picture left by the trace of a random walk on a large discrete torus when it runs up to times proportional to the volume of the torus, and has been since a popular object of study. In the first part of the talk we introduce random interlacements and give a brief account of some results surrounding this object. In the second part of the talk we study the event that random interlacements cover a substantial fraction of a macroscopic body. This allows to obtain an upper bound on the probability of the corresponding event for the random walk. Finally, by constructing a near-optimal strategy for the random walk to cover a macroscopic body, we discuss a matching large deviation lower bound. The talk is based on ongoing work with M. Nitzschner (NYU Courant). 

Martedì 6 Dicembre, ore 14:00
Dipartimento di Matematica, Sapienza Università di Roma, Sala di Consiglio

Speaker: Alberto Chiarini, Università di Padova