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