Emilio Corso: Large deviations for random walks induced by isometric group actions

Abstract: Starting with the pioneering work of Furstenberg and Kesten in the sixties, the quest for non-commutative analogues of classical limit theorems in probability theory has gathered continuous attention. We will discuss aspects of the study of random walks driven by actions of discrete countable groups by isometries of metric spaces, with a focus on those possessing certain hyperbolic-like features, which are most clearly on display for actions on Gromov-hyperbolic spaces and for the dynamics of the mapping class group on the Teichmüller space of a closed oriented surface. The emphasis is on large deviations for the process of renormalized distances from the origin: we prove existence of a large deviation principle, with identification of the rate function as an appropriate Fenchel-Legendre transform, for a broad class of isometric group actions encompassing the two aforementioned examples, as well as many others of relevance in geometric group theory and geometric topology. This at once extends recent work of Boulanger, Mathieu, Sert and Sisto, and provides a more streamlined argument for their result. If time permits, quantitative refinements of the main result, afforded by thermodynamic formalism on symbolic spaces in the non-additive regime, will also be presented.