Thematic Seminar: Probability Theory: Rare Behavior in Models of Random Geometry

Seminar | January 19 | 4:10-5 p.m. | 740 Evans Hall

 Shirshendu Ganguly, University of California, Berkeley

 Department of Mathematics

Models of random geometry have long been investigated in contexts such as the internet, fluid flow in porous media, and interface dynamics in statistical physics. To develop a refined understanding of such models, one often needs to study not only typical fluctuation theory but also the realm of atypical events. In this talk we describe such a program for two classical models of random geometry: percolation on the complete graph and random distortions of the Euclidean lattice. In particular, we will consider the large deviations behavior of the count of certain local structures in a sparse random network, and geodesics in a random metric space. The random geometry associated to typical instances of these rare events is an important topic of inquiry: this geometry can involve merely local structures, or more global ones. We will discuss some recent results concerning such phenomena, and connections to other areas of mathematics including random matrix theory, information theory, and algebraic and extremal combinatorics.