First-passage percolation and stochastic homogenization

Abstract: First-passage percolation on Z^d was originally introduced to model the spread of fluid in a random porous medium. It a generalization of classical percolation theory, a model for geodesics of a random metric, and a member of the Kardar-Parisi-Zhang (KPZ) universality class for growth models. The study of this universality class is intimately related to random matrices, random tilings and asymptotic representation theory. The main object of study in first-passage percolation is the passage-time T(x), the (random) time it takes for a fluid particle that starts at the origin to reach x, a point on the lattice. The limiting behavior of the time-constant may be viewed as a stochastic homogenization problem for a Hamilton-Jacobi partial differential equation (PDE). We will talk about a variational formula for the limiting PDE, the behavior of its special minimizers --- the so-called Busemann functions --- and their connection to universal behavior in the KPZ class.