In this talk we will consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on N vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy roughly N^alpha vertices, for some 0 < alpha < 1. The value of alpha is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. These results are in sharp contrast with the picture in the lattice case.
This is a joint work with Yael Dekel, Elchanan Mossel and Yuval Peres.

Abstract: I'll talk about the recently discovered strong negative dependence properties of the symmetric exclusion process, a model of non-intersecting random walkers. The negative dependence theory gives a simple way to show central limit theorems for the bulk motion of particles. Our results are general enough to deal with non-equilibrium systems of particles with inhomogeneous hopping rates.

Energy of the finite-volume ground state of a random Schroedinger operator is studied in the limit as the volume increases. We relate its fluctuations to a classical probability problem---extreme statistics of IID random variables---and describe the detailed behavior of its distribution. Surprisingly, the distributions do not converge---presence of two scales in the system leads to a chaotic volume dependence. A possible application to a sharp estimate of the Lifshits tail will be mentioned. The work presented is done jointly with Michael Bishop.

The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers, particle systems, matrices and stochastic PDEs, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian.