Probability

We will discuss non-Hermitian random matrix models, namely the universality problem for local eigenvalue statistics. The main result is universality in the bulk (i.e., away from the edge of the limiting spectrum) for complex eigenvalues of real non-symmetric matrices with i.i.d. entries. The…

A nodal domain of a Laplacian eigenvector of a graph is a maximal connected component where it does not change sign. Sparse random regular graphs have been proposed as discrete toy models of "quantum chaos", and it has accordingly been conjectured by Y. Elon and experimentally observed by Dekel…

We study the distribution of the maximum gap size in one-dimensional hard-core models. First, we sequentially pack rods of length 1 into an interval of length L at random, subject to the hard-core constraint that rods do not overlap. We find that in a saturated packing, with high probability…

We study the distribution of the maximum gap size in one-dimensional hard-core models. First, we sequentially pack rods of length 1 into an interval of length L at random, subject to the hard-core constraint that rods do not overlap. We find that in a saturated packing, with high probability…

For many random graph models, the analysis of a related birth process suggests local sampling algorithms for the size of, e.g., the giant connected component, the k-core, the size and probability of an epidemic outbreak, etc. In this talk, I consider the question of when these algorithms are…

I will talk about parallelization of sampling algorithms. The main focus of the talk will be a new result, where we show how to speed up sampling from an arbitrary distribution on a product space [q]^n, given oracle access to conditional marginals. Our algorithm takes roughly n^{2/3} polylog(n,…

A striking phenomenon in probability theory is universality, where different probabilistic models produce the same large-scale or long-time limits. One example is the Kardar-Parisi-Zhang (KPZ) universality class, which encompasses a wide range of natural models such as growth processes modeling…

I will talk about recent work which studies Wilson loop expectations in lattice Yang-Mills models. In particular, I will give a representation of these expectations as sums over embedded planar maps. Time permitting, I will also discuss alternate derivations, interpretations, and generalizations…

The stochastic heat equation is a fundamental model in statistical physics featuring noise scaled by the solution itself. In this talk, I will discuss the pointwise statistics of a family of nonlinear stochastic heat equations in the critical dimension two. Curiously, these statistics evoke a "…

Perceptron problems are a class of random constraint satisfaction problems with geometric structure. They arise as fundamental models in fields as diverse as statistical physics, information theory, combinatorial optimization, and Banach geometry. We study the sharpness of the satisfiability…