Probability Seminar

In stochastic homogenization, solutions to a heterogeneous equation converge to the solution to a homogeneous equation provided that the coefficients are stationary, ergodic and satisfy a sufficient ellipticity condition. I will explain why certain coarse-grained ellipticity constants appear naturally in homogenization, show that boundedness of the coarse-grained ellipticity constants implies quenched homogenization of the PDE, and compare this to recent results on the random conductance model and the case of a divergence-free drift.

Consider a connected finite graph in which each vertex carries a real number. At each step, an edge (u, v) is chosen uniformly at random, and the numbers at u and v are replaced by their average. This dynamics, known as the repeated averaging process, appears in many contexts, including thermal equilibration, opinion dynamics, wealth exchange, and quantum circuits. All numbers eventually converge to the global average, and we study the speed of convergence in the L1 distance (which is, for example, the Gini index in wealth distributions). On random d-regular graphs, we show a sharp phase transition in this decay, where the L1 distance drops abruptly to zero with a Gaussian profile. Our techniques are robust, and we expect them to extend to more general dynamics on expander graphs. This is joint work in preparation with Dong Yao.

In this talk, I will present the two-dimensional analogue of the asymptotics for Toeplitz determinants with Fisher-Hartwig singularities, for general real symbols. A key focus of the talk will be the surgery method we developed to handle these singularities and establish global asymptotics. I will also discuss applications of this result, including the convergence of the characteristic polynomial of random normal matrices to Gaussian Multiplicative Chaos measure. Based on joint work with Paul Bourgade, Guillaume Dubach, and Lisa Hartung.