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.

Coulomb and Riesz gases are interacting particle systems with a wide range of applications in random matrix theory, approximation theory, convex geometry, and diverse areas of physics. We study the statistical mechanics of general Riesz gases at mesoscopic and microscopic length scales, providing controls on fluctuations of linear statistics down to microscopic length scales and establishing for the first time a CLT for fluctuations of linear statistics for general two-dimensional Riesz gases.

A novel technical difficulty involves the development of a transport method for general Riesz gases, building on work of Leblé and Serfaty for Coulomb gases, to understand the behavior of the partition function under small perturbations of the external potential. Our study involves several questions concerning degenerate, singular elliptic PDE and fractional operators.

This is based on joint work with S. Serfaty.

The chemical distance is the observable that encapsulates the metric structure of percolation clusters. At criticality, heuristics suggest that the chemical distance between two connected points scales quadratically in the extrinsic distance, in line with the analogy to branching random walk. Our work presents an exact statement of this result, where the rescaled two-point chemical distance converges in distribution to a random variable whose density is expressible as a Brownian motion hitting time. The method relies on the robust incipient infinite cluster constructed in our previous work to enforce a decoupling argument that separates neighborhoods of distant pivotal edges. This decoupling tool yields further applications towards studying the mass structure of percolation clusters, i.e. k-point functions, which is necessary in the steps towards a full scaling limit result for the IIC. These projects are joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe. The preprint can be found at https://arxiv.org/abs/2509.06236.