We study the Brownian evolution of large non-Hermitian matrices and show that their log-determinant converges to a 2+1 dimensional Gaussian field in the Edwards-Wilkinson regularity class, i.e. logarithmically correlated for the parabolic distance. This gives a dynamical extension of the celebrated result by Rider and Virag (2006) proving that the fluctuations of the eigenvalues of Gaussian non-Hermitian matrices converge to the Gaussian Free Field.

We show that the fluctuations of the density of boundary driven, weakly asymmetric systems are described by energy solutions of the KPZ equation with corresponding boundary conditions. Conditioned on the uniqueness of these energy solutions, we show that the non-equilibrium stationary states (NESS) of these systems are described by the invariant measures of KPZ introduced by Barraquand, Bryc, Corwin, Knizel among others.
Joint work with Juan Arroyave (IMPA).

The Aztec diamond, under the uniform measure on domino tilings, is one of the classic examples of an exactly solvable model in probability and statistical mechanics. Its rich geometric features—such as limit shapes and arctic boundaries—have long made it a cornerstone of integrable probability. More recently, variants of this model with doubly periodic weights have revealed that much of the underlying structure persists far beyond the uniform case and can be used to uncover new behaviors—such as regions with smooth disorder—that were previously out of reach. At the heart of these models lies a birational map that encodes their integrable character. In this talk, I will describe how this map unifies different regimes of the Aztec diamond—from uniform and periodic settings to models in random environments. I will also discuss how integrable features survive (sometimes unexpectedly) in the presence of disorder, and how they connect to other probabilistic models such as directed polymers. The presentation is aimed at a broad mathematical audience, and no prior background in tilings or integrable systems will be assumed.

TBA

We consider fully connected and feedforward deep neural networks with dependent and possibly heavy-tailed weights. In a Bayesian framework, when the likelihood is Gaussian, we identify the posterior distribution of the sequential wide-width limit of the output, and provide an algorithm to sample from the posterior network. In the shallow case, we compute explicitly the posterior distribution of the output.
This is a joint work with Nicola Apollonio and Giovanni Franzina.

TBA