The large-$n$ behavior of certain random matrix models through free probability and non-commutative function theory

David Jekel University of California, Los Angeles (UCLA) Mathematics

We consider self-adjoint random $n \times n$ matrices $X_1^{(n)}$, ... ,$X_d^{(n)}$ given by a joint probability density of the form constant times $e^{-V^{(n)}(x)}\,dx$, where $V^{(n)}$ is a function $M_n(\mathbb{C})_{sa}^d \to \mathbb{R}$. If $V^{(n)}$ is uniformly convex and semi-concave, and if $\nabla V^{(n)}$ has asymptotic behavior described by certain ``non-commutative functions'' as $n \to \infty$, then the large-$n$ behavior of the random matrix tuple can be described using non-commutative random variables in a von Neumann algebra. This includes the large-$n$ behavior of moments such as $(1/n) \Tr(X_{i_1} \dots X_{i_k})$, of entropy and Fisher's information, of conditional expectations, and of transport of measure in certain cases. We will present these results as well as some open questions through the lens of looking at the large-$n$ behavior of functions and differential equations associated to the potential $V^{(n)}$, specifically the heat equation and Hamilton-Jacobi-Bellman equation.