College of Liberal Arts & Sciences

# Operator Theory Seminar

**Abstract: **

We study the large $N$ behavior of a random $d$-tuple of $N \times N$ self-adjoint matrices $X^{(N} = (X_1^{(N)}, \dots, X_d^{(N)})$ whose probability density is a constant times $e^{-N^2 V^{(N)}(x)}$, where $V^{(N)}: M_N(\mathbb{C}_{\mathrm{sa}}^d \to \mathbb{R}$ is a given potential. Assume $V^{(N)}$ is uniformly convex and semiconcave and the gradient $\nabla V^{(N)}$ has a large $N$ limit in a suitable sense. Past work has shown that the traces $(1/N) \operatorname{Tr}(X_{i_1}^{(N)} \dots X_{i_\ell}^{(N)})$ converge almost surely to a deterministic limit, which is given by the traces corresponding to operators $X_1$, \dots, $X_d$ in a tracial von Neumann algebra. The noncommutative distribution of $X_1$, \dots, $X_d$ is known as a free Gibbs law, and like classical Gibbs laws in statistical mechanics, it maximizes a certain entropy functional on the space of noncommutative distributions. In fact, we show that the classical entropy and Fisher information of $X^{(N)}$ converge to the appropriate limits as $N \to \infty$. Moreover, certain functions that produce transport of measure between the distribution of $X^{(N)}$ and a Gaussian also have a large $N$ limit as $N \to \infty$, which is an element in a certain space of ``tracial non-commutative smooth functions.'' These ideas lead to a triangular transport of measure in the noncommutative setting, meaning that we can express the distribution of $X_1$, \dots, $X_d$ as the pushforward of the noncommutative Gaussian distribution by a certain non-commutative function $f = (f_1,\dots,f_d)$ such that each $f_j$ only depends on the first $j$ coordinates.

Zoom: ID 952 1767 4506