College of Liberal Arts & Sciences

# Mathematics Colloquium

**Abstract: **

We study transport problems for joint probability distributions for several $n \times n$ random matrices as in the limit as $n \to \infty$; specifically we consider matrix models which have a smooth log-density. A transport problem refers to the problem of finding a way to rearrange one probability distribution $\mu$ into another probability distribution $\nu$ using a (hopefully smooth) function $f$ to assign where the mass at a given point $x$ should be moved to so that $f_* \mu = \nu$. For measures with smooth densities, one way to construct smooth transport functions is by interpolating with a continuous path between the two densities, then realing the transport function $f$ as a solution to a certain differential. For our $n \times n$ matrix models, we execute such a construction of a smooth transport function $f^{(n)}$ such that the asymptotic behavior of $f^{(n)}$ in the large-$n$ limit is well-controlled. The objects that describe the large-$n$ behavior of the random matrix models are certain operators in a von Neumann algebra, and the large-$n$ behavior of $f^{(n)}$ is described by certain functions $f$ on tuples from a von Neumann algebra.

The transport functions $f$ in the large-$n$ limit allow us to get highly non-obvious decompositions of the von Neumann algebra into a free product, which sheds new light on their structure.