The basic operations of free probability - additive free convolution, multiplicative free convolution, and free compression - describe how the eigenvalues of large random matrices interact under the basic matrix operations, such as addition, multiplication, and taking minors.

In this talk we discuss how these free probability operations can be formulated in terms of an “entropic optimal transport” problem – an optimal transport problem but with an entropy penalty for the coupling measure.

Our proof of this formulation uses the quadrature formulas of Marcus, Spielman and Srivastava, which relate the expected characteristic polynomial of matrices under random unitary vs symmetric conjugation. The approach involves an asymptotic analysis of the quadrature formulas using a large deviation principle on the symmetric group.

This is joint work with Octavio Arizmendi (CIMAT).