Free probability via entropic optimal transport

Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}$ with compact support, and let $\mu \boxplus \nu$ denote their additive free convolution. We show that for $z \in \mathbb{R}$ greater than the sum of essential suprema of $\mu$ and $\nu$, we have \begin{equation*} \int_{-\infty}^\infty \log(z - x) \mu \boxplus \nu (\mathrm{d}x) = \sup_{\Pi} \left\{ \mathbf{E}_\Pi[\log(z - (X+Y)] - H(\Pi|\mu \otimes \nu) \right\}, \end{equation*} where the supremum is taken over all couplings $\Pi$ of the probability measures $\mu$ and $\nu$, and $H(\Pi|\mu \otimes \nu)$ denotes the relative entropy of a coupling $\Pi$ against product measure. We prove similar formulas for the multiplicative free convolution $\mu \boxtimes \nu$ and the free compression $[\mu]_\tau$ of probability measures, as well as for multivariate free operations. Thus the integrals of a log-potential against the fundamental measure operations of free probability may be formulated in terms of entropic optimal transport problems. The optimal couplings in these variational descriptions of the free probability operations can be computed explicitly, and from these we can then deduce the standard $R$- and $S$-transform descriptions of additive and multiplicative free convolution. We use our optimal transport formulations to derive new inequalities relating free and classical operations on probability measures, such as the inequality \begin{equation*} \int_{-\infty}^\infty \log(z - x) \mu \boxplus \nu (\mathrm{d}x) \geq \int_{-\infty}^{\infty} \log(z-x) \mu \ast \nu( \mathrm{d}x) \end{equation*} relating free and classical convolution. Our approach is based on applying a large deviation principle on the symmetric group to the quadrature formulas of Marcus, Spielman and Srivastava.