Free probability via entropic optimal transport

Abstract Let $\mu $ μ and $\nu $ ν be compactly supported probability measures on ℝ, and let $\mu \boxplus \nu $ μ ⊞ ν denote their additive free convolution. We show that for all sufficiently large real $z$ z , $$ \int _{-\infty }^{\infty }\log (z - x) \, (\mu \boxplus \nu )( \mathrm{d}x) = \sup _{\Pi } \left \{ \mathbf{E}_{\Pi }[\log (z - (X+Y))] - H(\Pi \mid \mu \otimes \nu ) \right \} , $$ ∫ − ∞ ∞ log ( z − x ) ( μ ⊞ ν ) ( d x ) = sup Π { E Π [ log ( z − ( X + Y ) ) ] − H ( Π ∣ μ ⊗ ν ) } , where the supremum is taken over all couplings $\Pi $ Π of $\mu $ μ and $\nu $ ν . Analogous formulas hold for multiplicative free convolution $\mu \boxtimes \nu $ μ ⊠ ν and free compression $[\mu ]_{\tau }$ [ μ ] τ . In this way, integrals of a log-potential against free convolutions can be expressed as entropic optimal transport problems. The corresponding optimal couplings admit explicit formulas, from which the standard $R$ R - and $S$ S -transform descriptions of additive and multiplicative free convolution are recovered. In particular, the optimizer $\Pi _{z}$ Π z in (0.1) encodes the subordination equations via couplings of random variables. Our approach is based on a large deviation principle on the symmetric group, combined with the quadrature method of Marcus–Spielman–Srivastava.