Free probability via entropic optimal transport

Let μ μ and ν ν be compactly supported probability measures on ℝ, and let μ ν μ ⊞ ν denote their additive free convolution. We show that for all sufficiently large real z z , ∫ -∞∞log (z - x) \, (μ ν )( dx) = Π \ EΠ[log (z - (X+Y))] - H(Π μ ⊗ ν ) \ , ∫ − ∞ ∞ 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 Π Π of μ μ and ν ν . Analogous formulas hold for multiplicative free convolution μ ν μ ⊠ ν and free compression [μ ]τ [ 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 Π 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.