Free probability via entropic optimal transport

Abstract

<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\mu $</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\nu $</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> be compactly supported probability measures on ℝ, and let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\mu \boxplus \nu $</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> <mml:mo>⊞</mml:mo> <mml:mi>ν</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> denote their additive free convolution. We show that for all sufficiently large real <jats:inline-formula> <jats:alternatives> <jats:tex-math>$z$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>z</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> , <jats:disp-formula> <jats:alternatives> <jats:tex-math>$$ \int _{-\infty }^{\infty }\log (z - x) \, (\mu \boxplus \nu )( \mathrm{d}x) = \sup _{\Pi } \left \{ \mathbf{E}_{\Pi }[\log (z - (X+Y))] - H(\Pi \mid \mu \otimes \nu ) \right \} , $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> <mml:mo>log</mml:mo> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>−</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mspace/> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>⊞</mml:mo> <mml:mi>ν</mml:mi> <mml:mo>)</mml:mo> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:munder> <mml:mo>sup</mml:mo> <mml:mi>Π</mml:mi> </mml:munder> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>Π</mml:mi> </mml:msub> <mml:mo>[</mml:mo> <mml:mo>log</mml:mo> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>−</mml:mo> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>+</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> <mml:mo>]</mml:mo> <mml:mo>−</mml:mo> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Π</mml:mi> <mml:mo>∣</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>⊗</mml:mo> <mml:mi>ν</mml:mi> <mml:mo>)</mml:mo> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:math> </jats:alternatives> </jats:disp-formula> where the supremum is taken over all couplings <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\Pi $</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Π</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\mu $</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\nu $</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> . Analogous formulas hold for multiplicative free convolution <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\mu \boxtimes \nu $</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> <mml:mo>⊠</mml:mo> <mml:mi>ν</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> and free compression <jats:inline-formula> <jats:alternatives> <jats:tex-math>$[\mu ]_{\tau }$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>τ</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> . In this way, integrals of a log-potential against free convolutions can be expressed as entropic optimal transport problems. The corresponding optimal couplings admit explicit formulas, from which the standard <jats:inline-formula> <jats:alternatives> <jats:tex-math>$R$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> - and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$S$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> -transform descriptions of additive and multiplicative free convolution are recovered. In particular, the optimizer <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\Pi _{z}$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Π</mml:mi> <mml:mi>z</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> 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. </jats:p>