Fractional free convolution powers

Dimitri Shlyakhtenko; Terence Tao. With an appendix by David Jekel

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Fractional free convolution powers

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arxiv 2009.01882 v3 pith:VMOFPYSS submitted 2020-09-03 math.PR math.CVmath.OA

Dimitri Shlyakhtenko , Terence Tao. With an appendix by David Jekel This is my paper · ORCID

classification math.PRmath.CVmath.OA

keywords freeconvolutionpowerprocessbercovici-voiculescuboxpluscaseconcept

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abstract

The extension $k \mapsto \mu^{\boxplus k}$ of the concept of a free convolution power to the case of non-integer $k \geq 1$ was introduced by Bercovici-Voiculescu and Nica-Speicher, and related to the minor process in random matrix theory. In this paper we give two proofs of the monotonicity of the free entropy and free Fisher information of the (normalized) free convolution power in this continuous setting, and also establish an intriguing variational description of this process.

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