Fractional free convolution powers

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.