Free multiplicative convolution with an arbitrary measure on the real line

TL;DR

Addressing the problem of computing free multiplicative convolution between a general real-line measure and a nonnegative measure, the paper introduces subordination functions and an $S$-transform for general measures. The method relies on fixed-point analysis of analytic maps (via Denjoy--Wolff points) and on the extended $T$-transform, yielding multiplicativity and a cohesive framework for stability identities and regularity analysis. The main contributions are the existence of subordination functions for $(\mu,\nu)$, the $S$-transform for general measures, identities for stable laws and their mixtures, and a Lebesgue-decomposition type regularity result. This work broadens the applicability of free probability techniques to nonstandard measures, with implications for convolution identities, semigroup homomorphisms, and analytic density regularity.

Abstract

We develop analytic tools for studying the free multiplicative convolution of any measure on the real line and any measure on the nonnegative real line. More precisely, we construct the subordination functions and the $S$-transform of an arbitrary probability measure. The important multiplicativity of $S$-transform is proved with the help of subordination functions. We then apply the $S$-transform to establish convolution identities for stable laws, which had been considered in the literature only for the positive and symmetric cases. Subordination functions are also used in order to extend Belinschi--Nica's semigroup of homomorphisms, and to establish regularity properties of free multiplicative convolution, in particular, the absence of singular continuous part and analyticity of the density.

Figures (1)

• Figure 1: The density function of $(\frac{1}{2}\delta_1+\frac{1}{2}\delta_{-3})\boxtimes(\frac{1}{2}\delta_1+\frac{1}{2}\delta_3)$ and the scaled eigenvalue distribution of $BU^*AUB$, where $U$ is a realization of the Haar unitary random matrix, $A$ is a diagonal matrix with half of its eigenvalues equal to $1$ and the other half equal to $-3$, and $B$ is a diagonal matrix with half of its eigenvalues equal to $1$ and the other half equal to $\sqrt{3}$. The size of all matrices is $1500$. The black line represents $-\pi^{-1}\Im G_{\mu\boxtimes\nu}(x+10^{-10}\mathrm{i})$ for $x$. To improve clarity, we excluded data from a small neighborhood around the singularities.

Theorems & Definitions (104)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10