Monotone max-convolution and subordination functions for free max-convolution

TL;DR

This paper explores the relationship between spectral maxima of operators and max-convolution, introducing subordination functions for free max-convolution inspired by classical and free probability analogies.

Contribution

It introduces subordination functions for free max-convolution and establishes their existence and structural properties, extending free probability theory.

Findings

Spectral maximum distribution coincides with classical max-convolution.

Existence of subordination functions for free max-convolution is proven.

Structural properties of these subordination functions are characterized.

Abstract

We show that the distribution of the spectral maximum of monotonically independent self-adjoint operators coincides with the classical max-convolution of their distributions. In free probability, it was proven that for any probability measures $\sigma,\mu$ on $\mathbb{R}$ there is a unique probability measure $\mathbb{A}_\sigma(\mu)$ satisfying $\sigma\boxplus \mu = \sigma \triangleright \mathbb{A}_\sigma(\mu)$, where $\boxplus$ and $\triangleright$ are free and monotone additive convolutions, respectively. We recall that the reciprocal Cauchy transform of $\mathbb{A}_\sigma(\mu)$ is the subordination function for free additive convolution. Motivated by this analogy, we introduce subordination functions for free max-convolution and prove their existence and structural properties.