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

Yuki Ueda

Paper

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

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

there is a unique probability measure

satisfying

, where

and

are free and monotone additive convolutions, respectively. We recall that the reciprocal Cauchy transform of

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.