Variational formula for the logarithmic potential of free additive convolutions

TL;DR

This paper derives a general variational formula for the logarithmic potential of free additive convolutions of probability measures, linking it to the R-transform and simplifying for specific laws like semicircle and Marchenko-Pastur.

Contribution

It introduces a novel variational formula for the logarithmic potential of free additive convolutions, applicable to various important probability measures.

Findings

Provides a unified formula for the logarithmic potential of free additive convolutions.

Simplifies the formula for semicircle and Marchenko-Pastur laws.

Facilitates estimates of determinants of sums of independent random matrices.

Abstract

We establish a general variational formula for the logarithmic potential of the free additive convolution of two compactly supported probability measure on $\R$. The formula is given in terms of the $R$-transform of the first measure, and the logarithmic potential of second measure. The result applies in particular to the additive convolution with the semicircle or Marchenko-Pastur laws, for which the formula simplifies. The logarithmic potential of additive convolutions appears for instance in estimates of the determinant of sums of independent random matrices.