Density of the free additive convolution of multi-cut measures

TL;DR

This paper analyzes the local behavior of the density of free additive convolutions of multi-cut measures, revealing decay rates at endpoints and singular points, with implications for understanding measure singularities.

Contribution

It characterizes the local density behavior of free additive convolutions of multi-cut measures, including endpoint decay rates and singularity structures.

Findings

Density decays as square root or cubic root at support endpoints.

Endpoints exhibit power law behavior with exponents between -1 and 1.

Provides detailed local behavior at singular points and support endpoints.

Abstract

We consider the free additive convolution semigroup $\lbrace \mu^{\boxplus t}:\,t\ge 1\rbrace$ and determine the local behavior of the density of $\mu^{\boxplus t}$ at the endpoints and at any singular point of its support. We then study the free additive convolution of two multi-cut probability measures and show that its density decays either as a square root or as a cubic root at any endpoints of its support. The probability measures considered in this paper satisfy a power law behavior with exponents strictly between $-1$ and $1$ at the endpoints of their supports.