Density of the free additive convolution of multi-cut measures

Philippe Moreillon

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arXiv · 2209.15607

Density of the free additive convolution of multi-cut measures

Philippe Moreillon

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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.

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Philippe Moreillon. 2022-09-30. Density of the free additive convolution of multi-cut measures. https://arxiv.org/abs/2209.15607

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