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arXiv · 1804.11199

On the support of the free additive convolution

Zhigang Bao·Laszlo Erdos·Kevin Schnelli
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Abstract

We consider the free additive convolution of two probability measures $\mu$ and $\nu$ on the real line and show that $\mu\boxplus\nu$ is supported on a single interval if $\mu$ and $\nu$ each has single interval support. Moreover, the density of $\mu\boxplus\nu$ is proven to vanish as a square root near the edges of its support if both $\mu$ and $\nu$ have power law behavior with exponents between $-1$ and $1$ near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [4].

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Zhigang Bao, Laszlo Erdos, Kevin Schnelli. 2018-04-26. On the support of the free additive convolution. https://arxiv.org/abs/1804.11199

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