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arXiv · 1305.1920

Freely Independent Random Variables with Non-Atomic Distributions

Dimitri Shlyakhtenko·Paul Skoufranis
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Abstract

We examine the distributions of non-commutative polynomials of non-atomic, freely independent random variables. In particular, we obtain an analogue of the Strong Atiyah Conjecture for free groups thus proving that the measure of each atom of any $n \times n$ matricial polynomial of non-atomic, freely independent random variables is an integer multiple of $n^{-1}$. In addition, we show that the Cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real-analytic except at a finite number of points.

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Dimitri Shlyakhtenko, Paul Skoufranis. 2013-05-08. Freely Independent Random Variables with Non-Atomic Distributions. https://doi.org/10.1090/s0002-9947-2015-06434-4

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