Freely independent random variables with non-atomic distributions

Shlyakhtenko, Dimitri; Skoufranis, Paul

doi:10.1090/S0002-9947-2015-06434-4

Freely independent random variables with non-atomic distributions
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by Dimitri Shlyakhtenko and Paul Skoufranis PDF

Trans. Amer. Math. Soc. 367 (2015), 6267-6291 Request permission

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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