Freely Independent Random Variables with Non-Atomic Distributions
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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2013
- DOI:
- 10.48550/arXiv.1305.1920
- arXiv:
- arXiv:1305.1920
- Bibcode:
- 2013arXiv1305.1920S
- Keywords:
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- Mathematics - Operator Algebras
- E-Print:
- Trans. Amer. Math. Soc. 367 (2015), 6267-6291