## 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.## References

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## Additional Information

**Dimitri Shlyakhtenko**- Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095
- MR Author ID: 606307
- ORCID: 0000-0002-0221-7508
- Email: shlyakht@math.ucla.edu
**Paul Skoufranis**- Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095
- MR Author ID: 966934
- Email: pskoufra@math.ucla.edu
- Received by editor(s): June 12, 2013
- Published electronically: February 26, 2015
- Additional Notes: This research was supported in part by NSF grants DMS-090076, DMS-1161411, DARPA HR0011-12-1-0009, and by NSERC PGS
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**367**(2015), 6267-6291 - MSC (2010): Primary 46L54; Secondary 15B52
- DOI: https://doi.org/10.1090/S0002-9947-2015-06434-4
- MathSciNet review: 3356937