Critical points of random polynomials with independent identically distributed roots

Author:
Zakhar Kabluchko

Journal:
Proc. Amer. Math. Soc. **143** (2015), 695-702

MSC (2010):
Primary 30C15; Secondary 60G57, 60B10

Published electronically:
September 19, 2014

MathSciNet review:
3283656

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be independent identically distributed random

variables with values in . Denote by the probability distribution of . Consider a random polynomial . We prove a conjecture of Pemantle and Rivin [in: I. Kotsireas and E. V. Zima, eds., *Advances in Combinatorics*, Waterloo Workshop in Computer Algebra, ] that the empirical measure

**[1]**B. Derrida,*The zeroes of the partition function of the random energy model*, Phys. A**177**(1991), no. 1-3, 31–37. Current problems in statistical mechanics (Washington, DC, 1991). MR**1137017**, 10.1016/0378-4371(91)90130-5**[2]**I. M. Gel′fand and G. E. Shilov,*Generalized functions. Vol. 1*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR**0435831**- [3]
J.
Ben Hough, Manjunath
Krishnapur, Yuval
Peres, and Bálint
Virág,
*Zeros of Gaussian analytic functions and determinantal point processes*, University Lecture Series, vol. 51, American Mathematical Society, Providence, RI, 2009. MR**2552864** **[4]**Zakhar Kabluchko and Anton Klimovsky,*Complex random energy model: zeros and fluctuations*, Probab. Theory Related Fields**158**(2014), no. 1-2, 159–196. MR**3152783**, 10.1007/s00440-013-0480-5- [5]
Z. Kabluchko and D. Zaporozhets,
*Asymptotic distribution of complex zeros of random analytic functions*,

Ann. Probab., to appear, 2012.

Preprint available at http://arxiv.org/abs/1205.5355. **[6]**Olav Kallenberg,*Foundations of modern probability*, Probability and its Applications (New York), Springer-Verlag, New York, 1997. MR**1464694**- [7]
A.
I. Markushevich,
*Theory of functions of a complex variable. Vol. II*, Revised English edition translated and edited by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0181738** - [8]
R. Pemantle and I. Rivin,
*Advances in Combinatorics. Waterloo Workshop in Computer Algebra 2011,*

The distribution of zeros of the derivative of a random polynomial.

In I. Kotsireas and E. V. Zima, editors, Springer, New York, 2013.

Preprint available at http://arxiv.org/abs/1109.5975. **[9]**Valentin V. Petrov,*Limit theorems of probability theory*, Oxford Studies in Probability, vol. 4, The Clarendon Press, Oxford University Press, New York, 1995. Sequences of independent random variables; Oxford Science Publications. MR**1353441****[10]**Q. I. Rahman and G. Schmeisser,*Analytic theory of polynomials*, London Mathematical Society Monographs. New Series, vol. 26, The Clarendon Press, Oxford University Press, Oxford, 2002. MR**1954841****[11]**Sneha Dey Subramanian,*On the distribution of critical points of a polynomial*, Electron. Commun. Probab.**17**(2012), no. 37, 9. MR**2970701**, 10.1214/ECP.v17-2040**[12]**Terence Tao and Van Vu,*Random matrices: universality of ESDs and the circular law*, Ann. Probab.**38**(2010), no. 5, 2023–2065. With an appendix by Manjunath Krishnapur. MR**2722794**, 10.1214/10-AOP534

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Additional Information

**Zakhar Kabluchko**

Affiliation:
Institute of Stochastics, Ulm University, Helmholtzstr. 18, 89069 Ulm, Germany

Email:
zakhar.kabluchko@uni-ulm.de

DOI:
https://doi.org/10.1090/S0002-9939-2014-12258-1

Keywords:
Random polynomials,
empirical distribution,
critical points,
zeros of the derivative,
logarithmic potential

Received by editor(s):
July 4, 2012

Received by editor(s) in revised form:
February 12, 2013, and May 3, 2013

Published electronically:
September 19, 2014

Communicated by:
David Levin

Article copyright:
© Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.