## Critical values of Gaussian $SU(2)$ random polynomials

HTML articles powered by AMS MathViewer

- by Renjie Feng and Zhenan Wang PDF
- Proc. Amer. Math. Soc.
**144**(2016), 487-502 Request permission

## Abstract:

In this article we will get the estimate of the expected distribution of critical values of Gaussian $SU(2)$ random polynomials as the degree is large enough. The result about the expected density is a direct application of the Kac-Rice formula. The critical values will accumulate at infinity, then we will study the rate of this convergence and its rescaling limit as $n\rightarrow \infty$.## References

- Robert J. Adler and Jonathan E. Taylor,
*Random fields and geometry*, Springer Monographs in Mathematics, Springer, New York, 2007. MR**2319516** - Jean-Marc Azaïs and Mario Wschebor,
*On the distribution of the maximum of a Gaussian field with $d$ parameters*, Ann. Appl. Probab.**15**(2005), no. 1A, 254–278. MR**2115043**, DOI 10.1214/105051604000000602 - Pavel Bleher, Bernard Shiffman, and Steve Zelditch,
*Universality and scaling of correlations between zeros on complex manifolds*, Invent. Math.**142**(2000), no. 2, 351–395. MR**1794066**, DOI 10.1007/s002220000092 - Amir Dembo, Bjorn Poonen, Qi-Man Shao, and Ofer Zeitouni,
*Random polynomials having few or no real zeros*, J. Amer. Math. Soc.**15**(2002), no. 4, 857–892. MR**1915821**, DOI 10.1090/S0894-0347-02-00386-7 - Michael R. Douglas, Bernard Shiffman, and Steve Zelditch,
*Critical points and supersymmetric vacua. I*, Comm. Math. Phys.**252**(2004), no. 1-3, 325–358. MR**2104882**, DOI 10.1007/s00220-004-1228-y - Alan Edelman and Eric Kostlan,
*How many zeros of a random polynomial are real?*, Bull. Amer. Math. Soc. (N.S.)**32**(1995), no. 1, 1–37. MR**1290398**, DOI 10.1090/S0273-0979-1995-00571-9 - Philippe Flajolet and Robert Sedgewick,
*Analytic combinatorics*, Cambridge University Press, Cambridge, 2009. MR**2483235**, DOI 10.1017/CBO9780511801655 - Renjie Feng and Steve Zelditch,
*Critical values of random analytic functions on complex manifolds*, Indiana Univ. Math. J.**63**(2014), no. 3, 651–686. - Renjie Feng and Steve Zelditch,
*Critical values of fixed Morse Index of random analytic functions on Riemann surfaces*, arXiv:1503:08892. To appear in Indiana University Mathematics Journal. - Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR**507725** - J. H. Hannay,
*Chaotic analytic zero points: exact statistics for those of a random spin state*, J. Phys. A**29**(1996), no. 5, L101–L105. MR**1383056**, DOI 10.1088/0305-4470/29/5/004 - Boris Hanin,
*Correlations and pairing between zeros and critical points of random polynomials*, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–Northwestern University. MR**3295261** - 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**, DOI 10.1090/ulect/051 - M. Kac,
*On the average number of real roots of a random algebraic equation*, Bull. Amer. Math. Soc.**49**(1943), 314–320. MR**7812**, DOI 10.1090/S0002-9904-1943-07912-8 - S. O. Rice,
*Mathematical analysis of random noise*, Bell System Tech. J.**23**(1944), 282–332. MR**10932**, DOI 10.1002/j.1538-7305.1944.tb00874.x - M. Sodin,
*Zeros of Gaussian analytic functions*, Math. Res. Lett.**7**(2000), no. 4, 371–381. MR**1783614**, DOI 10.4310/MRL.2000.v7.n4.a2 - Mikhail Sodin and Boris Tsirelson,
*Random complex zeroes. I. Asymptotic normality*, Israel J. Math.**144**(2004), 125–149. MR**2121537**, DOI 10.1007/BF02984409

## Additional Information

**Renjie Feng**- Affiliation: Department of Mathematics and Statistics, Mcgill University, Montreal, Quebec, Canada
- MR Author ID: 939975
- Email: renjie@math.mcgill.ca
**Zhenan Wang**- Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
- Email: zn_wang@math.northwestern.edu
- Received by editor(s): July 26, 2014
- Received by editor(s) in revised form: January 12, 2015
- Published electronically: June 10, 2015
- Communicated by: Walter Van Assche
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 487-502 - MSC (2010): Primary 12E10
- DOI: https://doi.org/10.1090/proc/12765
- MathSciNet review: 3430828