Critical values of Gaussian 𝑆𝑈(2) random polynomials

Feng, Renjie; Wang, Zhenan

doi:10.1090/proc/12765

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