publication . Preprint . 2019

Roots of Kostlan polynomials: moments, strong Law of Large Numbers and Central Limit Theorem

Ancona, Michele; Letendre, Thomas;
Open Access English
  • Published: 26 Nov 2019
  • Publisher: HAL CCSD
  • Country: France
Abstract
We study the number of real roots of a Kostlan (or elliptic) random polynomial of degree d in one variable. More generally, we are interested in the distribution of the counting measure of the set of real roots of such a polynomial. We compute the asymptotics of the central moments of any order of these random variables, in the large degree limit. As a consequence, we prove that these quantities satisfy a strong Law of Large Numbers and a Central Limit Theorem. In particular, the real roots of a Kostlan polynomial almost surely equidistribute as the degree diverges. Moreover, the fluctuations of the counting measure of this random set around its mean converge in...
Subjects
free text keywords: Kostlan polynomials, Elliptic polynomials, Complex Fubini-Study model, Kac-Rice formula, Law of Large Numbers, Central Limit Theorem, Method of moments, MSC 2010: 14P99, 32L05, 60F05, 60F15, 60G15, 60G57, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Method of moments Mathematics Subject Classification 2010: 14P25, 32L05, 60F05, 60F15, 60G15, 60G57, Mathematics - Algebraic Geometry, Mathematics - Metric Geometry, Mathematics - Probability
Funded by
ANR| UNIRANDOM
Project
UNIRANDOM
Universality for random nodal domains
  • Funder: French National Research Agency (ANR) (ANR)
  • Project Code: ANR-17-CE40-0008
,
ANR| SpInQS
Project
SpInQS
Spectral geometry of intermediate quantum systems
  • Funder: French National Research Agency (ANR) (ANR)
  • Project Code: ANR-17-CE40-0011
26 references, page 1 of 2

[1] R. J. Adler and J. E. Taylor, Random fields and geometry, 1st ed., Monographs in Mathematics, Springer, New York, 2007.

[2] M. Ancona, Random sections of line bundles over real Riemann surfaces, arXiv: 1806.10481 (2018).

[3] D. Armentano, J.-M. Azaïs, F. Dalmao, and J. R. Leòn, Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems, arXiv: 1801.06331 (2018).

[4] , Central Limit Theorem for the volume of the zero set of Kostlan Shub Smale random polynomial systems, arXiv: 1808.02967 (2018).

[5] J.-M. Azaïs and M. Wschebor, Level sets and extrema of random processes and fields, 1st ed., John Wiley & Sons, Hoboken, NJ, 2009. [OpenAIRE]

[6] R. Basu, A. Dembo, N. Feldheim, and O. Zeitouni, Exponential concentration of zeroes of stationary Gaussian processes, arXiv: 1709.06760 (2017).

[7] P. Billingsley, Probability and measure, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 1995.

[8] P. Bleher, B. Shiffman, and S. Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000), no. 2, 351-395. [OpenAIRE]

[9] E. Bogomolny, O. Bohigas, and P. Leboeuf, Distribution of roots of random polynomials, Phys. Rev. Lett. 68 (1992), no. 18, 2726-2729.

[10] F. Dalmao, Asymptotic variance and CLT for the number of zeros of Kostlan-Shub-Smale random polynomials, C. R. Math. Acad. Sci. Paris 353 (2015), no. 12, 1141-1145. [OpenAIRE]

[11] X. Fernique, Processus linéaires, processus généralisés, Ann. Inst. Fourier 17 (1967), 1-92.

[12] D. Gayet and J.-Y. Welschinger, Exponential rarefaction of real curves with many components, Publ. Math. Inst. Hautes Études Sci. (2011), no. 113, 69-96. [OpenAIRE]

[13] , Betti numbers of random real hypersurfaces and determinants of random symmetric matrices, J. of Eur. Math. Soc. 18 (2016), no. 4, 733-772.

[14] Ph. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, New York, 1994, Reprint of the 1978 original.

[15] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218. [OpenAIRE]

26 references, page 1 of 2
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