Metadata
Abstract
We study the number of real roots of a Kostlan random polynomial of degree in one variable. More generally, we consider the counting measure of the set of real roots of such polynomials. We compute the large degree asymptotics of the central moments of these random variables. As a consequence, we obtain 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 their counting measure converge in distribution to the Standard Gaussian White Noise. More generally, our results hold for the real zeros of a random real section in the complex Fubini–Study model.
References
[AADL18] Central Limit Theorem for the volume of the zero set of Kostlan Shub Smale random polynomial systems (2018) (https://arxiv.org/abs/1808.02967v1)
[AADL21] Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems, Am. J. Math., Volume 143 (2021) no. 4, pp. 1011-1042 | DOI | Zbl
[AL21] Zeros of smooth stationary Gaussian processes, Electron. J. Probab., Volume 26 (2021), 68 | MR | Zbl
[Anc21] Random sections of line bundles over real Riemann surfaces, Int. Math. Res. Not., Volume 2021 (2021) no. 9, pp. 7004-7059 | DOI | MR | Zbl
[AT07] Random fields and geometry, Springer Monographs in Mathematics, Springer, 2007 | DOI | Zbl
[AW09] Level sets and extrema of random processes and fields, John Wiley & Sons, 2009 | DOI | Zbl
[BBL92] Distribution of roots of random polynomials, Phys. Rev. Lett., Volume 68 (1992) no. 18, pp. 2726-2729 | DOI | MR | Zbl
[BDFZ20] Exponential concentration of zeroes of stationary Gaussian processes, Int. Math. Res. Not. (2020) no. 23, pp. 9769-9796 | DOI | MR | Zbl
[Bil95] Probability and measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1995 | Zbl
[BSZ00] Universality and scaling of correlations between zeros on complex manifolds, Invent. Math., Volume 142 (2000) no. 2, pp. 351-395 | DOI | MR | Zbl
[Dal15] Asymptotic variance and CLT for the number of zeros of Kostlan–Shub–Smale random polynomials, C. R. Math. Acad. Sci. Paris, Volume 353 (2015) no. 12, pp. 1141-1145 | DOI | MR | Zbl
[Fer67] Processus linéaires, processus généralisés, Ann. Inst. Fourier, Volume 17 (1967) no. 1, pp. 1-92 | DOI | Numdam | MR | Zbl
[GH94] Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, 1994 (reprint of the 1978 original) | DOI | Zbl
[GW11] Exponential rarefaction of real curves with many components, Publ. Math., Inst. Hautes Étud. Sci. (2011) no. 113, pp. 69-96 | DOI | Numdam | MR | Zbl
[GW16] Betti numbers of random real hypersurfaces and determinants of random symmetric matrices, J. Eur. Math. Soc., Volume 18 (2016) no. 4, pp. 733-772 | DOI | MR | Zbl
[KL01] Central limit theorems for level functionals of stationary Gaussian processes and fields, J. Theor. Probab., Volume 14 (2001) no. 3, pp. 639-672 | DOI | MR | Zbl
[Kos93] On the distribution of roots of random polynomials, From topology to computation: Proceedings of the Smalefest. Papers presented at the conference “From topology to computation: Unity and diversity in the mathematical sciences” held at the University of California at Berkeley, USA, August 5-9, 1990 in honor of Stephen Smale’s 60th birthday (Hirsch, M. W. et al., eds.), Springer (1993), pp. 419-431 | Zbl
[Let16] Expected volume and Euler characteristic of random submanifolds, J. Funct. Anal., Volume 270 (2016) no. 8, pp. 3047-3110 | DOI | MR | Zbl
[Let19] Variance of the volume of random real algebraic submanifolds, Trans. Am. Math. Soc., Volume 371 (2019) no. 6, p. 4129-–4192 | DOI | MR | Zbl
[LP19] Variance of the volume of random real algebraic submanifolds II, Indiana Univ. Math. J., Volume 68 (2019) no. 6, pp. 1649-1720 | DOI | MR | Zbl
[LS19] Maximal and typical topology of real polynomial singularities (2019) (https://arxiv.org/abs/1906.04444, to be published at the Annales de l’Institut Fourier)
[MM07] Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, 254, Birkhäuser, 2007 | Zbl
[MM15] Exponential estimate for the asymptotics of Bergman kernels, Math. Ann., Volume 362 (2015) no. 3–4, pp. 1327-1347 | DOI | MR | Zbl
[NS16] Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions, J. Math. Phys. Anal. Geom., Volume 12 (2016) no. 3, pp. 205-278 | DOI | MR | Zbl
[Olv01] Geometric foundations of numerical algorithms and symmetry, Appl. Algebra Eng. Commun. Comput., Volume 11 (2001) no. 5, pp. 417-436 | DOI | MR | Zbl
[Ros19] Random nodal lengths and Wiener chaos, Probabilistic methods in geometry, topology and spectral theory. CRM workshops on probabilistic methods in spectral geometry and PDE, August 22–26, 2016, and on probabilistic methods in topology, November 14–18, 2016, Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada (Contemporary Mathematics), Volume 739, American Mathematical Society; Centre de Recherches Mathématiques (CRM), Montréal, 2019, pp. 155-169 | DOI | MR | Zbl
[ST04] Random complex zeroes I: Asymptotic normality, Isr. J. Math., Volume 144 (2004), pp. 125-149 | DOI | MR | Zbl
[SZ99] Distribution of zeros of random and quantum chaotic sections of positive line bundles, Commun. Math. Phys., Volume 200 (1999) no. 3, pp. 661-683 | DOI | MR | Zbl
[SZ10] Number variance of random zeros on complex manifolds. II: Smooth statistics, Pure Appl. Math. Q., Volume 6 (2010) no. 4, pp. 1145-1167 | DOI | MR | Zbl