## Asymptotic normality of linear statistics of zeros of random polynomials

HTML articles powered by AMS MathViewer

- by Turgay Bayraktar
- Proc. Amer. Math. Soc.
**145**(2017), 2917-2929 - DOI: https://doi.org/10.1090/proc/13441
- Published electronically: December 30, 2016
- PDF | Request permission

## Abstract:

In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted $L^2$-space of polynomials endowed with varying measures of the form $e^{-2n\varphi _n(z)}dz$ under suitable assumptions on the weight functions $\varphi _n$.## References

- T. Bayraktar, Equidistribution of zeros of random holomorphic sections, Indiana Univ. Math. J., 65 (2016), no. 5, 1759-1793, DOI 10.1512/iumj.2016.65.5910.
- T. Bayraktar, Zero distribution of random polynomials with fixed newton polytope, to appear in Michigan Math. J., arXiv:1503.00630.
- T. Bayraktar, D. Coman, and G. Marinescu, Equidistribution of zeros of random holomorphic sections and asymptotic normality, in preparation.
- Bo Berndtsson,
*Uniform estimates with weights for the $\overline \partial$-equation*, J. Geom. Anal.**7**(1997), no. 2, 195–215. MR**1646760**, DOI 10.1007/BF02921720 - Bo Berndtsson,
*Bergman kernels related to Hermitian line bundles over compact complex manifolds*, Explorations in complex and Riemannian geometry, Contemp. Math., vol. 332, Amer. Math. Soc., Providence, RI, 2003, pp. 1–17. MR**2016088**, DOI 10.1090/conm/332/05927 - Robert J. Berman,
*Bergman kernels for weighted polynomials and weighted equilibrium measures of $\Bbb C^n$*, Indiana Univ. Math. J.**58**(2009), no. 4, 1921–1946. MR**2542983**, DOI 10.1512/iumj.2009.58.3644 - T. Bloom and N. Levenberg,
*Random polynomials and pluripotential-theoretic extremal functions*, Potential Anal.**42**(2015), no. 2, 311–334. MR**3306686**, DOI 10.1007/s11118-014-9435-4 - Thomas Bloom and Bernard Shiffman,
*Zeros of random polynomials on $\Bbb C^m$*, Math. Res. Lett.**14**(2007), no. 3, 469–479. MR**2318650**, DOI 10.4310/MRL.2007.v14.n3.a11 - 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 - Michael Christ,
*On the $\overline \partial$ equation in weighted $L^2$ norms in $\textbf {C}^1$*, J. Geom. Anal.**1**(1991), no. 3, 193–230. MR**1120680**, DOI 10.1007/BF02921303 - D. Coman, X. Ma, and G. Marinescu, Equidistribution for sequences of line bundles on normal Kaehler spaces, arXiv:1412.8184, 2014.
- D. Coman, G. Marinescu, and V.-A. Nguyên,
*Hölder singular metrics on big line bundles and equidistribution,*arXiv:1506.01727, 2015. - Henrik Delin,
*Pointwise estimates for the weighted Bergman projection kernel in $\mathbf C^n$, using a weighted $L^2$ estimate for the $\overline \partial$ equation*, Ann. Inst. Fourier (Grenoble)**48**(1998), no. 4, 967–997 (English, with English and French summaries). MR**1656004**, DOI 10.5802/aif.1645 - Jean-Pierre Demailly,
*Estimations $L^{2}$ pour l’opérateur $\bar \partial$ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète*, Ann. Sci. École Norm. Sup. (4)**15**(1982), no. 3, 457–511 (French). MR**690650**, DOI 10.24033/asens.1434 - Tien-Cuong Dinh, George Marinescu, and Viktoria Schmidt,
*Equidistribution of zeros of holomorphic sections in the non-compact setting*, J. Stat. Phys.**148**(2012), no. 1, 113–136. MR**2950760**, DOI 10.1007/s10955-012-0526-6 - Tien-Cuong Dinh and Nessim Sibony,
*Distribution des valeurs de transformations méromorphes et applications*, Comment. Math. Helv.**81**(2006), no. 1, 221–258 (French, with English summary). MR**2208805**, DOI 10.4171/CMH/50 - Niklas Lindholm,
*Sampling in weighted $L^p$ spaces of entire functions in ${\Bbb C}^n$ and estimates of the Bergman kernel*, J. Funct. Anal.**182**(2001), no. 2, 390–426. MR**1828799**, DOI 10.1006/jfan.2000.3733 - F. Nazarov and M. Sodin,
*Correlation functions for random complex zeroes: strong clustering and local universality*, Comm. Math. Phys.**310**(2012), no. 1, 75–98. MR**2885614**, DOI 10.1007/s00220-011-1397-4 - Edward B. Saff and Vilmos Totik,
*Logarithmic potentials with external fields*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR**1485778**, DOI 10.1007/978-3-662-03329-6 - Bernard Shiffman and Steve Zelditch,
*Number variance of random zeros on complex manifolds*, Geom. Funct. Anal.**18**(2008), no. 4, 1422–1475. MR**2465693**, DOI 10.1007/s00039-008-0686-3 - Bernard Shiffman and Steve Zelditch,
*Number variance of random zeros on complex manifolds, II: smooth statistics*, Pure Appl. Math. Q.**6**(2010), no. 4, Special Issue: In honor of Joseph J. Kohn., 1145–1167. MR**2742043**, DOI 10.4310/PAMQ.2010.v6.n4.a10 - Mikhail Sodin and Boris Tsirelson,
*Random complex zeroes. I. Asymptotic normality*, Israel J. Math.**144**(2004), 125–149. MR**2121537**, DOI 10.1007/BF02984409

## Bibliographic Information

**Turgay Bayraktar**- Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244
- Address at time of publication: Faculty of Engineering and Natural Sciences, Sabanci University, Istanbul, Turkey
- MR Author ID: 1009679
- Email: tbayrakt@syr.edu, tbayraktar@sabanciuniv.edu
- Received by editor(s): March 2, 2016
- Received by editor(s) in revised form: March 18, 2016, March 28, 2016, June 14, 2016, and August 2, 2016
- Published electronically: December 30, 2016
- Communicated by: Franc Forstneric
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 2917-2929 - MSC (2010): Primary 32A60, 32A25; Secondary 60F05, 60D05
- DOI: https://doi.org/10.1090/proc/13441
- MathSciNet review: 3637941