Advertisement

Mass Equidistribution for Random Polynomials

  • Turgay BayraktarEmail author
Article
  • 4 Downloads

Abstract

The purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of associated orthogonal polynomials with subgaussian coefficients. This class of probability distributions contains a wide range of random variables including standard Gaussian and all bounded random variables. We prove that for almost every sequence of random polynomials their normalized zero currents become equidistributed with respect to a deterministic extremal current. The main ingredients of the proof are Bergman kernel asymptotics, mass equidistribution of random polynomials and concentration inequalities for subgaussian quadratic forms.

Keywords

Random polynomial Equidistribution of zeros Equilibrium measure Global extremal function Bergman kernel asymptotics 

Mathematics Subject Classification (2010)

32A60 32A25 60D05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Bayraktar, T.: Equidistribution of zeros of random holomorphic sections. Indiana Univ. Math. J. 65(5), 1759–1793 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bayraktar, T.: Asymptotic normality of linear statistics of zeros of random polynomials. Proc. Amer. Math. Soc. 145(7), 2917–2929 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bayraktar, T.: Zero distribution of random sparse polynomials. Michigan Math. J. 66(2), 389–419 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bayraktar, T.: On global universality for zeros of random polynomials. Hacet. J. Math.Stat. 48(2), 384–398 (2019)MathSciNetGoogle Scholar
  5. 5.
    Bayraktar, T., Coman, D., Herrmann, H., Marinescu, G.: A survey on zeros of random holomorphic sections. Dolomites Res. Notes Approx. 11(4), 1–19 (2018)MathSciNetGoogle Scholar
  6. 6.
    Bayraktar, T., Coman, D., Marinescu, G.: Universality results for zeros of random holomorphic sections. Trans. Amer. Math. Soc.  https://doi.org/10.1090/tran/7807
  7. 7.
    Berman, R.J.: Berman kernels equilibrium measures for line bundles over projective manifolds. Amer. J. Bergman Math. 131(5), 1485–1524 (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    Berman, R.J.: Bergman kernels for weighted polynomials and weighted equilibrium measures of \(\mathbb {C}^{n}\). Indiana Univ. Math. J. 58(4), 1921–1946 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bloom, T., Dauvergne, D.: Asymptotic zero distribution of random orthogonal polynomials. Ann Probab. 47(5), 3202–3230 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bloom, T., Levenberg, N.: Random polynomials and pluripotential-theoretic extremal functions. Potential Anal. 42(2), 311–334 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bloom, T.: On families of polynomials which approximate the pluricomplex green function. Indiana Univ. Math. J. 50(4), 1545–1566 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bloom, T.: Random polynomials and green functions. Int. Math. Res. Not. 28, 1689–1708 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bloom, T., Levenberg, N., Piazzon, F., Wielonsky, F.: Bernstein-markov: a survey. Dolomites Research Notes on Approximation 8(Special_Issue) (2015)Google Scholar
  14. 14.
    Bloom, T., Shiffman, B.: Zeros of random polynomials on \(\mathbb {C}^{m}\). Math. Res Lett. 14(3), 469–479 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37(1), 1–44 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149(1-2), 1–40 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bedford, E., Taylor, B.A.: Fine topology, šilov boundary, and (dd c)n. J. Funct. Anal. 72(2), 225–251 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Demailly, J.-P.: Complex analytic and differential geometry. http://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf (2009)
  19. 19.
    Dinew, S.: Uniqueness in \({\mathcal{E}}(X,\omega )\). J. Funct. Anal. 256(7), 2113–2122 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dinh, T.-C., Sibony, N.: Distribution des valeurs de transformations méromorphes et applications. Comment. Math. Helv. 81(1), 221–258 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Erdös, P., Turán, P.: On the distribution of roots of polynomials. Ann. of Math. (2), 51:105–119 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Guedj, V., Zeriahi, A.: The weighted Monge-Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal. 250(2), 442–482 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hörmander, L.: Notions of convexity, volume 127 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA (1994)Google Scholar
  24. 24.
    Hammersley, J.M.: The zeros of a random polynomial. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II, pp 89–111. University of California Press, Berkeley and Los Angeles (1956)Google Scholar
  25. 25.
    Hughes, C.P., Nikeghbali, A.: The zeros of random polynomials cluster uniformly near the unit circle. Compos. Math. 144(3), 734–746 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hanson, D.L., Wright, F.T.: A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math Statist. 42, 1079–1083 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ibragimov, I., Zaporozhets, D.: On distribution of zeros of random polynomials in complex plane. In: Prokhorov and Contemporary Probability Theory, pp 303–323. Springer (2013)Google Scholar
  28. 28.
    Kac, M.: On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 49, 314–320 (1943)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Klimek, M.: Pluripotential Theory, volume 6 of London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, New York. Oxford Science Publications (1991)Google Scholar
  30. 30.
    Littlewood, J.E., Offord, A.C.: On the number of real roots of a random algebraic equation. III. Rec. Math. N.S. 12(54), 277–286 (1943)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Nonnenmacher, S., Voros, A.: Chaotic eigenfunctions in phase space. J. Statist. Phys. 92(3-4), 431–518 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Pritsker, I.E.: Zero distribution of random polynomials. J. Anal. Math. 134(2), 719–745 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Rudnick, Z.: On the asymptotic distribution of zeros of modular forms. Int. Math. Res. Not., (34), 2059–2074 (2005)Google Scholar
  34. 34.
    Rudelson, M., Vershynin, R.: Hanson-Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18(82), 9 (2013)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Siciak, J.: Extremal plurisubharmonic functions in c n. Ann. Polon. Math. 39, 175–211 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Saff, E.B., V. Totik.: Logarithmic Potentials with external Fields, volume 316 of Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1997). Appendix B by Thomas BloomGoogle Scholar
  37. 37.
    Shepp, L.A., Vanderbei, R.J.: The complex zeros of random polynomials. Trans. Amer. Math. Soc. 347(11), 4365–4384 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Shiffman, B., Zelditch, S.: Distribution of zeros of random and quantum chaotic sections of positive line bundles. Comm. Math. Phys. 200(3), 661–683 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. In: Compressed Sensing, pp 210–268. Cambridge Univ. Press, Cambridge (2012)Google Scholar
  40. 40.
    Walters, P.: An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics. Springer, New York (1982)CrossRefGoogle Scholar
  41. 41.
    S. Zelditch.: Quantum ergodic sequences and equilibrium measures. Constr. Approx. 47(1), 89–118 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Engineering and Natural SciencesSabanci UniversityİstanbulTurkey

Personalised recommendations