Mass Equidistribution for Random Polynomials

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.

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References

  1. 1.

    Bayraktar, T.: Equidistribution of zeros of random holomorphic sections. Indiana Univ. Math. J. 65(5), 1759–1793 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Bayraktar, T.: Asymptotic normality of linear statistics of zeros of random polynomials. Proc. Amer. Math. Soc. 145(7), 2917–2929 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Bayraktar, T.: Zero distribution of random sparse polynomials. Michigan Math. J. 66(2), 389–419 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Bayraktar, T.: On global universality for zeros of random polynomials. Hacet. J. Math.Stat. 48(2), 384–398 (2019)

    MathSciNet  Google 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)

    MathSciNet  Google 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)

    MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Bloom, T., Dauvergne, D.: Asymptotic zero distribution of random orthogonal polynomials. Ann Probab. 47(5), 3202–3230 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Bloom, T., Levenberg, N.: Random polynomials and pluripotential-theoretic extremal functions. Potential Anal. 42(2), 311–334 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Bloom, T.: On families of polynomials which approximate the pluricomplex green function. Indiana Univ. Math. J. 50(4), 1545–1566 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Bloom, T.: Random polynomials and green functions. Int. Math. Res. Not. 28, 1689–1708 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Bloom, T., Levenberg, N., Piazzon, F., Wielonsky, F.: Bernstein-markov: a survey. Dolomites Research Notes on Approximation 8(Special_Issue) (2015)

  14. 14.

    Bloom, T., Shiffman, B.: Zeros of random polynomials on \(\mathbb {C}^{m}\). Math. Res Lett. 14(3), 469–479 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37(1), 1–44 (1976)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149(1-2), 1–40 (1982)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Bedford, E., Taylor, B.A.: Fine topology, šilov boundary, and (ddc)n. J. Funct. Anal. 72(2), 225–251 (1987)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Dinh, T.-C., Sibony, N.: Distribution des valeurs de transformations méromorphes et applications. Comment. Math. Helv. 81(1), 221–258 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Erdös, P., Turán, P.: On the distribution of roots of polynomials. Ann. of Math. (2), 51:105–119 (1950)

  22. 22.

    Guedj, V., Zeriahi, A.: The weighted Monge-Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal. 250(2), 442–482 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Hörmander, L.: Notions of convexity, volume 127 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA (1994)

  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)

  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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

  28. 28.

    Kac, M.: On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 49, 314–320 (1943)

    MathSciNet  MATH  Article  Google 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)

  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)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Nonnenmacher, S., Voros, A.: Chaotic eigenfunctions in phase space. J. Statist. Phys. 92(3-4), 431–518 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Pritsker, I.E.: Zero distribution of random polynomials. J. Anal. Math. 134(2), 719–745 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Rudnick, Z.: On the asymptotic distribution of zeros of modular forms. Int. Math. Res. Not., (34), 2059–2074 (2005)

  34. 34.

    Rudelson, M., Vershynin, R.: Hanson-Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18(82), 9 (2013)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Siciak, J.: Extremal plurisubharmonic functions in cn. Ann. Polon. Math. 39, 175–211 (1981)

    MathSciNet  MATH  Article  Google 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 Bloom

    Google Scholar 

  37. 37.

    Shepp, L.A., Vanderbei, R.J.: The complex zeros of random polynomials. Trans. Amer. Math. Soc. 347(11), 4365–4384 (1995)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

  40. 40.

    Walters, P.: An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics. Springer, New York (1982)

    Book  Google Scholar 

  41. 41.

    S. Zelditch.: Quantum ergodic sequences and equilibrium measures. Constr. Approx. 47(1), 89–118 (2018)

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Turgay Bayraktar.

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T. Bayraktar is partially supported by TÜ BİTAK grants BİDEB-2232/118C006, ARDEB-3501/118F049 and Science Academy BAGEP grant.

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Bayraktar, T. Mass Equidistribution for Random Polynomials. Potential Anal 53, 1403–1421 (2020). https://doi.org/10.1007/s11118-019-09811-w

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Keywords

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

Mathematics Subject Classification (2010)

  • 32A60
  • 32A25
  • 60D05