Research Article
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Year 2019, Volume: 48 Issue: 2, 384 - 398, 01.04.2019

Abstract

References

  • T. Bayraktar, Equidistribution of zeros of random holomorphic sections, Indiana Univ. Math. J. 65 (5), 1759–1793, 2016.
  • T. Bayraktar, Asymptotic normality of linear statistics of zeros of random polynomi- als, Proc. Amer. Math. Soc. 145 (7), 2917–2929, 2017.
  • T. Bayraktar, Zero distribution of random sparse polynomials, Michigan Math. J. 66 (2), 389–419, 2017.
  • T. Bayraktar, Expected number of real roots for random linear combinations of or- thogonal polynomials associated with radial weights, Potential Anal. 48 (4), 459–471, 2018.
  • T. Bayraktar, D. Coman and G. Marinescu, Universality results for zeros of random holomorphic sections, to appear in Trans. Amer. Math. Soc., DOI: 10.1090/tran/7807, ArXiv:1709.10346.
  • E. Bogomolny, O. Bohigas, and P. Leboeuf, Quantum chaotic dynamics and random polynomials, J. Statist. Phys. 85 (5-6), 639–679, 1996.
  • R. Berman, Bergman kernels for weighted polynomials and weighted equilibrium mea- sures of $\mathbb{C}^n$, Indiana Univ. Math. J. 58 (4), 1921–1946, 2009.
  • T. Bloom and N. Levenberg, Random Polynomials and Pluripotential-Theoretic Ex- tremal Functions, Potential Anal. 42 (2), 311–334, 2015.
  • T. Bloom, Random polynomials and Green functions, Int. Math. Res. Not. (28), 1689– 1708, 2005.
  • T. Bloom, Random polynomials and (pluri)potential theory, Ann. Polon. Math. 91 (2-3), 131–141, 2007.
  • T. Bloom and B. Shiffman, Zeros of random polynomials on $\mathbb{C}^m$, Math. Res. Lett. 14 (3), 469–479, 2007.
  • J.-D. Deuschel and D. W. Stroock, Large deviations, in: Pure and Applied Mathe- matics, 137, Academic Press, Inc., Boston, MA, 1989.
  • T.-C. Dinh and N. Sibony, Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv. 81 (1), 221–258, 2006.
  • C. G. Esseen, On the concentration function of a sum of independent random vari- ables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 9, 290–308, 1968.
  • O. Friedland and S. Sodin, Bounds on the concentration function in terms of the Diophantine approximation, C. R. Math. Acad. Sci. Paris 345 (9), 513–518, 2007.
  • J. M. Hammersley, The zeros of a random polynomial, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II (Berkeley and Los Angeles), University of California Press, pp. 89–111, 1956.
  • C. P. Hughes and A. Nikeghbali, The zeros of random polynomials cluster uniformly near the unit circle, Compos. Math. 144 (3), 734–746, 2008
  • I. Ibragimov and D. Zaporozhets, On distribution of zeros of random polynomials in complex plane, in: Prokhorov and Contemporary Probability Theory, Springer, pp. 303–323, 2013.
  • M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49, 314–320, 1943.
  • M. Klimek, Pluripotential theory, in: London Mathematical Society Monographs. New Series, 6, The Clarendon Press, Oxford University Press, New York, 1991.
  • Z. Kabluchko and D. Zaporozhets, Asymptotic distribution of complex zeros of random analytic functions, Ann. Probab. 42 (4), 1374–1395, 2014.
  • J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. III, Rec. Math. [Mat. Sbornik] N.S. 12 (54), 277–286, 1943.
  • I. Pritsker and K. Ramachandran, Equidistribution of zeros of random polynomials, J. Approx. Theory 215, 106–117, 2017.
  • R. T. Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Prince- ton University Press, Princeton, NJ, 1997, Reprint of the 1970 original, Princeton Paperbacks.
  • M. Rudelson and R. Vershynin, The Littlewood-Offord problem and invertibility of random matrices, Adv. Math. 218 (2), 600–633, 2008.
  • M. Rudelson and R. Vershynin, Smallest singular value of a random rectangular ma- trix, Comm. Pure Appl. Math. 62 (12), 1707–1739, 2009.
  • J. Siciak, Extremal plurisubharmonic functions in $C^n$, Ann. Polon. Math. 39, 175– 211, 1981.
  • H. Stahl and V. Totik, General orthogonal polynomials, in: Encyclopedia of Mathe- matics and its Applications, 43, Cambridge University Press, Cambridge, 1992.
  • E. B. Saff and V. Totik, Logarithmic potentials with external fields, in: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sci- ences], 316, Springer-Verlag, Berlin, 1997, Appendix B by Thomas Bloom.
  • L. A. Shepp and R. J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (11), 4365–4384, 1995.
  • B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys. 200 (3), 661–683, 1999.
  • B. Shiffman and S. Zelditch, Equilibrium distribution of zeros of random polynomials, Int. Math. Res. Not. (1), 25–49. 2003.
  • T. Tao and V. H. Vu, Inverse Littlewood-Offord theorems and the condition number of random discrete matrices, Ann. of Math. 169 (2), 595–632, 2009.

On global universality for zeros of random polynomials

Year 2019, Volume: 48 Issue: 2, 384 - 398, 01.04.2019

Abstract

In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations $\sum_{j}a_jP_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j\}_j$ induced by a multi-circular weight function $Q$ defined on $\mathbb{C}^m$ satisfying suitable smoothness and growth conditions. In complex dimension $m\geq3$, we prove that $\mathbb{E}[(\log(1+|a_j|))^m]<\infty$ is a necessary and sufficient condition for normalized zero currents of random polynomials to be almost surely asymptotic to the (deterministic) extremal current $\frac{i}{\pi}\partial\overline{\partial}V_{Q}.$ In addition, in complex dimension one, we consider random linear combinations of orthonormal polynomials with respect to a regular measure in the sense of Stahl & Totik and we prove analogous results in this setting.

References

  • T. Bayraktar, Equidistribution of zeros of random holomorphic sections, Indiana Univ. Math. J. 65 (5), 1759–1793, 2016.
  • T. Bayraktar, Asymptotic normality of linear statistics of zeros of random polynomi- als, Proc. Amer. Math. Soc. 145 (7), 2917–2929, 2017.
  • T. Bayraktar, Zero distribution of random sparse polynomials, Michigan Math. J. 66 (2), 389–419, 2017.
  • T. Bayraktar, Expected number of real roots for random linear combinations of or- thogonal polynomials associated with radial weights, Potential Anal. 48 (4), 459–471, 2018.
  • T. Bayraktar, D. Coman and G. Marinescu, Universality results for zeros of random holomorphic sections, to appear in Trans. Amer. Math. Soc., DOI: 10.1090/tran/7807, ArXiv:1709.10346.
  • E. Bogomolny, O. Bohigas, and P. Leboeuf, Quantum chaotic dynamics and random polynomials, J. Statist. Phys. 85 (5-6), 639–679, 1996.
  • R. Berman, Bergman kernels for weighted polynomials and weighted equilibrium mea- sures of $\mathbb{C}^n$, Indiana Univ. Math. J. 58 (4), 1921–1946, 2009.
  • T. Bloom and N. Levenberg, Random Polynomials and Pluripotential-Theoretic Ex- tremal Functions, Potential Anal. 42 (2), 311–334, 2015.
  • T. Bloom, Random polynomials and Green functions, Int. Math. Res. Not. (28), 1689– 1708, 2005.
  • T. Bloom, Random polynomials and (pluri)potential theory, Ann. Polon. Math. 91 (2-3), 131–141, 2007.
  • T. Bloom and B. Shiffman, Zeros of random polynomials on $\mathbb{C}^m$, Math. Res. Lett. 14 (3), 469–479, 2007.
  • J.-D. Deuschel and D. W. Stroock, Large deviations, in: Pure and Applied Mathe- matics, 137, Academic Press, Inc., Boston, MA, 1989.
  • T.-C. Dinh and N. Sibony, Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv. 81 (1), 221–258, 2006.
  • C. G. Esseen, On the concentration function of a sum of independent random vari- ables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 9, 290–308, 1968.
  • O. Friedland and S. Sodin, Bounds on the concentration function in terms of the Diophantine approximation, C. R. Math. Acad. Sci. Paris 345 (9), 513–518, 2007.
  • J. M. Hammersley, The zeros of a random polynomial, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II (Berkeley and Los Angeles), University of California Press, pp. 89–111, 1956.
  • C. P. Hughes and A. Nikeghbali, The zeros of random polynomials cluster uniformly near the unit circle, Compos. Math. 144 (3), 734–746, 2008
  • I. Ibragimov and D. Zaporozhets, On distribution of zeros of random polynomials in complex plane, in: Prokhorov and Contemporary Probability Theory, Springer, pp. 303–323, 2013.
  • M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49, 314–320, 1943.
  • M. Klimek, Pluripotential theory, in: London Mathematical Society Monographs. New Series, 6, The Clarendon Press, Oxford University Press, New York, 1991.
  • Z. Kabluchko and D. Zaporozhets, Asymptotic distribution of complex zeros of random analytic functions, Ann. Probab. 42 (4), 1374–1395, 2014.
  • J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. III, Rec. Math. [Mat. Sbornik] N.S. 12 (54), 277–286, 1943.
  • I. Pritsker and K. Ramachandran, Equidistribution of zeros of random polynomials, J. Approx. Theory 215, 106–117, 2017.
  • R. T. Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Prince- ton University Press, Princeton, NJ, 1997, Reprint of the 1970 original, Princeton Paperbacks.
  • M. Rudelson and R. Vershynin, The Littlewood-Offord problem and invertibility of random matrices, Adv. Math. 218 (2), 600–633, 2008.
  • M. Rudelson and R. Vershynin, Smallest singular value of a random rectangular ma- trix, Comm. Pure Appl. Math. 62 (12), 1707–1739, 2009.
  • J. Siciak, Extremal plurisubharmonic functions in $C^n$, Ann. Polon. Math. 39, 175– 211, 1981.
  • H. Stahl and V. Totik, General orthogonal polynomials, in: Encyclopedia of Mathe- matics and its Applications, 43, Cambridge University Press, Cambridge, 1992.
  • E. B. Saff and V. Totik, Logarithmic potentials with external fields, in: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sci- ences], 316, Springer-Verlag, Berlin, 1997, Appendix B by Thomas Bloom.
  • L. A. Shepp and R. J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (11), 4365–4384, 1995.
  • B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys. 200 (3), 661–683, 1999.
  • B. Shiffman and S. Zelditch, Equilibrium distribution of zeros of random polynomials, Int. Math. Res. Not. (1), 25–49. 2003.
  • T. Tao and V. H. Vu, Inverse Littlewood-Offord theorems and the condition number of random discrete matrices, Ann. of Math. 169 (2), 595–632, 2009.
There are 33 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Turgay Bayraktar 0000-0002-1364-9728

Publication Date April 1, 2019
Published in Issue Year 2019 Volume: 48 Issue: 2

Cite

APA Bayraktar, T. (2019). On global universality for zeros of random polynomials. Hacettepe Journal of Mathematics and Statistics, 48(2), 384-398.
AMA Bayraktar T. On global universality for zeros of random polynomials. Hacettepe Journal of Mathematics and Statistics. April 2019;48(2):384-398.
Chicago Bayraktar, Turgay. “On Global Universality for Zeros of Random Polynomials”. Hacettepe Journal of Mathematics and Statistics 48, no. 2 (April 2019): 384-98.
EndNote Bayraktar T (April 1, 2019) On global universality for zeros of random polynomials. Hacettepe Journal of Mathematics and Statistics 48 2 384–398.
IEEE T. Bayraktar, “On global universality for zeros of random polynomials”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 2, pp. 384–398, 2019.
ISNAD Bayraktar, Turgay. “On Global Universality for Zeros of Random Polynomials”. Hacettepe Journal of Mathematics and Statistics 48/2 (April 2019), 384-398.
JAMA Bayraktar T. On global universality for zeros of random polynomials. Hacettepe Journal of Mathematics and Statistics. 2019;48:384–398.
MLA Bayraktar, Turgay. “On Global Universality for Zeros of Random Polynomials”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 2, 2019, pp. 384-98.
Vancouver Bayraktar T. On global universality for zeros of random polynomials. Hacettepe Journal of Mathematics and Statistics. 2019;48(2):384-98.