Rocky Mountain Journal of Mathematics

Zeros of random orthogonal polynomials with complex Gaussian coefficients

Aaron Yeager

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Abstract

Let $\{f_j\}_{j=0}^n$ be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line or on the unit circle. We study zero distribution of random linear combinations of the form $$P_n(z)=\sum _{j=0}^n\eta _jf_j(z),$$ where $\eta _0,\ldots ,\eta _n$ are complex-valued iid standard Gaussian random variables. Using the Christoffel-Darboux formula, the density function for the expected number of zeros of $P_n$ in these cases takes a very simple shape. From these expressions, under the mere assumption that the orthogonal polynomials are from the Nevai class, we give the limiting value of the density function away from their respective sets where the orthogonality holds. In the case when $\{f_j\}$ are orthogonal polynomials on the unit circle, the density function shows that the expected number of zeros of $P_n$ are clustering near the unit circle. To quantify this phenomenon, we give a result that estimates the expected number of complex zeros of $P_n$ in shrinking neighborhoods of compact subsets of the unit circle.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2385-2403.

Dates
First available in Project Euclid: 14 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1544756814

Digital Object Identifier
doi:10.1216/RMJ-2018-48-7-2385

Mathematical Reviews number (MathSciNet)
MR3892137

Zentralblatt MATH identifier
06999267

Subjects
Primary: 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05] 30B20: Random power series 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 60B99: None of the above, but in this section

Keywords
Random polynomials orthogonal polynomials Christoffel-Darboux formula Nevai class

Citation

Yeager, Aaron. Zeros of random orthogonal polynomials with complex Gaussian coefficients. Rocky Mountain J. Math. 48 (2018), no. 7, 2385--2403. doi:10.1216/RMJ-2018-48-7-2385. https://projecteuclid.org/euclid.rmjm/1544756814


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References

  • T. Bayraktar, Equidistribution of zeros of random holomorphic sections, Indiana Univ. Math. J. 65 (2016), 1759–1793.
  • A. Bharucha-Reid and M. Sambandham, Random polynomials, Academic Press, New York, 1986.
  • A. Bloch and G. Pólya, On the roots of a certain algebraic equation, Proc. Lond. Math. Soc. 33 (1932), 102–114.
  • T. Bloom, Random polynomials and Green functions, Int. Math. Res. Not. 28 (2005), 1689–1708.
  • ––––, Random polynomials and $($pluri$)$ potential theory, Ann. Polon. Math. 91 (2007), 131–141.
  • T. Bloom and N. Levenberg, Random polynomials and pluripotential-theoretic extremal functions, Potential Anal. 42 (2015), 311–334.
  • T. Bloom and B. Shiffman, Zeros of random polynomials on $\mathbb C^m$, Math. Res. Lett. 14 (2007), 469–479.
  • M. Das, Real zeros of a random zum of orthogonal polynomials, Proc. Amer. Math. Soc. 1 (1971), 147–153.
  • M. Das and S. Bhatt, Real roots of random harmonic equations, Indian J. Pure Appl. Math. 13 (1982), 411–420.
  • A. Edelman and E. Kostlan, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. 32 (1995), 1–37.
  • P. Erdős and A. Offord, On the number of real roots of a random algebraic equation, Proc. Lond. Math. Soc. 6 (1956), 139–160.
  • K. Farahmand, Complex roots of a random algebraic polynomial, J. Math. Anal. App. 210 (1997), 724–730.
  • ––––, Random polynomials with complex coefficients, Stat. Prob. Lett. 27 (1996), 347–355.
  • ––––, Topics in random polynomials, Pitman Res. Notes Math. 393 (1998).
  • K. Farahmand and A. Grigorash, Complex zeros of trignometric polynomials with standard normal random coefficients, J. Math. Anal. App. 262 (2001), 554–563.
  • K. Farahmand and J. Jahangiri, Complex roots of a class of random algebraic polynomials, J. Math. Anal. App. 226 (1998), 220–228.
  • N. Feldheim, Zeros of Gaussian analytic functions with translation-invariant distribution, Israel J. Math. 195 (2012), 317–345.
  • J. Hammersley, The zeros of a random polynomial, University of California Press, Berkeley, 1956.
  • J. Hough, M. Krishnapur, Y. Peres and B. Virág, Zeros of Gaussian analytic functions and determinantal point processes, American Mathematical Society, Providence, RI, 2009.
  • I. Ibragimov and O. Zeitouni, On the roots of random polynomials, Trans. Amer. Math. Soc. 349 (1997), 2427–2441.
  • M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 314–320.
  • ––––, On the average number of real roots of a random algebraic equation, II, Proc. Lond. Math. Soc. 50 (1948), 390–408.
  • A. Ledoan, Explicit formulas for the distribution of complex zeros of a family of random sums, J. Math. Anal. Appl. 444 (2016), 1304–1320.
  • E. Levin and D. Lubinsky, Universality limits involving orthogonal polynomials on the unit circle, Comp. Meth. Funct. Th. 7 (2007), 543–561.
  • J. Littlewood and A. Offord, On the number of real roots of a random algebraic equation, J. Lond. Math. Soc. 13 (1938), 288–295.
  • D. Lubinsky, I. Pritsker and X. Xie, Expected number of real zeros for random linear combinations of orthogonal polynomials, Proc. Amer. Math. Soc. 144 (2016), 1631–1642.
  • ––––, Expected number of real zeros for random orthogonal polynomials, Math. Proc. Cambr. Philos. Soc. 164 (2018), 47–66.
  • P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979).
  • Y. Peres and B. Virág, Zeros of the iid Gaussian power series: A conformally invariant determinantal process, Acta Math. 194 (2005), 1–35.
  • S. Rice, Mathematical theory of random noise, Bell Syst. Tech. J. 25 (1945), 46–156.
  • L.A. Shepp and R.J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (1995), 4365–4384.
  • B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys. 200 (1999), 661–683.
  • ––––, Equilibrium distribution of zeros of random polynomials, Int. Math. Res. Not. 1 (2003), 25–29.
  • ––––, Random complex fewnomials, I, Notions of positivity and the geometry of polynomials, Birkhäuser/Springer, Basel, 2001.
  • B. Simon, Orthogonal polynomials on the unit circle, Amer. Math. Soc. Colloq. Publ. 54 (2005).
  • G. Szegő, Orthogonal polynomials, American Mathematical Society, Providence, RI, 1975.
  • V. Totik, Othogonal polynomials, Surv. Approx. Th. 1 (2005), 70–125.
  • R.J. Vanderbei, The complex zeros of random sums, arXiv: 1508.05162v1.
  • Y. Wang, Bounds on the average number of real roots of a random algebraic equation, Chinese Ann. Math. 4 (1983), 601–605 (in Chinese).
  • J. Wilkins, Jr., An asymptotic expansion for the expected number of real zeros of a random polynomial, Proc. Amer. Math. Soc. 103 (1988), 1249–1258.
  • M. Yattselev and A. Yeager, Zeros of real random polynomials spanned by OPUC, Indiana Univ. Math. J., to appear.
  • A. Yeager, Zeros of random linear combinations of entire functions with complex Gaussian coefficients, arXiv: 1605.06836v1.