Natural Boundary and Zero Distribution of Random Polynomials in Smooth Domains
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Natural Boundary and Zero Distribution of Random Polynomials in Smooth Domains Igor Pritsker1 · Koushik Ramachandran2 Received: 16 October 2017 / Revised: 17 September 2018 / Accepted: 20 September 2018 / Published online: 21 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We of random polynomials of the form Pn (z) = n consider the zero distribution ∞ are non-trivial i.i.d. complex random variables with a B (z), where {a } k k=0 k=0 k k mean 0 and finite variance. Polynomials {Bk }∞ k=0 are selected from a standard basis such as Szeg˝o, Bergman, or Faber polynomials associated with a Jordan domain G whose boundary is C 2,α smooth. We show that the zero counting measures of Pn converge almost surely to the equilibrium measure on the boundary of G. We also show that if {ak }∞ k=0 are i.i.d. random variables, and the domain G has analytic boundary, then for a random series of the form f (z) = ∞ k=0 ak Bk (z), ∂G is almost surely the natural boundary for f (z). Keywords Random polynomials · Orthogonal polynomials · Zero distribution · Natural boundary Mathematics Subject Classification MSC 60F05 · 31A15 · 30B20 · 30B30
1 Introduction This work is a sequel to [8] where we showed that zeros of a sequence of random polynomials {Pn }n (spanned by an appropriate basis) associated to a Jordan domain
Dedicated to Prof. R. S. Varga on his 90th birthday. Communicated by Doron Lubinsky.
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Koushik Ramachandran [email protected] Igor Pritsker [email protected]
1
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA
2
Present Address: Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Bengaluru 560065, India
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I. Pritsker, K. Ramachandran
G with analytic boundary L, equidistributed near L, i.e., distribute according to the equilibrium measure of L. We refer the reader to [8] for references to the literature on random polynomials. In this note, we extend the above result to Jordan domains with lesser regularity, namely domains with C 2,α boundary, see Theorem 1.1 below. To state our results we need to set up some notation. Let G ⊂ C be a Jordan domain. We set = C\G, the exterior of G and the exterior of the closed unit disc. By the Riemann mapping theorem there is a unique conformal mapping : → , (∞) = ∞, (∞) > 0. We denote the equilibrium measure of E = G by 1 n δZ μ E . For a polynomial Pn of degree n, with zeros at {Z k,n }nk=1 , let τn = n k=1 k,n denote its normalized zero counting measure. For a sequence of positive measures w {μn }∞ n=1 , we write μn → μ to denote weak convergence of these measures to μ. A random variable X is called non-trivial if P(X = 0) < 1. Theorem 1.1 Let G be a Jordan domain in C whose boundary L is C 2,α smooth for some 0 0 such that lim inf n→∞
1/n max
n−b log n 1, and find a subsequence n m , m ∈ N, such that 1/n
lim sup Pn m L R m < R,
(2.8)
m→∞
holds with positive probability. It follows from a result of Suetin [11, Ch. 1], that for Bergman polynomials, Bn (z) =
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