Mass Equidistribution for Random Polynomials
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Mass Equidistribution for Random Polynomials Turgay Bayraktar1 Received: 10 January 2019 / Accepted: 4 November 2019 / © Springer Nature B.V. 2019
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
1 Introduction Let ϕ : Cm → R be a C 1,1 weight function (i.e. ϕ is differentiable and all of its first partial derivatives are locally Lipschitz continuous) satisfying ϕ(z) ≥ (1 + ) log z for z 1
(1.1)
for some fixed > 0. We define an inner product on the space Pn of multi-variable polynomials of degree at most n by setting p(z)q(z)e−2nϕ(z) dVm (z) (1.2) p, qn := Cm
¨ B˙ITAK grants B˙IDEB-2232/118C006, T. Bayraktar is partially supported by TU ARDEB-3501/118F049 and Science Academy BAGEP grant. Turgay Bayraktar
[email protected] 1
Faculty of Engineering and Natural Sciences, Sabanci University, ˙Istanbul, Turkey
T. Bayraktar
where dVm denotes the Lebesgue measure on Cm . We also let {Pjn }dj n=1 be a fixed orthonormal basis (ONB) for Pn with respect to the inner product Eq. 1.2. A random polynomial is of the form fn (z) =
dn
cjn Pjn (z)
j =1
cjn
where are independent identically distributed (iid) real or complex subgaussian random variables (see Section 3.3) and dn := dim(Pn ) = n+m n . This allows us to endow Pn with a dn -fold product probability measure P robn induced by the probability law of cjn . We also consider the product probability space ∞ n=1 (Pn , P robn ) whose elements are sequences of random polynomials of increasing degree. We are interested in limiting distribution of zeros of random polynomials. In the present setting, the choice of weight function ϕ determines a weighted global extremal function ϕe (see Eq. 2.2) which induces a weighted equilibrium measure μe (see Eq. 3.3) whose support is a compact set denoted by Sϕ . The following result indicates that for a typical (in the sense of probability) sequence {fn }∞ n=1 of random polynomials the masses (respectively, normalized zero currents) are asymptotic to the equilibrium measure (respectively, the extremal current): Theorem 1.1 Let ϕ : Cm → R be a C 1,1 -weight function satisfyin
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