A Universality Property of Gaussian Analytic Functions

Ledoan, Andrew; Merkli, Marco; Starr, Shannon

doi:10.1007/s10959-011-0356-5

Cited by 9 publications

(12 citation statements)

References 15 publications

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“…We also remark that a result similar to Theorem 5.1 has recently been established (by a rather different method) by Ledoan, Merkli & Starr [2012]. In our language, the results in Ledoan, Merkli & Starr [2012] establish universality for the distribution of the random variable…”

Section: Flat Polynomialssupporting

confidence: 70%

Local Universality of Zeroes of Random Polynomials

Tao

2014

Int Math Res Notices

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In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials f = n i=1 ciξiz i and f = n i=1 ciξiz i , where the ξi andξi are iid random variables that match moments to second order, the coefficients ci are deterministic, and the degree parameter n is large. Our results show, under some light conditions on the coefficients ci and the tails of ξi,ξi, that the correlation functions of the zeroes of f andf are approximately the same. As an application, we give some answers to the classical question "How many zeroes of a random polynomials are real ?" for several classes of random polynomial models. Our analysis relies on a general replacement principle, motivated by some recent work in random matrix theory. This principle enables one to compare the correlation functions of two random functions f andf if their log magnitudes log |f |, log |f | are close in distribution, and if some non-concentration bounds are obeyed.

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Section: Flat Polynomialssupporting

confidence: 70%

Local Universality of Zeroes of Random Polynomials

Tao

2014

Int Math Res Notices

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“…The only ensemble of random polynomials for which the independence of the limiting distribution of zeros on the distribution of the coefficients is well understood is the Kac ensemble; see [1,15,16,36]. In the context of random polynomials, there were many results on the universal character of local correlations between close zeros [3,19,29,30]. In this work, we focus on the global distribution of zeros.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic distribution of complex zeros of random analytic functions

Kabluchko¹,

Zaporozhets²

2014

Ann. Probab.

104

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Let $\xi_0,\xi_1,\ldots$ be independent identically distributed complex- valued random variables such that $\mathbb{E}\log(1+|\xi _0|)<\infty$. We consider random analytic functions of the form \[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}\xi_kf_{k,n}z^k,\] where $f_{k,n}$ are deterministic complex coefficients. Let $\mu_n$ be the random measure counting the complex zeros of $\mathbf{G}_n$ according to their multiplicities. Assuming essentially that $-\frac{1}{n}\log f_{[tn],n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $\frac{1}{n}\mu_n$ converges in probability to some deterministic measure $\mu$ which is characterized in terms of the Legendre-Fenchel transform of $u$. The limiting measure $\mu$ does not depend on the distribution of the $\xi_k$'s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.Comment: Published in at http://dx.doi.org/10.1214/13-AOP847 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: substantial text overlap with arXiv:1205.535

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“…It uses several tools from probability theory and complex analysis such as the Lindeberg-Feller condition and Hurwitz's theorem. Since these are well-known to probabilists, these results were merely referred to in an implicit way in the version of our paper [5] in order to shorten the presentation. But here, we will also briefly review those tools.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Universality of correlations for random analytic functions

Starr¹

2011

Entropy and the Quantum II

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We review a result obtained with Andrew Ledoan and Marco Merkli. Consider a random analytic function f (z) = ∞ n=0 anXnz n , where the Xn's are i.i.d., complex valued random variables with mean zero and unit variance, and the coefficients an are non-random and chosen so that the variance transforms covariantly under conformal transformations of the domain. If the Xn's are Gaussian, this is called a Gaussian analytic function (GAF). We prove that, even if the coefficients are not Gaussian, the zero set converges in distribution to that of a GAF near the boundary of the domain.

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A Universality Property of Gaussian Analytic Functions

Cited by 9 publications

References 15 publications

Local Universality of Zeroes of Random Polynomials

Local Universality of Zeroes of Random Polynomials

Asymptotic distribution of complex zeros of random analytic functions

Universality of correlations for random analytic functions

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