Random complex zeroes, I. Asymptotic normality

Sodin, Mikhail; Tsirelson, Boris

doi:10.1007/bf02984409

Cited by 117 publications

(161 citation statements)

References 24 publications

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“…We want to mention here the pioneering contributions of Chambers and Slud [14], Slud [38,39], Kratz and León [24], Sodin and Tsirelson [40].…”

Section: Thenmentioning

confidence: 99%

Critical points of multidimensional random Fourier series: Variance estimates

Nicolaescu

2016

Journal of Mathematical Physics

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Abstract. We investigate certain families X , 0 < ≪ 1 of stationary Gaussian random smooth functions on the m-dimensional torus T m := R m /Z m approaching the white noise as → 0. We show that there exists universal constants c1, c2 > 0 such that for any cube B ⊂ R m of size r ≤ 1/2 and centered at the origin, the number of critical points of X in the region B mod Z m ⊂ T m has mean ∼ c1 vol(B) −m , variance ∼ c2 vol(B) −m , and satisfies a central limit theorem as ց 0.

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“…We want to mention here the pioneering contributions of Chambers and Slud [14], Slud [38,39], Kratz and León [24], Sodin and Tsirelson [40].…”

Section: Thenmentioning

confidence: 99%

Critical points of multidimensional random Fourier series: Variance estimates

Nicolaescu

2016

Journal of Mathematical Physics

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“…We may estimate the dependence on L using (7) to see that the integral decays polynomially in L. If the measure µ is locally flat (see Definition 3) then we see that the variance decays as L −3 , just as in [ST04].…”

Section: Theorem 2 For Any Smooth Function ψ With Compact Support Inmentioning

confidence: 99%

“…is an orthonormal basis for the corresponding Fock space, so that the construction just given corresponds to the GAF studied in [ST04] and [ST05]. More generally if φ(z) = |z| α /2 and α > 0 then the set (…”

Section: Theorem 2 For Any Smooth Function ψ With Compact Support Inmentioning

confidence: 99%

“…In fact in [ST04] the authors prove a more general result, but we shall only require the form we have stated. We have also slightly modified the denominator in condition (11), but it is easy to verify that this does not affect the proof (cf.…”

Section: Then the Distributions Of The Random Variablesmentioning

confidence: 99%

“…This problem has already been considered when µ is the Lebesgue measure on the plane ( [ST04] and [ST05]) and the resultant zero-sets are invariant in distribution under plane isometries. We are interested in generalising this construction to other measures, where we cannot expect any such invariance to hold.…”

Section: Introductionmentioning

confidence: 99%

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Inhomogeneous random zero sets

Buckley¹,

Massaneda

Ortega-Cerdà

2014

Indiana Univ. Math. J.

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ABSTRACT. We construct random point processes in C that are asymptotically close to a given doubling measure. The processes we construct are the zero sets of random entire functions that are constructed through generalised Fock spaces. We offer two alternative constructions, one via bases for these spaces and another via frames, and we show that for both constructions the average distribution of the zero set is close to the given doubling measure, and that the variance is much less than the variance of the corresponding Poisson point process. We prove some asymptotic large deviation estimates for these processes, which in particular allow us to estimate the 'hole probability', the probability that there are no zeroes in a given open bounded subset of the plane. We also show that the 'smooth linear statistics' are asymptotically normal, under an additional regularity hypothesis on the measure. These generalise previous results by Sodin and Tsirelson for the Lebesgue measure.

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The forbidden region for random zeros: Appearance of quadrature domains

Nishry,

Wennman

2023

Comm Pure Appl Math

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Our main discovery is a surprising interplay between quadrature domains on the one hand, and the zeros of the Gaussian Entire Function (GEF) on the other. Specifically, consider the GEF conditioned on the rare hole event that there are no zeros in a given large Jordan domain. We show that in the natural scaling limit, a quadrature domain enclosing the hole emerges as a forbidden region, where the zero density vanishes. Moreover, we give a description of the class of holes for which the forbidden region is a disk. The connecting link between random zeros and potential theory is supplied by a constrained extremal problem for the Zeitouni‐Zelditch functional. To solve this problem, we recast it in terms of a seemingly novel obstacle problem, where the solution is forced to be harmonic inside the hole.

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Random complex zeroes, I. Asymptotic normality

Abstract: We consider three models (elliptic, flat and hyperbolic) of Gaussian random analytic functions distinguished by invariance of their zeroes distribution. Asymptotic normality is proven for smooth functionals (linear statistics) of the set of zeroes.

Cited by 117 publications

References 24 publications

Critical points of multidimensional random Fourier series: Variance estimates

Critical points of multidimensional random Fourier series: Variance estimates

Inhomogeneous random zero sets

The forbidden region for random zeros: Appearance of quadrature domains

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