Random complex zeroes, III. Decay of the hole probability

Sodin, Mikhail; Tsirelson, Boris

doi:10.1007/bf02785373

Cited by 58 publications

(94 citation statements)

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“…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%

“…When we take φ(z) = |z| 2 /2 then the asymptotic decay of the hole probability for the zero set of the GAF defined via bases was computed in [ST05], and the more precise version…”

Section: Corollarymentioning

confidence: 99%

“…We borrow many of the ideas used here from [ST05] and [SZZ05], but some modifications are necessary to deal with the fact that φ is non-radial and we are in a non-compact setting. The key ingredient in the proof of Theorem 3 is the following lemma.…”

Section: Large Deviationsmentioning

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

confidence: 99%

“…When we take φ(z) = |z| 2 /2 then the asymptotic decay of the hole probability for the zero set of the GAF defined via bases was computed in [ST05], and the more precise version…”

Section: Corollarymentioning

confidence: 99%

Section: Large Deviationsmentioning

confidence: 99%

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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“…Theorem (Sodin, Tsirelson [7] Theorem 1). Let ψ(z) = In [9], the authors considered the case of Gaussian random sections: let M be a compact Kähler manifold with complex dimension m and (L, h) → M be a positive holomorphic line bundle.…”

Section: Introductionmentioning

confidence: 98%

Hole probabilities of SU(m+ 1) Gaussian random polynomials

Zhu

2014

Anal. PDE

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Abstract. In this paper, we study hole probabilities P 0,m (r, N ) of SU (m + 1) Gaussian random polynomials of degree N over a polydisc (D(0, r)) m . When r ≥ 1, we find asymptotic formulas and decay rate of log P 0,m (r, N ). In dimension one, we also consider hole probabilities over some general open sets and compute asymptotic formulas for the generalized hole probabilities P k,1 (r, N ) over a disc D(0, r). IntroductionHole probability is the probability that some random field never vanishes over some set. The case of Gaussian random entire functions was studied by Sodin and Tsirelson:In [9], the authors considered the case of Gaussian random sections: let M be a compact Kähler manifold with complex dimension m and (L, h) → M be a positive holomorphic line bundle. γ N denotes the Gaussian probability measure on H Therefore, it is natural to ask: can we find sharp constants C 1 , C 2 in the above two theorems and furthermore, is it possible to obtain an asymptotic formula and a decay rate for the hole probability? Using Cauchy's integral estimates, Nishry answered this question in the random entire function case:Gaussian random variables. ThenThis inspires us that for those line bundles with polynomial sections, maybe it is possible to find an asymptotic formula for the hole probability.If P 0,m (r, N ) denotes the hole probability of SU (m+1) Gaussian random polynomials over the polydisc D(0, r) m , d m x is the Lebesgue measure on R m andis a continuous function defined over the standard simplex Σ m ∶= {x = (x 1 , . . . ,x i ≤ 1}(here we adopt the convention that 0 log 0 = 0), we have the following results:where,Here when m = 1, we takeRemark 0.3. Theorem 0.1 can be derived from Theorem 0.2 as when r ≥ 1, {x ∈ Σ m ∶ E r (x) ≥ 0} = Σ m and α 0 (r, m) = 1. In fact we could have proved this general case directly. But the idea of the proof would turn out to be extremely difficult to follow.Corollary 0.4. In the case of m = 1, the asymptotic formula for the logarithm of the hole probability over a disc exists for all r > 0:α 0 (2 log r + 1 − log α 0 ),Because of the simplicity of one dimensional case, we can obtain more about the hole probability of SU (2) Gaussian random polynomials:Theorem 0.5. If U ⊂ C is a bounded simply connected domain containing 0 and ∂U is a Jordan curve. Let φ ∶ D(0, 1) → U be a biholomorphism given by the Riemmann mapping theorem such that φ(0) = 0(thus φ is unique up to the composition of a unitary transformation of C). Then the hole probability P 0,1 (U, N ) of SU (2) Gaussian random polynomials of degree N over U satisfiesAlso in dimension one, it makes sense to study the number of zeros in some set. So let a generalized hole probability P k,1 (r, N ) be the probability that an SU (2) Gaussian random polynomial of degree N has no more than k zeros in D(0, r), then the following theorem shows that asymptotic formula of log P k,1 (r, N ) exists: Theorem 0.6. For all k ≥ 0 and r > 0:where α 0 = α 0 (r, 1) ∈ (0, 1] is given in Theorem 0.2.We should remark here that in all the cases w...

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Gaussian Complex Zeros on the Hole Event: The Emergence of a Forbidden Region

Ghosh

Nishry

2018

Comm Pure Appl Math

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Consider the Gaussian entire function Fℂ(z)=∑k=0∞ξkzkk!, z∈ℂ, where {ξk} is a sequence of independent standard complex Gaussians. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the plane ℂ. It has been of considerable interest to study the statistical properties of the zero set, particularly in comparison to other planar point processes. We show that the law of the zero set, conditioned on the function Fℂ having no zeros in a disk of radius r and normalized appropriately, converges to an explicit limiting Radon measure on ℂ as r → ∞. A remarkable feature of this limiting measure is the existence of a large “forbidden region” between a singular part supported on the boundary of the (scaled) hole and the equilibrium measure far from the hole. In particular, this answers a question posed by Nazarov and Sodin, and is in stark contrast to the corresponding result of Jancovici, Lebowitz, and Manificat in the random matrix setting: there is no such forbidden region for the Ginibre ensemble. © 2018 Wiley Periodicals, Inc.

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Random complex zeroes, III. Decay of the hole probability

Abstract: The 'hole probability' that a random entire functionwhere ζ 0 , ζ 1 , . . . are Gaussian i.i.d. random variables, has no zeroes in the disc of radius r decays as exp(−cr 4 ) for large r.

Cited by 58 publications

References 5 publications

Inhomogeneous random zero sets

Inhomogeneous random zero sets

Hole probabilities of SU(m+ 1) Gaussian random polynomials

Gaussian Complex Zeros on the Hole Event: The Emergence of a Forbidden Region

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