Download Zeros of Gaussian Analytic Functions and Determinantal Point by Manjunath Krishnapur, Yuval Peres, and Balint Virag J. Ben PDF

By Manjunath Krishnapur, Yuval Peres, and Balint Virag J. Ben Hough

ISBN-10: 0821843737

ISBN-13: 9780821843734

The publication examines in a few intensity very important sessions of aspect techniques, determinantal procedures and ``Gaussian zeros'', i.e., zeros of random analytic features with Gaussian coefficients. those methods proportion a estate of ``point-repulsion'', the place designated issues are much less more likely to fall with reference to one another than in tactics, reminiscent of the Poisson approach, that come up from autonomous sampling. however, the remedy within the ebook emphasizes using independence: for random energy sequence, the independence of coefficients is vital; for determinantal strategies, the variety of issues in a website is a sum of autonomous signs, and this yields a lovely clarification of the important restrict theorem (CLT) for this aspect count number. one other unifying topic of the e-book is invariance of thought of aspect procedures lower than usual transformation teams. The publication strives for stability among basic thought and urban examples. at the one hand, it offers a primer on glossy recommendations at the interface of likelihood and research. however, a wealth of determinantal strategies of intrinsic curiosity are analyzed; those come up from random spanning bushes and eigenvalues of random matrices, in addition to from unique energy sequence with determinantal zeros. the fabric within the publication shaped the root of a graduate direction given on the IAS-Park urban summer season tuition in 2007; the single historical past wisdom assumed might be obtained in first-year graduate classes in research and chance

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We show the details in the planar case (Λ = C) with L = 1. The proof is 1 2 virtually identical in the hyperbolic case. For β ∈ C, let fβ (z) = f(z + β)e− zβ− 2 |β| . 4 that fβ = f. We compute 1 2 + zw+β w E fβ (z)f(w) = e− zβ− 2 |β| . As β → ∞ this goes to 0 uniformly for z, w in any compact set. By Cauchy’s formula, the coefficients of the power series expansion of fβ around 0 are given by 1 2π i fβ (ζ) C ζ n +1 d ζ, where C(t) = e it , 0 ≤ t ≤ 2π. Therefore, for any n, the first n coefficients in the power series of f and the first n coefficients in the power series of fβ become uncorrelated and hence (by joint Gaussianity) independent, as β → ∞.

Thus δ | Z + U |−η 1(| Z + U | > 3 ) ≤ | Z/2|−η , so we have | Z + U |−η ≤ | Z + U |−η 1(| Z + U | ≤ 3 ) + | Z/2|−η . After taking expectations we get E| Z + U |−η ≤ B3 | z|−η P(Z + U ∈ dz) + E| Z/2|−η . Given Z, the conditional probability that Z + U ∈ dz is 1(z + Z ∈ D)/(π 2 ). So the first term can be written as | z|−η dz | z|−η dz P(Z ∈ z + D) ≤ P(| Z | < 4 ) 2 π π 2 B3 B3 −η = c η P(| Z | < 4 )(4 ) ≤ c η E| Z |−η , the last inequality is Markov’s. We conclude E| Z + U |−η ≤ c η E| Z |−η , as required.

1 holds for random analytic functions satisfying the following condition. 4) k B n f+ z (K) dz are uniformly integrable as → 0. P ROOF. Let ϕ : Ck → R be a continuous test function with compact support. 2 with the notation F = (f, . . 5) 1 k Vol(B ) Ck ϕ(x)|F (x)|2 1B k (F(x)) dx = 1 Vol(B k ) Bk ϕ[F −1 (y)]d y. 6) 1 Vol(B ) k B n f+ z (supp ϕ) dz , which is uniformly integrable by assumption. 5), let → 0 and use uniform integrability. 7) ϕ(z1 , . . , z k ) = ϕ d µk . E z∈Z k The left hand side, by the Fubini argument, becomes ϕ(x) E |F (x)|2 1(F(x) ∈ B k ) Vol(B k ) = ϕ(x) dx E |f (x1 ) · · · f (xk )|2 1(f(x1 ), .

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