Geometrically, zeroes of a Gaussian analytic function are intersection points of an analytic curve in a Hilbert space with a randomly chosen hyperplane. Mathematical physics provides another interpretation as a gas of interacting particles. In the last decade, these interpretations influenced progress in understanding statistical patterns in the zeroes of Gaussian analytic functions, and led to the discovery of canonical models with invariant zero distribution. We shall discuss some of recent results in this area and mention several open questions.
Correlation functions for random complex zeroes: strong clustering and local universality
证明了高斯整函数零点的k点关联函数的强聚类性。在证明过程中，我们还得到了任意非退化高斯解析函数零点的k点函数的泛局部界。在本文的第二部分中，我们证明了强聚类产生的高斯整函数的零点的一些线性统计量的波动的渐近正态性，特别是，在可测量的大区域的零点的数量。这补充了我们最近在论文“Fluctuations in random complex zeroes”（arXiv：1003.4251v1）中的结果。
We study the zeroes of a family of random holomorphic functions on the unit disc, distinguished by their invariance with respect to the hyperbolic geometry. Our main finding is a transition in the limiting behaviour of the number of zeroes in a large hyperbolic disc. We find a normal distribution if the covariance decays faster than a certain critical value. In contrast, in the regime of 'long-range dependence' when the covariance decays slowly, the limiting distribution is skewed. For a closely related model we emphasise a link with Gaussian multiplicative chaos.
It is known that the Cauchy's argument principle, applied to an holomorphic function $f$, requires that $f$ has no zeros on the curve of integration. In this short note, we give a generalization of such a principle which covers the case when $f$ has zeros on the curve, as well as an application.
We show that the analytic continuations of Helson zeta functions $ ζ_χ(s)= \sum_1^{\infty}χ(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole critical strip $ 1/2 < \Re s <1 $ under Riemann Hypothesis.
The geometric Kannan-Lovasz-Simonovits lemma, dimension-free estimates for volumes of sublevel sets of polynomials, and distribution of zeroes of random analytic functions
The Moderate Deviations Principle (MDP) is well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. Here it is established for some planary random fields of the form $ X_t = ψ(G_t) $ obtained from a Gaussian random field $ G_t $ via a function $ ψ$, and consequently, for zeroes of the Gaussian Entire Function. Version 2: Appendix "Reader's guide to parts I-V" added. Minor changes, as follows. Formulations corrected: (2.1), 3.12(b), 4.5. Proofs corrected: 2.5, 2.6, 3.12, 3.16, 3.19, 4.13, 4.14, 4.15, 5…
We investigate the complex Gaussian as well as non-Gaussian distributed random analytical and entire functions (complex entire random field) and calculate their domain of definiteness (radius of convergence) as well as some important characteristics: order and type. As a consequence we deduce that all the mentioned characteristics, under very natural conditions, are deterministic (non-random) with probability one and we calculate them. Moreover we exhibit some examples to show the exactness of the obtained results.
We present the continuous wavelet transform (WT) of white Gaussian noise and establish a connection to the theory of Gaussian analytic functions. Based on this connection, we propose a methodology that detects components of a signal in white noise based on the distribution of the zeros of its continuous WT. To illustrate that the continuous theory can be employed in a discrete setting, we establish a uniform convergence result for the discretized continuous WT and apply the proposed method to a variety of acoustic signals.
