Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

双曲高斯解析函数零点过度拥挤现象

Overcrowding for zeros of Hyperbolic Gaussian analytic functions

本文研究单位圆盘内双曲不变高斯解析函数零点过度拥挤稀有事件的渐近概率，揭示当 L<1 时过度拥挤概率的指数衰减阶远小于零点亏缺概率，与平面模型形成鲜明对比。

Continuum Percolation for Gaussian zeroes and Ginibre eigenvalues

We study continuum percolation on certain negatively dependent point processes on \R^2. Specifically, we study the Ginibre ensemble and the planar Gaussian zero process, which are the two main natural models of translation invariant point processes on the plane exhibiting local repulsion. For the Ginibre ensemble, we establish the uniqueness of infinite cluster in the supercritical phase. For the Gaussian zero process, we establish that a non-trivial critical radius exists, and we prove the uniqueness of infinite cluster in the supercritical regime.

Large deviations for the zero set of an analytic function with diffusing coefficients

The "hole probability" that the zero set of the time dependent planar Gaussian analytic function f(z,t) = sum_(n=0)^infty a_n(t) z^n/sqrt(n!), where a_n(t) are i.i.d. complex valued Ornstein-Uhlenbeck processes, does not intersect a disk of radius R for all 0<t<T decays like exp(-Te^(cR^2)). This result sharply differentiates the zero set of f from a number of canonical evolving planar point processes. For example, the hole probability of the perturbed lattice model {sqrtπ(m,n) + c zeta_{m,n}: m,n integers} where zeta_(m,n) are i.i.d. Ornstein-Uhlenbeck processes decays like exp(-cTR^4). This …

Rigidity and Tolerance in point processes: Gaussian zeroes and Ginibre eigenvalues

Let X be a translation invariant point process on the complex plane and let D be a bounded open set whose boundary has zero Lebesgue measure. We ask what does the point configuration obtained by taking the points of X outside D tell us about the point configuration inside D? We show that for the Ginibre ensemble, it determines the number of points in D. For the translation-invariant zero process of a planar Gaussian Analytic Function, we show that it determines the number as well as the centre of mass of the points in D. Further, in both models we prove that the outside says "nothing more" abo…

Linear response and moderate deviations: hierarchical approach. V

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…