数值计算中随机数据的普遍性。案例研究、分析结果及一些推测

A Universality Property of Gaussian Analytic Functions

We consider random analytic functions defined on the unit disk of the complex plane as power series such that the coefficients are i.i.d., complex valued random variables, with mean zero and unit variance. For the case of complex Gaussian coefficients, Peres and Virág showed that the zero set forms a determinantal point process with the Bergman kernel. We show that for general choices of random coefficients, the zero set is asymptotically given by the same distribution near the boundary of the disk, which expresses a universality property. The proof is elementary and general.

Universality of Correlations for Random Analytic Functions

We review a result obtained with Andrew Ledoan and Marco Merkli. Consider a random analytic function $f(z) = \sum_{n=0}^{\infty} a_n X_n z^n$, where the $X_n$'s are i.i.d., complex valued random variables with mean zero and unit variance, and the coefficients $a_n$ are non-random and chosen so that the variance transforms covariantly under conformal transformations of the domain. If the $X_n$'s are Gaussian, this is called a Gaussian analytic function (GAF). We prove that, even if the coefficients are not Gaussian, the zero set converges in distribution to that of a GAF near the boundary of th…

Universality for zeros of random analytic functions

Let $ξ_0,ξ_1,...$ be independent identically distributed (i.i.d.) random variables such that $\E \log (1+|ξ_0|)<\infty$. We consider random analytic functions of the form $$ G_n(z)=\sum_{k=0}^{\infty} ξ_k f_{k,n} z^k, $$ where $f_{k,n}$ are deterministic complex coefficients. Let $ν_n$ be the random measure assigning the same weight $1/n$ to each complex zero of $G_n$. Assuming essentially that $-\frac 1n \log f_{[tn], n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $ν_n$ converges weakly to some deterministic measure which is characterized in terms of the …

Universality of General $β$-Ensembles

We prove the universality of the $β$-ensembles with convex analytic potentials and for any $β>0$, i.e. we show that the spacing distributions of log-gases at any inverse temperature $β$ coincide with those of the Gaussian $β$-ensembles.

非解析随机信号零点期望个数的全局普适性

Global universality of the expected number of zeros of non-analytic random signals

该研究证明了非解析随机周期信号零点期望个数的全局普适性，扩展了相关局部普适性结果，采用新的中心极限定理及一致可积性等估计完成推导。

Two-Point Correlation Functions and Universality for the Zeros of Systems of SO(n+1)-invariant Gaussian Random Polynomials

We study the two-point correlation functions for the zeroes of systems of $SO(n+1)$-invariant Gaussian random polynomials on $\mathbb{RP}^n$ and systems of ${\rm isom}(\mathbb{R}^n)$-invariant Gaussian analytic functions. Our result reflects the same "repelling," "neutral," and "attracting" short-distance asymptotic behavior, depending on the dimension, as was discovered in the complex case by Bleher, Shiffman, and Zelditch. For systems of the ${\rm isom}(\mathbb{R}^n)$-invariant Gaussian analytic functions we also obtain a fast decay of correlations at long distances. We then prove that the c…