Number variance of random zeros on complex manifolds

Asymptotic expansion of the variance of random zeros on complex manifolds

Linear statistics of random zero sets are integrals of smooth differential forms over the zero set and as such are smooth analogues of the volume of the random zero set inside a fixed domain. We derive an asymptotic expansion for the variance of linear statistics of the zero divisors of random holomorphic sections of powers of a positive line bundle on a compact Kähler manifold. This expansion sharpens the leading-order asymptotics (in the codimension one case) given by Shiffman--Zelditch in 2010.

紧Kahler流形上一列线丛中随机全纯截面系统的方差估计

Variance Estimate of Systems of Random Holomorphic Sections in a Sequence of Line Bundles on Compact Kahler Manifolds

本文研究了紧Kahler流形上一列线丛的随机全纯截面系统零点的方差估计，在比现有文献更一般的概率测度条件下给出了方差估计，并涵盖了Gaussian和Fubini-Study等已知测度。

Number variance of random zeros on complex manifolds, II: smooth statistics

We consider the zero sets $Z_N$ of systems of $m$ random polynomials of degree $N$ in $m$ complex variables, and we give asymptotic formulas for the random variables given by summing a smooth test function over $Z_N$. Our asymptotic formulas show that the variances for these smooth statistics have the growth $N^{m-2}$. We also prove analogues for the integrals of smooth test forms over the subvarieties defined by $k<m$ random polynomials. Such linear statistics of random zero sets are smooth analogues of the random variables given by counting the number of zeros in an open set, which we proved…

Variance of the spectral numbers

A formula for the variance of the spectrum of a quasihomogeneous singularity is proved, using the G-function of a semisimple Frobenius manifold.

Number variance of random zeros

The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree $N$ in an open set $U \subset C$ as the degree $N \to \infty$, and more generally for the zeros of random holomorphic sections of high powers of any positive line bundle over any Riemann surface. The formulas were conjectured in special cases by Forrester and Honner. In higher dimensions, we give similar formulas for the variance of the volume inside a domain $U$ of the zero hypersurface of a random holomorphic section of a high power of a positive line bun…

Variance of real zeros of random orthogonal polynomials

We determine the asymptotics for the variance of the number of zeros of random linear combinations of orthogonal polynomials of degree $\leq n$ in subintervals $\left [ a,b\right ] $ of the support of the underlying orthogonality measure $μ$. We show that, as $n\rightarrow \infty $, this variance is asymptotic to $cn$, for some explicit constant $c>0$.