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

TL;DR

This paper derives asymptotic variance formulas for smooth statistics of zeros of random polynomials on complex manifolds, extending normality results to higher codimension cases.

Contribution

It provides new asymptotic formulas for variances of smooth linear statistics of random zeros and extends normality results to higher codimension zero sets.

Findings

Variance of smooth statistics grows as N^{m-2}

Asymptotic normality holds for smooth linear statistics in higher dimensions

Results extend previous variance and normality findings to codimension one zero sets

Abstract

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 elsewhere to have variances of order $N^{m-1/2}$. We use the variance asymptotics and off-diagonal estimates of Szego kernels to extend an asymptotic normality result of Sodin-Tsirelson to the case of smooth linear statistics for zero sets of codimension one in any dimension $m$.