On the variance of linear statistics of Hermitian random matrices

TL;DR

This paper derives a formula for the variance of linear eigenvalue statistics of Hermitian random matrices when the test function is a polynomial, linking variance to polynomial coefficients and orthogonal polynomial expansions.

Contribution

It provides an explicit variance formula for polynomial functions of eigenvalues in Hermitian ensembles, connecting variance to orthogonal polynomial coefficients and weights.

Findings

Variance of tr f(M) is a sum involving polynomial coefficients and orthogonal polynomial degrees.

Variance formula depends on the specific weight function associated with the orthogonal polynomials.

The result applies to various weight functions, including those with singularities at endpoints.

Abstract

Linear statistics, a random variable build out of the sum of the evaluation of functions at the eigenvalues of a N times N random matrix,sum[j=1 to N]f(xj) or tr f(M), is an ubiquitous statistical characteristics in random matrix theory. Hermitian random matrix ensembles, under the eigenvalue-eigenvector decomposition give rise to the joint probability density functions of N random variables. We show that if f(.) is a polynomial of degree K, then the variance of trf(M), is of the form,sum[n=1 to K] n(d[n])square, and d[n] is related to the expansion coefficients c[n] of the polynomial f(x) =sum[n=0 to K] c[n] b Pn(x), where Pn(x) are polynomials of degree n, orthogonal with respect to the weights 1/[(b-x)(x-a)]^(1/2), [(b -x)(x -a)]^(1/2), [(b-x)(x-a)]^(1/2)/x; (0 < a < x < b), [(b-x)(x-a)]^(1/2)/[x(1-x)] ; (0 < a < x < b < 1), respectively.