ON THE VARIANCE OF LINEAR STATISTICS OF HERMITIAN RANDOM MATRICES

Linear statistics, a random variable built out of the sum of the evaluation of functions at the eigenvalues of a N x N random matrix, Sigma(N)(j=1) f (x(j)) or tr f (M), is an ubiquitous statistical characteristics in random matrix theory. Hermitian random matrix ensembles, under the eigenvalue-eigenvector decompositions 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 tr f (M) is of the form of Sigma(K)(n=1) n (d(n))(2) and d(n) is related to the expansion coefficients c(n) of the polynomial f (x) = Sigma(K)(n=1) c(n) (P) over cap (n) (x), where (P) over cap (n) (x) are polynomials of degree n, orthogonal with respect to the weights 1/root(b-x) (x-a), root(b-x) (x-a), root(b-x) (x-a)/x, (0 < a < x < b), root(b-x) (x-a)/x(1-x), (0 < a < x < b < 1), respectively.