Linear statistics of matrix ensembles in classical background

摘要

Given a joint probability density function of N real random variables, {x(j)}(j=1)(N), obtained from the eigenvector-eigenvalue decomposition of N x N random matrices, one constructs a random variable, the linear statistics, defined by the sum of smooth functions evaluated at the eigenvalues or singular values of the random matrix, namely, Sigma(N)(j=1) F(x(j)). For the joint PDFs obtained from the Gaussian and Laguerre ensembles, we compute, in this paper, the moment-generating function E-beta(exp(-lambda Sigma(j) F(x(j)))), where E-beta denotes expectation value over the orthogonal (beta = 1) and symplectic (beta = 4) ensembles, in the form one plus a Schwartz function, none vanishing over R for the Gaussian ensembles and R+ for the Laguerre ensembles. These are ultimately expressed in the form of the determinants of identity plus a scalar operator, from which we obtained the large N asymptotic of the linear statistics from suitably scaled F(center dot). Copyright (C) 2016 John Wiley & Sons, Ltd.