Linear statistics of matrix ensembles in classical background |
| |
| Authors: | Chao Min Yang Chen |
| |
| Affiliation: | Department of Mathematics, University of Macau, Macau, China |
| |
| Abstract: | Given a joint probability density function of N real random variables,  , obtained from the eigenvector–eigenvalue decomposition of N × 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,  . For the joint PDFs obtained from the Gaussian and Laguerre ensembles, we compute, in this paper, the moment‐generating function  , where  denotes expectation value over the orthogonal (β = 1) and symplectic (β = 4) ensembles, in the form one plus a Schwartz function, none vanishing over  for the Gaussian ensembles and  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(·). Copyright © 2016 John Wiley & Sons, Ltd. |
| |
| Keywords: | random matrices linear spectral statistics orthogonal polynomials asymptotics |
|
|
