Random matrix ensembles with an effective extensive external charge

Abstract

Recent theoretical studies of chaotic scattering have encounted ensembles of random matrices in which the eigenvalue probability density function contains a one-body factor with an exponent proportional to the number of eigenvalues. Two such ensembles have been encounted: an ensemble of unitary matrices specified by the so-called Poisson kernel, and the Laguerre ensemble of positive definite matrices. Here we consider various properties of these ensembles. Jack polynomial theory is used to prove a reproducing property of the Poisson kernel, and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the Dyson circular ensemble. For the Laguerre ensemble, the scaled global density is calculated exactly for all even values of the parameter $\beta$, while for $\beta = 2$ (random matrices with unitary symmetry), the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class.