Random-matrix models with the logarithmic-singular level confinement: Method of fictitious fermions

Abstract

A joint distribution function of N eigenvalues of a U(N) invariant random-matrix ensemble can be interpreted as a probability density of finding N fictitious non-interacting fermions to be confined in a one-dimensional space. Within this picture, a general formalism is developed to study the eigenvalue correlations in non-Gaussian ensembles of large random matrices possessing non-monotonic log-singular level confinement. An effective one-particle Schrödinger equation for wavefunctions of fictitious fermions is derived. It is shown that eigenvalue correlations are completely determined by the Dyson density of states and by the parameter of the logarithmic singularity. Closed analytical expressions for the two-point kernel in the origin, bulk and soft-edge scaling limits are deduced in a unified way, and novel universal correlations are predicted near the end point of the single spectrum support.