9 - Circular Ensembles. Correlation Functions, Spacing Distribution, etc.

Abstract

This chapter discusses circular ensembles, correlation functions, and spacing distribution. It presents a derivation of the joint probability density function for the eigenvalues of a unitary self-dual random matrix taken from the symplectic ensemble. The chapter presents a theorem that states that the statistical properties of N alternate angles θj, where eiθj are the eigenvalues of a symmetric unitary random matrix of order (2N 2N) taken from the orthogonal ensemble, are identical to those of the N angles φj, where eiφj are the eigenvalues of an N N quaternion self-dual unitary random matrix taken from the symplectic ensemble. The probability density function for the spacings of the eigenvalues in a self-dual, quaternion, and unitary random matrix is taken from the symplectic ensemble. Systems with no time-reversal invariance are characterized by unitary random matrices. The joint probability density function for the eigenvalue angles of such matrices is taken from the unitary ensemble.