10 - Circular Ensembles (Continued)

Abstract

Systems with no time reversal invariance may be characterized by unitary random matrices. This chapter discusses the orthogonal ensemble which is the most important from a practical point of view. It is also one of the most complicated from the mathematical point of view. The theorem presented in the chapter 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.