Statistical Theory of the Energy Levels of Complex Systems. V

Abstract

This paper is divided into three disconnected parts. (i) An identity is proved which establishes an intimate connection between the statistical behavior of eigenvalues of random unitary matrices over the real field and over the quaternion field. It is proved that the joint distribution function of all the eigenvalues of a random unitary self-dual quaternion matrix of order N is identical with the joint distribution function of a set of N alternate eigenvalues of a random unitary symmetric matrix of order 2N. A corollary of this result is the following: the distribution of spacings between next-nearest-neighbor eigenvalues in a real symmetric matrix of large order is identical with the distribution of nearest-neighbor spacings in a self-dual Hermitian quaternion matrix of large order. (ii) A conjecture is made which gives an exact analytic formula for the partition function of a finite gas of N point charges free to move on an infinite straight line under the influence of an external harmonic potential. This conjecture is at the same time a statement about the statistical properties of the eigenvalues of Hermitian matrices whose elements are Gaussian random variables. (iii) A list is made of several other problems which remain unsolved in the statistical theory of eigenvalues of random matrices.