Exact Wigner surmise type evaluation of the spacing distribution in the bulk of the scaled random matrix ensembles

Abstract

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices respectively. We show that the probability density function for the corresponding spacings between consecutive eigenvalues can be written exactly in the Wigner surmise type form $a(s) e^{-b(s)}$ for $a$ simply related to a Painlev\'e transcendent and $b$ its anti-derivative. A formula consisting of the sum of two such terms is given for the symplectic case (Hermitian matrices with real quaternion elements).