The probabilities for several consecutive eigenvalues of a random matrix

Abstract

We study statistical properties of the eigenvalues of a random matrix belonging to the so-called Gaussian unitary and Gaussian orthogonal ensembles. An expression is given for the probability that an interval of length 2t contains exactly n levels (their positions are specified or not). This result is used to derive the probability density of the spacing between a level and its nth neighbour (the positions of the (n-1) intermediate levels are specified or not).