CORRELATION-FUNCTIONS, CLUSTER FUNCTIONS, AND SPACING DISTRIBUTIONS FOR RANDOM MATRICES

The usual formulas for the correlation functions in orthogonal and sym plectic matrix models express them as quaternion determinants. From th is representation one can deduce formulas for spacing probabilities in terms of Fredholm determinants of matrix-valued kernels. The derivati ons of the various formulas are somewhat involved. In this article we present a direct approach which leads immediately to scalar kernels fo r the unitary ensembles and matrix kernels for the orthogonal and symp lectic ensembles, and the representations of the correlation functions , cluster functions, and spacing distributions in terms of them.