THE SPECTRUM EDGE OF RANDOM-MATRIX ENSEMBLES

The scaled n-level distribution and scaled level spacing distribution for the small and large eigenvalues of various ensembles of random matrices are considered. Exact results for both these quantities are obtained for various special values of the parameters in the gaussian and Laguerre ensembles. On the basis of a Coulomb gas analogy, an asymptotic formula for the distribution of the smallest and largest eigenvalue is given in terms of the eigenvalue density at the spectrum edge.