Level spacing of random matrices in an external source

In an earlier work we considered a Gaussian ensemble of random matrices in the presence of a given external matrix source. The measure is no longer un itary invariant, and the usual techniques based on orthogonal polynomials, or on the Coulomb gas representation, are not available. Nevertheless the n -point correlation functions are still given in terms of the determinant of a kernel, known through an explicit integral representation. This kernel i s no longer symmetric, however, and is not readily accessible to standard m ethods. In particular, finding the level spacing probability is always a de licate problem in Fredholm theory, and we have to reconsider the problem wi thin our model. We find a class of universality for the level spacing distr ibution when the spectrum of the source is adjusted to produce a vanishing gap in the density of the state. The problem is solved through coupled nonl inear differential equations, which turn out to form a Hamiltonian system. As a result we find that the level spacing probability p(s) behaves like ex p[- Cs-8/3] for large spacing s; this is consistent with the asymptotic beh avior exp[- Cs2 beta+2], whenever the density of state behaves near the edg e as rho(lambda) similar to lambda(beta). [S1063-651X(98)00112-3].