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 unitary 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 is no longer symmetric, however, and is not readily accessible to standard methods. In particular, finding the level spacing probability is always a delicate problem in Fredholm theory, and we have to reconsider the problem within our model. We find a class of universality for the level spacing distribution 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 nonlinear differential equations, which turn out to form a Hamiltonian system. As a result we find that the level spacing probability $p\left(s\right)\mathrm{}$ behaves like $\exp \left[-{\mathrm{Cs}}^{8/3}\right]$ for large spacing s; this is consistent with the asymptotic behavior $\exp \left[-{\mathrm{Cs}}^{2\beta +2\mathrm{}}\right],$ whenever the density of state behaves near the edge as $\rho \left(\lambda \right)\sim {\lambda }^{\beta }.$