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In an earlier work we had considered a Gaussian ensemble of random matricesin the presence of a given external matrix source. The measure is no longerunitary invariant and the usual techniques based on orthogonal polynomials, oron the Coulomb gas representation, are not available. Nevertheless the n-pointcorrelation functions are still given in terms of the determinant of a kernel,known through an explicit integral representation. This kernel is no longersymmetric though and is not readily accessible to standard methods. Inparticular finding the level spacing probability is always a delicate problemin Fredholm theory, and we have to reconsider the problem within our model. Wefind a new class of universality for the level spacing distribution when thespectrum of the source is ajusted to produce a vanishing gap in the density ofthe state. The problem is solved through coupled non-linear differentialequations, which turn out to form a Hamiltonian system. As a result we findthat the level spacing probability $p(s)$ behaves like $\exp[ - Cs^{{8\over{3}}}]$ for large spacing $s$; this is consistent with the asymptoticbehavior $\exp[ - C s^{2 \beta + 2}]$, whenever the density of state behavesnear the edge as $\rho(\lambda)\sim \lambda^{\beta}$.Comment: 25 pages, late

Level spacing of random matrices in an external source