Random hermitian matrices in an external field

In this article, a model of random hermitian matrices is considered, in which the measure $\exp(-S)$ contains a general U(N)-invariant potential and an external source term: $S=N\tr(V(M)+MA)$. The generalization of known determinant formulae leads to compact expressions for the correlation functions of the energy levels. These expressions, exact at finite $N$, are potentially useful for asymptotic analysis.