Gaussian matrix model in an external field and non-intersecting Brownian motions

We study the Gaussian hermitian random matrix ensemble with an external matrix which has an arbitrary number of eigenvalues with arbitrary multiplicity. We compute the limiting eigenvalues correlations when the size of the matrix goes to infinity in non-critical regimes. We show that they exhibit universal behavior and can be expressed with the Sine and Airy kernels. We also briefly review the implication of these results in terms of non-intersecting Brownian motions.