  In this paper, we consider the addition of two matrices in generic position,
namely A + U BU * , where U is drawn under the Haar measure on the unitary or
the orthogonal group. We show that, under mild conditions on the empirical
spectral measures of the deterministic matrices A and B, the law of the largest
eigenvalue satisfies a large deviation principle, in the scale N, with an
explicit rate function involving the limit of spherical integrals. We cover in
particular all the cases when A and B have no outliers.
