Large deviations asymptotics for spherical integrals

Consider the spherical integral I-N((beta))(D-N, E-N) := integral exp{N tr(UDNU*E-N)} dm(N)(beta)(U), where m(N)(beta) denote the Haar measure on the orthogonal group O-N when beta = 1 and on the unitary group U-N when beta = 2, and D-N, E-N are diagonal real matrices whose spectral measures converge to mu(D), mu(E) In this paper we prove the existence and represent as solution to a variational problem the limit I-(beta)(mu(D), mu(E)):= lim N-2 log I-N((beta))(D-N, E-N). This limit appears in so-called "matrix models" but also in the evaluation of large deviations of the spectral measure of generalized Wishart matrices. Our technique is based on stochastic calculus, large deviations. and elements from free probability. (C) 2002 Elsevier Science (USA).