Large Deviations Asymptotics for Spherical Integrals

Consider the spherical integral I(β)N(DN, EN)≔∫ exp{N tr(UDNU*EN)} dmβN(U), where mβN denote the Haar measure on the orthogonal group ON when β=1 and on the unitary group UN when β=2, and DN, EN are diagonal real matrices whose spectral measures converge to μD, μE. In this paper we prove the existence and represent as solution to a variational problem the limit I(β)(μD, μE)≔lim N−2 log I(β)N(DN, EN). 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.