摘要: Consider the spherical integral I(β)N(DN, EN)≔∫ exp{N tr(UDNU*EN)} dmβN(U), where mβN denote Haar measure on orthogonal group ON when β=1 and unitary UN β=2, DN, EN are diagonal real matrices whose spectral measures converge to μD, μE. In this paper we prove existence represent as solution a variational problem limit I(β)(μD, μE)≔lim N−2 log EN). This appears in so-called “matrix models” but also evaluation of large deviations generalized Wishart matrices. Our technique is based stochastic calculus, deviations, elements from free probability.
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