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Large deviations asymptotics for spherical integrals. (English) Zbl 1002.60021

Authors’ summary: Consider the spherical integral \[ I^{(\beta)}_N(D_N,E_N):=\int\exp\{N\text{tr}(UD_NU^*E_N)\} dm^\beta_N(U), \] where \(m^\beta_N\) denotes the Haar measure on the orthogonal group \({\mathcal O}_N\) when \(\beta=1\) and on the unitary group \({\mathcal U}_N\) when \(\beta=2\), and \(D_N\), \(E_N\) are diagonal real matrices whose spectral measures converge to \(\mu_D\), \(\mu_E\). We prove the existence and represent as solution to a variational problem the limit \(I^{(\beta)}(\mu_D,\mu_E):=\lim N^{-2}\log I^{(\beta)}_N(D_N,E_N)\). 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.

MSC:

60F10 Large deviations
46L51 Noncommutative measure and integration
46L54 Free probability and free operator algebras
15B52 Random matrices (algebraic aspects)
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