Large Deviations Asymptotics for Spherical Integrals

By Alice Guionnet, Ofer Zeitouni. 2002

152 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/63e7055fa5407c24975002b1c7dc8dceb5acff72

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

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Citations in the corpus

Second order asymptotics for matrix models (2006, background): “This is due to Klein’s lemma (see [15]) which states that the trace of a real convex function of a self-adjoint matrix is a convex function of the entries of the matrix.”
A Fourier view on the R-transform and related asymptotics of spherical integrals (2005, background): “1 of [13] that lim N→∞ 1 N2 log I (β) N (DN , EN ) = I (μD, μE) (1) exists under some technical assumptions and a (complicated) formula for this limit is given.”
Mean field games (2007, background): “It turns out that when d = 1 a particular case of such systems has been studied by A. Guionnet [13] and A. Guionnet and O. Zeitouni [ 14 ] in the context of large deviations theory (and their applications to Physics).”
Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices (2002, methodology): “This result in turn provides a large deviation upper bound for the spectral measure of generalized Gaussian sample covariance matrices, but a full large deviation principle was very recently obtained in [ 15 ] by O. Zeitouni and myself in this restricted context.”
Cleaning large correlation matrices: tools from random matrix theory (2016, background): “Another soluble case is when one of the two matrix has a Flat distribution [180].”
Moments and cumulants of polynomial random variables on unitarygroups, the Itzykson-Zuber integral, and free probability (2002, background): “that the distributions of X and Y admit a smooth limit. The existence of a large d limit remains a non-obvious mathematical problem. It has been solved very recently by A. Guionnet and O. Zeitouni in [GZ02] under the assumption that X and Y admits a limit bounded distribution and are Hermitian. Howe”
Large deviations of the extreme eigenvalues of random deformations of matrices (2010, background): “A large deviation principle was also established in [25] for the law of the spectral measure of a random matrix given as the sum of a self-adjoint Gaussian Wigner random matrix and a deterministic self-adjoint matrix (or as a Gaussian Wishart matrix with non trivial covariance matrix).”
Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles (2006, background): “As in (14), we denote by J0 the set of indices i such that we have such a pairing.”
Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails (2015, methodology): “In the setting of the so-called Wigner deformed ensemble, the large deviations of the empirical spectral measure were studied, first in [10] and then in [17], in which a large deviations principle was established for the empirical spectral measure of the sum of Gaussian Wigner matrix and a determini”
Asymptotics of unitary and orthogonal matrix integrals (2006, methodology): “Then, the limit of N − 2 log HCIZ ( A, B ) is given, according to [18], by where the inﬁmum is taken over m, ρ so that the measure-valued process µ t ( dx ) = ρ t ( x ) dx is a continuous process, µ 0 = µ A , µ 1 = µ B and The inf over ( ρ t , m t ) is taken (see [14]) at the solution of an Euler…”
Large Deviations Upper Bounds for the Laws of Matrix-Valued Processes and Non-Communicative Entropies (2001, background): “Zeitouni and one of the authors were able to improve into a full large deviation lower bound (see [17]).”