Large Deviation Principles via Spherical Integrals

Abstract

In this article, we develop a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the limits of spherical integrals obtained in [46,47]. As examples, we obtain 1. a large deviation principle for the empirical distribution of the diagonal entries of $U{B}_N{U}^{\ast }$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary Haar distributed matrices $U$; 2. a large deviation upper bound for the empirical eigenvalue distribution of $A_N+U{B}_N{U}^{\ast }$, for two sequences of $N\times N$ diagonal matrices $A_N,B_N$, and their complementary lower bounds at measures which are described by the free product with amalgamation; 3. a large deviation principle for the Kostka number $K_{{\mathbf{\lambda }}_N{\mathbf{\eta }}_N}$, for two sequences of partitions ${\mathbf{\lambda }}_N,{\mathbf{\eta }}_N$ with at most $N$ rows; 4. a large deviation upper bound for the Littlewood-Richardson coefficients $c_{{\mathbf{\lambda }}_N{\mathbf{\eta }}_N}^{{\mathbf{\kappa }}_N}$, for three sequences of partitions ${\mathbf{\lambda }}_N,{\mathbf{\eta }}_N,{\mathbf{\kappa }}_N$ with at most $N$ rows, and their complementary lower bounds at nice measures.

arXiv comment70 pages, journal version

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Cited by in corpus (4)

• On the Complex Asymptotics of the HCIZ and BGW Integrals
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• Edge Universality for Nonintersecting Brownian Bridges
• Spectral deconvolution of unitarily invariant matrix models