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 UBNU*, for a sequence of N× N diagonal matrices BN and unitary Haar distributed matrices U; 2. a large deviation upper bound for the empirical eigenvalue distribution of AN+UBNU*, for two sequences of N× N diagonal matrices AN, BN, 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 KλN ηN, for two sequences of partitions λN, ηN with at most N rows; 4. a large deviation upper bound for the Littlewood-Richardson coefficients cλN ηN N, for three sequences of partitions λN, ηN, N with at most N rows, and their complementary lower bounds at nice measures.