Large Deviation Principles via Spherical Integrals

70 pages, we added the description of measures which we have large deviation lower bound using the free product with amalgamationIn 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 $UB_NU^*$, 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 $A_N+UB_NU^*$, 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 $K_{\boldsymbol\lambda_N \boldsymbol\eta_N}$, for two sequences of partitions $\boldsymbol\lambda_N, \boldsymbol\eta_N$ with at most $N$ rows; 4. a large deviation upper bound for the Littlewood-Richardson coefficients $c_{\boldsymbol\lambda_N \boldsymbol \eta_N}^{\boldsymbol \kappa_N}$, for three sequences of partitions $\boldsymbol\lambda_N, \boldsymbol \eta_N, \boldsymbol \kappa_N$ with at most $N$ rows, and their complementary lower bounds at nice measures