Large deviation principles via spherical integrals

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