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 $UB_NU^*$, 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 $K_{\boldsymbolλ_N \boldsymbolη_N}$, for two sequences of partitions $\boldsymbolλ_N, \boldsymbolη_N$ with at most $N$ rows; 4. a large deviation upper bound for the Littlewood-Richardson coefficients $c_{\boldsymbolλ_N \boldsymbol η_N}^{\boldsymbol κ_N}$, for three sequences of partitions $\boldsymbolλ_N, \boldsymbol η_N, \boldsymbol κ_N$ with at most $N$ rows, and their complementary lower bounds at nice measures.