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 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 $K_{\boldsymbol\lambda_N \boldsymbol\eta_N}$,fortwosequencesofpartitions$\boldsymbol\lambda_N, \boldsymbol\eta_N$withatmost$N$ rows;
4. a large deviation upper bound for the Littlewood-Richardson coefficients $c_{{\mathit{\lambda }}_N{\mathit{\eta }}_N}^{{\mathit{\kappa }}_N}$, for three sequences of partitions $\boldsymbol\lambda_N, \boldsymbol \eta_N, \boldsymbol \kappa_N$withatmost$N$ rows, and their complementary lower bounds at nice measures.