Large deviation principles via spherical integrals (Q2693722)

This is an extended and detailed work, in which the authors develop a framework to study the large deviation principle for matrix models and their quantized versions. The method is tilting the measures using the limits of spherical integrals. In particular, one approach to the large deviation principles is to tilt the measure by an exponential moment generating function. This technique, for instance, allows to prove the celebrated Cramer's theorem for the empirical distribution of independent variables taking their values in Polish spaces. The authors make a step ahead and study the large deviations for the empirical measures of different models of random matrices. The first model concerns the diagonal entries of Hermitian matrices with given eigenvalues. Then the authors mention that the approach by tilting the original measure using the spherical integrals, requires only the derivatives for the limiting spherical integral. As a consequence, it gives new understandings of the Schur-Horn theorem and Horn's problem, as well as the evaluation of the asymptotics of Kostka numbers and Littlewood-Richardson coefficients. As the first application of the spherical integral approach, the authors study large deviations of the randomized Schur-Horn theorem and Horn's problem. As the second application of the spherical integral approach, they derive the large deviation principle of Kostka numbers and the large deviation upper bound for the Littlewood-Richardson coefficients.