Sum rules via large deviations: extension to polynomial potentials and\n the multi-cut regime

A sum rule is an identity connecting the entropy of a measure with\ncoefficients involved in the construction of its orthogonal polynomials (Jacobi\ncoefficients). Our paper is an extension of Gamboa, Nagel and Rouault (2016),\nwhere we have showed sum rules by using only probabilistic tools (namely the\nlarge deviations theory). Here, we prove large deviation principles for the\nweighted spectral measure of unitarily invariant random matrices in two general\nsituations: firstly, when the equilibrium measure is not necessarily supported\nby a single interval and secondly, when the potential is a nonnegative\npolynomial. The rate functions can be expressed as functions of the Jacobi\ncoefficients. These new large deviation results lead to original sum rules both\nfor the one and the multi-cut regime and also answer a conjecture stated in\nGamboa, Nagel and Rouault (2016) concerning general sum rules.\n