Sum rules via large deviations: Extension to polynomial potentials and the multi-cut regime

Abstract

A sum rule is an identity connecting the entropy of a measure with coefficients involved in the construction of its orthogonal polynomials, the Jacobi coefficients. Our paper is an extension of [15] where we showed sum rules by using only probabilistic tools, namely the large deviations theory. Here, we prove large deviation principles for the weighted spectral measure of unitarily invariant random matrices in two general situations: first, when the equilibrium measure is not necessarily supported by a single interval and second, when the potential is a nonnegative polynomial. The rate functions can be expressed as functions of the Jacobi coefficients. These new large deviation results lead to new sum rules both for the one and the multi-cut regime and also answer a conjecture stated in [15] concerning general sum rules.