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A sum rule is an identity connecting the entropy of a measure withcoefficients involved in the construction of its orthogonal polynomials (Jacobicoefficients). Our paper is an extension of Gamboa, Nagel and Rouault (2016),where we have showed sum rules by using only probabilistic tools (namely thelarge deviations theory). Here, we prove large deviation principles for theweighted spectral measure of unitarily invariant random matrices in two generalsituations: firstly, when the equilibrium measure is not necessarily supportedby a single interval and secondly, when the potential is a nonnegativepolynomial. The rate functions can be expressed as functions of the Jacobicoefficients. These new large deviation results lead to original sum rules bothfor the one and the multi-cut regime and also answer a conjecture stated inGamboa, Nagel and Rouault (2016) concerning general sum rules.

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