A sum rule is an identity connecting the entropy of a measure with the coefficients involved in the construction of its orthogonal polynomials. It is possible to prove sum rules using large deviation theory. We consider the weighted spectral measure of random matrices and prove a large deviation principle when the size of the matrix tends to infinity. The measure may be described by its spectral information or its recursion coefficients. This allows to write the rate function in two different ways, which leads to the sum rule.

In this talk I present an extension to unitary random matrices in the multi-cut case, when the limit of the spectral measure is supported by several arcs of the unit circle. In this case the rate function cannot be given explicitly. We can still state a sum rule under additional conditions on the recursion coefficients, which are related to finding a specific representative in the Aleksandrov class of measures. The talk is based on a joint work with Fabrice Gamboa and Alain Rouault.