  In this paper, a sum rule means a relationship between a functional defined
on a subset of all probability measures on $\mathbb{R}$ involving the reverse
Kullback-Leibler divergence with respect to a particular distribution and
recursion coefficients related to the orthogonal polynomial construction. The
first sum rule is the Szeg\H{o}-Verblunsky theorem (see Simon (2011) Theorem
1.8.6 p. 29) and concerns the case of trigonometrical polynomials on the torus.
More recently, Killip and Simon (2003) have given a revival interest to this
subject by showing a quite surprising sum rule for measures dominating the
semicircular distribution. Indeed, this sum rule includes a contribution of the
atomic part of the measure away from the support of the circular law. In this
paper, we recover this sum rules by using only large deviations tools on random
matrices. Furthermore, we obtain new (up to our knowledge) magic sum rules for
the reverse Kullback-Leibler divergence with respect to the Pastur-Marchenko or
Kesten-McKay distributions. As in the semicircular case, these formulas
generally include a contribution of the atomic part appearing away from the
support of the reference measure. From the point of view of large deviations,
we obtain these formulas by showing a large deviation principle for the point
mass measure concentrated on random eigenvalues of classical ensembles and
combine it with a large deviation principle due to Gamboa Rouault (2011) for
the recursion coefficients.
