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The estimation of various matrix integrals as the size of the matrices goesto infinity is motivated by theoretical physics, geometry and free probabilityquestions. On a rigorous ground, only integrals of one matrix or of severalmatrices with simple quadratic interaction (called AB interaction) could beevaluated so far. In this article, we follow an idea widely developped in thephysics litterature, which is based on character expansion, to study morecomplex interaction. We more specifically consider a model defined in thespirit of the 'dually weighted graph model' studied by V. A. Kazakov, M.Staudacher and T. Wynter, but with a cutoff function such that the matrixintegral and its character expansion converge. We prove that the free energy ofthis model converges as the size of the matrices go to infinity and study thesaddle points of the limit.Comment: Revised version : the organisation of the paper was changed quite  deeply, namely we show a large deviation result for the empirical measures of  Young tableaux in a bit more general context than dually weighted graphs  models so that it can be used more easily to treat other models. In the  concluding remarks, we provide another possible application of our result and  a discussion about the links with maps enumeration. Some references are adde

Character expansion method for the first order asymptotics of a matrix integral