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

Abstract

The estimation of various matrix integrals as the size of the matrices goes to infinity is motivated by theoretical physics, geometry and free probability questions. On a rigorous ground, only integrals of one matrix or of several matrices with simple quadratic interaction (called AB interaction) could be evaluated so far. In this article, we follow an idea widely developped in the physics litterature, which is based on character expansion, to study more complex interaction. We more specifically consider a model defined in the spirit of the 'dually weighted graph model' studied by V. A. Kazakov, M. Staudacher and T. Wynter, but with a cutoff function such that the matrix integral and its character expansion converge. We prove that the free energy of this model converges as the size of the matrices go to infinity and study the saddle points of the limit.

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 added