  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.
