  We continue our study of matrix models of dually weighted graphs. Among the
attractive features of these models is the possibility to interpolate between
ensembles of regular and random two-dimensional lattices, relevant for the
study of the crossover from two-dimensional flat space to two-dimensional
quantum gravity. We further develop the formalism of large $N$ character
expansions. In particular, a general method for determining the large $N$ limit
of a character is derived. This method, aside from being potentially useful for
a far greater class of problems, allows us to exactly solve the matrix models
of dually weighted graphs, reducing them to a well-posed Cauchy-Riemann
problem. The power of the method is illustrated by explicitly solving a new
model in which only positive curvature defects are permitted on the surface, an
arbitrary amount of negative curvature being introduced at a single insertion.
