  Our main result is an explicit operator-theoretic formula for the number of
colored planar maps with a fixed set of stars each of which has a fixed set of
half-edges with fixed coloration. The formula bounds the number of such colored
planar maps well enough to prove convergence near the origin of generating
functions arising naturally in the matrix model context. Such convergence is
known but the proof of convergence proceeding by way of our main result is
relatively simple. Besides Voiculescu's generalization of Wigner's semicircle
law, our main technical tool is an integration identity representing the joint
cumulant of several functions of a Gaussian random vector. The latter identity
in the case of cumulants of order 2 reduces to one well-known as a means to
prove the Poincare inequality. We derive the identity by combining the heat
equation with the so-called BKAR formula from constructive quantum field theory
and rigorous statistical mechanics.
