Counting colored planar maps free-probabilistically

Details Counting colored planar maps free-probabilistically Counting colored planar maps free-probabilistically

2012-03-14

arxiv.org/abs/1203.3185v1

Mathematics

Probability

24 pages, no figures, LaTeX

Scientific paper

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 transparently bounds the number of such colored planar maps and does so 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. Our main technical tool is an integration identity representing the joint cumulant of several functions of a Gaussian random vector. In the case of cumulants of order 2 the identity reduces to one well-known as a means to prove the Poincar\'{e} inequality.

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