Counting colored planar maps free-probabilistically

TL;DR

This paper provides an explicit operator-theoretic formula for counting colored planar maps, enabling simpler proofs of convergence for related generating functions in matrix models using advanced probabilistic and quantum field theory techniques.

Contribution

It introduces a novel operator-theoretic formula for colored planar maps and applies it to prove convergence of generating functions in matrix models more straightforwardly.

Findings

Derived an explicit operator formula for colored planar maps

Proved convergence of matrix model generating functions near the origin

Connected cumulant identities with quantum field theory methods

Abstract

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.