Counting colored planar maps: algebraicity results

TL;DR

This paper proves that the generating functions for counting properly q-colored planar maps are algebraic for specific q values, extending to maps weighted by Potts polynomials and triangulations, thus advancing the understanding of the Potts model on random lattices.

Contribution

It establishes algebraicity of generating functions for colored planar maps and triangulations for specific q values, extending Tutte's work to more general models.

Findings

Generating functions are algebraic for q=2,3, and certain non-integer q values.

Extended algebraicity results to maps weighted by Potts polynomials.

Solved complex non-linear equations with catalytic variables for the first time since Tutte.

Abstract

We address the enumeration of properly q-colored planar maps, or more precisely, the enumeration of rooted planar maps M weighted by their chromatic polynomial \chi_M(q) and counted by the number of vertices and faces. We prove that the associated generating function is algebraic when q\not=0,4 is of the form 2+2 cos (j\pi/m), for integers j and m. This includes the two integer values q=2 and q=3. We extend this to planar maps weighted by their Potts polynomial P_M(q,\nu), which counts all q-colorings (proper or not) by the number of monochromatic edges. We then prove similar results for planar triangulations, thus generalizing some results of Tutte which dealt with their proper q-colorings. In statistical physics terms, the problem we study consists in solving the Potts model on random planar lattices. From a technical viewpoint, this means solving non-linear equations with two "catalytic" variables. To our knowledge, this is the first time such equations are being solved since Tutte's remarkable solution of properly q-colored triangulations.