The two-point function of bicolored planar maps

TL;DR

This paper derives explicit formulas for the two-point function of vertex-bicolored planar maps, incorporating color-dependent weights, using slice decomposition and continued fractions, with special cases for quadrangulations and hexangulations.

Contribution

It introduces a novel approach to compute the two-point function of bicolored maps with color-dependent weights, extending previous methods to include vertex coloring.

Findings

Explicit formulas for bicolored maps with bounded face degrees

Two-point functions for quadrangulations and hexangulations obtained

Extension to vertex-tricolored maps discussed

Abstract

We compute the distance-dependent two-point function of vertex-bicolored planar maps, i.e., maps whose vertices are colored in black and white so that no adjacent vertices have the same color. By distance-dependent two-point function, we mean the generating function of these maps with both a marked oriented edge and a marked vertex which are at a prescribed distance from each other. As customary, the maps are enumerated with arbitrary degree-dependent face weights, but the novelty here is that we also introduce color-dependent vertex weights. Explicit expressions are given for vertex-bicolored maps with bounded face degrees in the form of ratios of determinants of fixed size. Our approach is based on a slice decomposition of maps which relates the distance-dependent two-point function to the coefficients of the continued fraction expansions of some distance-independent map generating functions. Special attention is paid to the case of vertex-bicolored quadrangulations and hexangulations, whose two-point functions are also obtained in a more direct way involving equivalences with hard dimer statistics. A few consequences of our results, as well as some extension to vertex-tricolored maps, are also discussed.