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Eulerian orientations and the six-vertex model on planar map

Mireille Bousquet-Mélou, Andrew Elvey Price, Paul Zinn-Justin

TL;DR

This work bridges bijective combinatorics and matrix-model methods to enumerate rooted planar quartic maps with Eulerian orientations. It provides an exact, unified description of the generating functions in terms of a single auxiliary series R(t) for the edge-counted case and a theta-function parametrization for the weighted six-vertex model, yielding G(t) = (1/4t^2)(t - 2t^2 - R(t)) and Q(t,1) = (1/(3t^2))(t - 3t^2 - R(t)), with a general formula Q(t,γ) = (1/((γ+2)t^2))(t - (γ+2)t^2 - R(t,γ)). The results demonstrate that Kostov’s framework generalizes the bijective constructions and open pathways to further generalizations (general γ, refined counting by vertices), thereby connecting combinatorial decompositions with matrix-model analytic techniques for the six-vertex model on planar maps.

Abstract

We address the enumeration of planar 4-valent maps equipped with an Eulerian orientation by two different methods, and compare the solutions we thus obtain. With the first method we enumerate these orientations as well as a restricted class which we show to be in bijection with general Eulerian orientations. The second method, based on the work of Kostov, allows us to enumerate these 4-valent orientations with a weight on some vertices, corresponding to the six vertex model. We prove that this result generalises both results obtained using the first method, although the equivalence is not immediately clear.

Eulerian orientations and the six-vertex model on planar map

TL;DR

This work bridges bijective combinatorics and matrix-model methods to enumerate rooted planar quartic maps with Eulerian orientations. It provides an exact, unified description of the generating functions in terms of a single auxiliary series R(t) for the edge-counted case and a theta-function parametrization for the weighted six-vertex model, yielding G(t) = (1/4t^2)(t - 2t^2 - R(t)) and Q(t,1) = (1/(3t^2))(t - 3t^2 - R(t)), with a general formula Q(t,γ) = (1/((γ+2)t^2))(t - (γ+2)t^2 - R(t,γ)). The results demonstrate that Kostov’s framework generalizes the bijective constructions and open pathways to further generalizations (general γ, refined counting by vertices), thereby connecting combinatorial decompositions with matrix-model analytic techniques for the six-vertex model on planar maps.

Abstract

We address the enumeration of planar 4-valent maps equipped with an Eulerian orientation by two different methods, and compare the solutions we thus obtain. With the first method we enumerate these orientations as well as a restricted class which we show to be in bijection with general Eulerian orientations. The second method, based on the work of Kostov, allows us to enumerate these 4-valent orientations with a weight on some vertices, corresponding to the six vertex model. We prove that this result generalises both results obtained using the first method, although the equivalence is not immediately clear.

Paper Structure

This paper contains 5 sections, 6 theorems, 44 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Bijection for general Eulerian orientations
  3. The number of planar Eulerian orientations
  4. The matrix integral approach to the six-vertex model
  5. Relationships between results and further problems

Key Result

Theorem 1

Let ${\sf R}(t)\equiv {\sf R}$ be the unique formal power series with constant term $0$ satisfying Then the generating function of rooted planar Eulerian orientations, counted by edges, is

Figures (4)

  • Figure 1: The two types of vertices in the six-vertex model.
  • Figure 2: A rooted Eulerian orientation (solid edges; the root edge is shown with a double arrow, and its orientation is chosen canonically) and the corresponding dual labelled map (dashed edges). The labelling rule is shown on the right.
  • Figure 3: A labelled quadrangulation $Q$ (black edges) and the corresponding labelled map $L$ (red edges). The rule for drawing red edges is shown on the right. Note that the two local minima of $Q$, both labelled $-2$, disappear in the construction.
  • Figure 4: The two vertex types allowed as non-root vertices in the Eulerian partial orientations counted by ${\sf W}(t,\omega,x)$ and ${\sf H}(t,\omega,x,y)$.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7