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Some combinatorial problems arising in the dimer model

Richard Kenyon

TL;DR

The most fundamental problem in the dimer model is the following: giving a dimer cover a probability proportional to its weight.

Abstract

We discuss some diverse open problems in the dimer model, motivated by a geometric viewpoint. This is part of a conference proceedings for the OPAC 2022 conference.

Some combinatorial problems arising in the dimer model

TL;DR

The most fundamental problem in the dimer model is the following: giving a dimer cover a probability proportional to its weight.

Abstract

We discuss some diverse open problems in the dimer model, motivated by a geometric viewpoint. This is part of a conference proceedings for the OPAC 2022 conference.
Paper Structure (19 sections, 3 theorems, 16 equations, 7 figures)

This paper contains 19 sections, 3 theorems, 16 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vector bundles and connections on graphs
  4. Planar bipartite graphs and the Kasteleyn matrix
  5. From edge weights to probabilities
  6. Fractional matchings
  7. Cycles
  8. Gauge equivalence
  9. Dimer walks
  10. Magnetic double dimer model
  11. Double dimers
  12. $\text{SL}_2$ connections
  13. Kasteleyn matrix
  14. Flat connections and simple laminations
  15. $n$-fold dimer model
  16. ...and 4 more sections

Key Result

Theorem 1

$|\det K| = \sum_{m\in\mathcal{M}}c(m).$

Figures (7)

  • Figure 1: A graph and its expected fractional matching (for uniform edge weights). This graph has three dimer covers; two of these use the leftmost vertical edge, so its probability is $2/3$.
  • Figure 2: A degenerate graph. The dimension of the cycle space is $3$ but $\Omega$ is of dimension $2$. Note that the two diagonal edges are "unused", that is, do not appear in any dimer cover.
  • Figure 3: A double dimer cover of a grid. Edges of multiplicity $1$ or $2$ are shown.
  • Figure 4: Connections with monodromy $q$ per face.
  • Figure 5: Simple laminations on a pair of pants consist in three kinds of loops: those surrounding $p_1,p_2$ or both.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Theorem 1: KastKuperberg
  • Theorem 2
  • proof
  • Theorem 3
  • proof