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Local Statistics of the $M_n$-Dimer Model

Nickolas Anderson, Moriah Elkin, Elizabeth Kelley, Nicholas Ovenhouse, Kayla Wright

TL;DR

This work extends the classical dimer model to the $M_n$-dimer setting by introducing matrix edge weights and cilia, defining a probability measure on $\Omega_n(G)$ and establishing a generalized Kasteleyn framework. Central to the analysis is the probability matrix $P_e$ for each edge, whose spectral data governs local statistics and correlations; the authors derive generating functions, moments, and joint moments for edge multiplicities, and show gauge invariance with respect to edge-weight transformations. They also develop a suite of local moves that preserve the partition function up to a scalar, enabling simplifications via Schur-complement techniques; these tools yield explicit formulas for edge distributions and covariances in terms of $P_e$ and related matrices. The paper substantiates the theory with concrete examples, notably the $2\times N$ grid via noncommutative continued fractions, and demonstrates connections to mixed dimer covers and the six-vertex model in the free-fermionic regime, illustrating the broad applicability to vertex-model mappings and combinatorial statistics.

Abstract

The classical dimer model is concerned with the (weighted) enumeration of perfect matchings of a graph. An $n$-dimer cover is a multiset of edges that can be realized as the disjoint union of $n$ individual matchings. For a probability measure recently defined by Douglas, Kenyon, and Shi, which we call the $M_n$-dimer model, we study random $n$-dimer covers on bipartite graphs with matrix edge weights and produce formulas for local edge statistics and correlations. We also classify local moves that can be used to simplify the analysis of such graphs.

Local Statistics of the $M_n$-Dimer Model

TL;DR

This work extends the classical dimer model to the -dimer setting by introducing matrix edge weights and cilia, defining a probability measure on and establishing a generalized Kasteleyn framework. Central to the analysis is the probability matrix for each edge, whose spectral data governs local statistics and correlations; the authors derive generating functions, moments, and joint moments for edge multiplicities, and show gauge invariance with respect to edge-weight transformations. They also develop a suite of local moves that preserve the partition function up to a scalar, enabling simplifications via Schur-complement techniques; these tools yield explicit formulas for edge distributions and covariances in terms of and related matrices. The paper substantiates the theory with concrete examples, notably the grid via noncommutative continued fractions, and demonstrates connections to mixed dimer covers and the six-vertex model in the free-fermionic regime, illustrating the broad applicability to vertex-model mappings and combinatorial statistics.

Abstract

The classical dimer model is concerned with the (weighted) enumeration of perfect matchings of a graph. An -dimer cover is a multiset of edges that can be realized as the disjoint union of individual matchings. For a probability measure recently defined by Douglas, Kenyon, and Shi, which we call the -dimer model, we study random -dimer covers on bipartite graphs with matrix edge weights and produce formulas for local edge statistics and correlations. We also classify local moves that can be used to simplify the analysis of such graphs.

Paper Structure

This paper contains 15 sections, 22 theorems, 57 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. The Dimer Model
  4. The $M_n$-Dimer Model
  5. Schur Reduction
  6. The Probability Matrix
  7. Local Moves
  8. Formulas for Local Edge Statistics
  9. Correlations for Multiple Edges
  10. Examples and Applications
  11. Product Measures
  12. $2 \times N$ Grid and Noncommutative Continued Fractions
  13. An Application to Vertex Models
  14. Mixed Dimer Covers
  15. The Six Vertex Model

Key Result

Theorem 1

kasteleyn For a bipartite planar graph $G$, with Kasteleyn matrix $K$,

Figures (6)

  • Figure 1: An example of a graph with matrix edge weights, and its Kasteleyn matrix.
  • Figure 2: The $2 \times 5$ grid graph $G_4$ with arbitrary matrix edge weights (Kasteleyn signs are included). Cilia are drawn as small blue lines.
  • Figure 3: (Left) A snake graph with matrix edge weights. (Right) The result of applying a contraction at the upper-left vertex, followed by a parallel edge reduction.
  • Figure 4: (Top) A graph with vertex labels $\vec{n}$ and a choice of matrix edge weights. (Middle) The five mixed dimer covers of this graph, with their weights given below each picture. (Bottom) The four half-edge colorings of the last mixed dimer cover. The weight of this configuration is the sum of the four contributions from these colorings.
  • Figure 5: Boltzmann weights for the six-vertex model.
  • ...and 1 more figures

Theorems & Definitions (60)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Definition 4
  • Theorem 2
  • Corollary 1
  • Definition 5
  • Definition 6
  • Example 1
  • ...and 50 more