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The multinomial dimer model

Richard Kenyon, Catherine Wolfram

Abstract

The dimer model is a classical statistical mechanics model which is exactly solvable in two dimensions, but about which little is known in higher dimensions. In analogy with large $N$ limits in lattice gauge theory, we study a large $N$ limit of the dimer model in any dimension $d$. The dependence on $N$ comes from the multinomial tiling model introduced by Kenyon and Pohoata, which gives a general framework for adding a dependence on $N$ to a tiling model. We study the behavior of this model on periodic bipartite graphs in ${\mathbb R}^d$, in the scaling limit as the multiplicity $N$ and then the size of the graph go to infinity. In this iterated limit, in any dimension $d$, we prove a variational principle and show that random configurations concentrate on a limit shape which is the unique solution to an associated system of Euler-Lagrange equations. The rate function of the variational principle is the integral of a surface tension function, which we can compute explicitly for lattices in any dimension $d$ as the Legendre dual of the free energy for the model on the torus. We give a unified methodology for computing the surface tension and Euler-Lagrange equations in any dimension $d$. A new structure called the critical gauge also emerges in the large $N$ limit. We show that the critical gauge functions converges in the scaling limit to a limiting gauge function which is the unique solution to a dual Euler-Lagrange equation. This limiting gauge function determines the limit shape and vice versa. We further use our techniques to compute explicit limit shapes in some two and three dimensional examples, such as the Aztec diamond and "Aztec cuboid". This is one of the first stat mech models in dimensions $d\ge3$ where limit shapes can be computed explicitly.

The multinomial dimer model

Abstract

The dimer model is a classical statistical mechanics model which is exactly solvable in two dimensions, but about which little is known in higher dimensions. In analogy with large limits in lattice gauge theory, we study a large limit of the dimer model in any dimension . The dependence on comes from the multinomial tiling model introduced by Kenyon and Pohoata, which gives a general framework for adding a dependence on to a tiling model. We study the behavior of this model on periodic bipartite graphs in , in the scaling limit as the multiplicity and then the size of the graph go to infinity. In this iterated limit, in any dimension , we prove a variational principle and show that random configurations concentrate on a limit shape which is the unique solution to an associated system of Euler-Lagrange equations. The rate function of the variational principle is the integral of a surface tension function, which we can compute explicitly for lattices in any dimension as the Legendre dual of the free energy for the model on the torus. We give a unified methodology for computing the surface tension and Euler-Lagrange equations in any dimension . A new structure called the critical gauge also emerges in the large limit. We show that the critical gauge functions converges in the scaling limit to a limiting gauge function which is the unique solution to a dual Euler-Lagrange equation. This limiting gauge function determines the limit shape and vice versa. We further use our techniques to compute explicit limit shapes in some two and three dimensional examples, such as the Aztec diamond and "Aztec cuboid". This is one of the first stat mech models in dimensions where limit shapes can be computed explicitly.

Paper Structure

This paper contains 57 sections, 59 theorems, 277 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Scalings limits in general dimension
  3. Large deviations and variational principle
  4. Explicit formulas: free energy and surface tension
  5. Limit shape regularity
  6. Euler-Lagrange equations
  7. Critical gauge
  8. Background and preliminaries
  9. Definition of the model
  10. The dimer model.
  11. Blow-up graph.
  12. The multinomial dimer model.
  13. Lattices and multinomial dimers on subgraphs
  14. Asymptotics as ${\mathbf N}\to \infty$ on a fixed graph
  15. Limiting boundary conditions
  16. ...and 42 more sections

Key Result

Theorem 1.1

Let $(R,b)$, $R_n, \rho_{n,N}$ be as above. In the iterated limit as $N$ and then $n$ go to infinity, the measures $\rho_{n,N}$ satisfy a large deviation principle in the weak topology on flows with good rate function $I_b(\cdot)$ which up to an additive constant is if $\omega$ is an asymptotic flow, where $\sigma$ is the surface tension determined by $\Lambda$, and is given explicitly as the Leg

Figures (9)

  • Figure 1: The limit shape height functions on the Aztec diamond for the dimer model (left) and the multinomial dimer model in the large $N$ limit (right).
  • Figure 2: Flow lines for the Aztec cuboid limit shape divergence free flow $\omega = (-\frac{2x}{A}+1, \frac{2y}{B}-1, \frac{2z}{C}-1)$ with $A=1$ and $B=C=2$.
  • Figure 3: The surface tension functions for $\mathbb{Z}^2$ for the standard dimer model (left) and the multinomial dimer model in the large $N$ limit (right). Notice that the multinomial surface tension is strictly convex everywhere, whereas the standard dimer one is strictly convex in the interior. Both have maximum value zero.
  • Figure 4: An application of Sinkhorn's algorithm gives a "discrete approximation" of the limit shape divergence-free flow for the hexagon graph $G_{n}$ in the honeycomb lattice, $n=30$. Namely, we embed the $G_{30}$ in the Newton polygon $\mathcal{N}$ for the lattice, where each vertex $v$ of the graph is embedded at the corresponding average slope $s(v)\in \mathcal{N}$, which is, up to a multiplicative constant, $\sum_{e\ni v} c(e) e$. For the hexagon graph, this embedding is a double cover.
  • Figure 5: The numerically computed exact critical gauge for the octahedron graph in $\mathbb{Z}^3$, with side length $n=10$, is used to define an embedding of the graph into the Newton polytope for $\mathbb{Z}^3$ (which is also the octahedron). The left is the image of the vertices and the right is the image of the edges of the graph in this embedding. This gives an approximation to the $n=\infty$ limit shape flow.
  • ...and 4 more figures

Theorems & Definitions (122)

  • Theorem 1.1: Rough statement; see Theorem \ref{['thm:ldp']}
  • Corollary 1.2: Rough statement; see Corollary \ref{['cor:limit_shape']}
  • Theorem 1.3: See Theorem \ref{['thm:no_facets']}
  • Corollary 1.4: See Corollary \ref{['cor:2Dsmooth']}
  • Theorem 1.5: See Theorem \ref{['thm:EL_equations']}
  • Theorem 1.6: See Theorem \ref{['thm:EL_equations_gauge']}, Corollary \ref{['cor:EL_gauge_cor']}
  • Theorem 1.7: See Theorem \ref{['thm:critical_gauge_scaling_limit']}
  • Definition 2.1
  • Theorem 2.2: KenyonPohoata
  • Theorem 2.3: See KenyonPohoata
  • ...and 112 more