Dimers and Amoebae

TL;DR

This work analyzes random dimer models on weighted, doubly periodic bipartite planar graphs by linking combinatorial structures to algebraic geometry. The authors derive explicit formulas for surface tension and local Gibbs measures through the spectral curve $P(z,w)=\det K(z,w)$, show that the Amoeba of $P$ governs the phase diagram, and prove that the spectral curve is always a real Harnack curve, which yields universal height fluctuations in the liquid phase and robust phase structure. By establishing the Legendre dual relationship between the Ronkin function and the surface tension, and proving Monge-Ampère relations, they unify variational and probabilistic descriptions of crystal facets and height profiles. The results give a comprehensive, exactly solvable framework for dimer-based random surfaces, with implications for phase behavior, universality of fluctuations, and facet formation in crystal-like models.

Abstract

We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on an weighted, bipartite, doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of the spectral curve. The amoeba of the spectral curve represents the phase diagram of the dimer model. Further, we prove that the spectral curve of a dimer model is always a real curve of special type, namely it is a Harnack curve. This implies many qualitative and quantitative statement about the behavior of the dimer model, such as existence of smooth phases, decay rate of correlations, growth rate of height function fluctuations, etc.

Figures (9)

• Figure 1: On the left is the height function of a random volume-constrained dimer configuration on the honeycomb lattice. The boundary conditions here are that of a crystal corner: all dimers are aligned the same way deep enough in each of the 3 sectors. On the right is (the boundary of) the amoeba of a straight line.
• Figure 2: On the left are the level sets of perimeter 10 or longer of the height function of a random volume-constrained dimer configuration on ${\mathbb Z}^2$ (with $2\times 2$ fundamental domain). The height function is essentially constant in the middle --- a facet is developing there. The intermediate region, in which the height function is not approximately linear, converges to the amoeba of the spectral curve, which can be seen on the right. The spectral curve in this case is a genus $1$ curve with the equation $z+z^{-1}+w+w^{-1}=6.25$.
• Figure 3: The fundamental domain of the square-octagon graph.
• Figure 4: The operator $K(z,w)$
• Figure 5: The curved part of minus the Ronkin function of $z+\frac{1}{z}+w+\frac{1}{w}+5$. This is the limit height function shape for square-octagon dimers with crystal corner boundary conditions.

Theorems & Definitions (33)

• Theorem 2.1: Sheffield
• Lemma 2.2
• Proposition 3.1
• Proposition 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• Theorem 3.5
• Theorem 3.6