Dimer Models and Conformal Structures

TL;DR

The paper develops a unified variational–PDE framework for two-dimensional dimer models, coupling a Monge–Ampère surface-tension functional with gradient constraints to a degenerate Beltrami equation via the Lewy transform. It establishes a deep conformal structure on liquid regions, proves that frozen boundaries are algebraic with at most cusp or tacnode singularities, and rigorously derives the Pokrovsky–Talapov regularity law in full generality. A key contribution is a universality principle: frozen boundaries in general dimer models match those in the lozenge model, with local and, in simply connected domains, global universality. The approach also provides explicit representation formulas for limiting height functions in terms of harmonic measures and complex-analytic maps, connects to prior complex-Burgers frameworks, and yields a complete description of simply and multiply connected frozen boundaries via rational/Blaschke parametrizations. Collectively, these results advance understanding of phase boundaries, boundary regularity, and potential universal scaling limits in random tilings and related stochastic interfaces.

Abstract

In this work we study the variational problem associated to dimer models, a class of models from integrable probability and statistical mechanics in dimension two which have been the focus of intense research efforts over the last decades. These models give rise to an infinite family of non-differentiable functionals on Lipschitz functions with gradient constraint, determined by solutions of the Dirichlet problem on compact convex polygons for a class of Monge-Ampère equations. We settle a number or outstanding open questions for this infinite class functionals. In particular we prove a complete classification of the regularity of minimizers, also known as height functions, for all dimer models for a natural class of polygonal (simply or multiply connected) domains much studied in numerical simulations and elsewhere. Our classification in particular implies that the Pokrovsky-Talapov law holds for all dimer models at a generic point on the frozen boundary and in addition shows a very strong local rigidity of dimer models which can be interpreted as a geometric universality result. Furthermore, we give a complete classification of the regularity of the associated free boundary, also known in the literature as frozen boundary or arctic curves and prove that they are all algebraic curves. The lack of differentiability of the functionals is intimately connected to the boundary behaviour of the solutions to the Monge-Ampère equations and we prove a complete classification for these, of independent interest.

Figures (23)

• Figure 1: Left: A uniform domino tiling. Right: Weighted domino tiling with gas phases. Image courtesy of S. Chhita and T. Berggren.
• Figure 2: The figure on the right displays the gradient constraint $N$. The figure on the left displays the image of $\nabla \sigma(N^\circ\setminus\mathscr{G})$, also known as the amoeba in the literature. The vector $i(p_2-p_3)$ indicates the direction of one of the asymptotes of the amoeba. Furthermore, the vectors at $p_3$ and $g$ in $N$, c.f. Theorem \ref{['sigma1']}, show directional limits that get mapped to the boundary of the amoeba under the map $\nabla \sigma$.
• Figure 3: Quasifrozen domains with different surface tensions but with the same boundary height function $h_0$. In the middle a 'quasi-particle'. Image courtesy of M. Duits.
• Figure 4: Simulation of random tiling of the Aztec diamond with two gas domains. Graph of the height function pictured from above (left) and from side (right). Courtesy of Tomas Berggren.
• Figure 5: On left: Cusp, where frozen facet $\mathcal{F} \not\subset \Lambda$. In middle: $\Lambda_m$. On right: $\Lambda_M$.

Theorems & Definitions (157)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5: Pokrovsky-Talapov law
• Theorem 1.6
• Theorem 1.7: Universality of frozen boundaries
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3: Local regularity of frozen boundaries