Conformally invariant boundary arcs in double dimers

TL;DR

The paper develops two arc-bearing variants of the double-dimer model on symmetric planar domains (folded and shifted) and proves that arc statistics nε(z) and oε(z) converge, in the small-mesh limit, to conformally invariant laws matching the Arc Loop Ensemble (ALE). It extends Kenyon's double-dimer formula to include arcs via an SL2 connection, and then shows that the arc observables have universal scaling limits independent of the folding/shifting construction, with limiting means and moments expressed in terms of contour integrals of the continuum Green function. The analysis links the discrete arc picture to couplings between the ALE and Gaussian Free Field (Dirichlet/Neumann boundary conditions) in the Qian–Werner framework, providing explicit mean formulas on the infinite strip and a detailed rectangle example. These results provide evidence for the conjectured ALE scaling limits for both folded and shifted double-dimer models and illuminate their connections to conformal field theory via discrete/continuous Green function technology and Kasteleyn-based Pfaffian techniques.

Abstract

We consider two different versions of the double dimer model on a planar domain, where we either fold a single dimer cover on a symmetric domain onto itself across the line of symmetry, or we superimpose two independent dimer covers on two, almost identical, domains that differ only on a certain portion of the boundary. This results in a collection of loops and doubled edges that, unlike in the classical double dimer case of Kenyon, are accompanied by arcs emanating from the line of symmetry or the chosen portion of the boundary. We argue that these arcs together with the associated height function satisfy a discrete version of the coupling of Qian and Werner between the Arc loop ensemble (ALE) and two different variants of the Gaussian free field (with Dirichlet and Neumann boundary conditions). We also show that certain statistics of the arcs (when the loops are disregarded from the picture) converge to conformally invariant quantities in the small-mesh scaling limit, and moreover the limits are the same for the two versions of the model, and equal to the corresponding statistics of the arc loop ensemble (ALE). This gives evidence to the conjecture of [7] (that concerns one of these models).

Figures (16)

• Figure 1: Left: a (partial) dimer cover of $\mathcal{G}_{\varepsilon}^r$ is represented in red, and the lower part is folded on the upper part in dotted red. A folded dimer configuration appears on the upper part: loops and arcs alternate between full red and dotted red. The graph $\mathcal{G}_{\varepsilon}^r$ has Temperleyan boundary conditions: all its corners are black squares, and one of the black squares on the boundary (in this case on the right-hand side of the intersection with the real line) is removed. Right: (partial) dimer covers of $\mathcal{G}_{\varepsilon}^r \cap (\mathbb{R} \times \mathbb{R}_{\geq 0})$ and $\mathcal{G}_{\varepsilon}^r \cap (\mathbb{R} \times \mathbb{R}_{<0})$ are represented in solid red. The lower part is superimposed on the upper part in dotted red. A superimposed configuration appears on the upper part: loops and arcs alternate between full red and dotted red. The upper part has again Temperleyan boundary conditions, and the lower part has piecewise Temperleyan boundary conditions with two white bullet corners. The sets of vertices $W_0$ and $B_0$ are represented by white and black bullets, $W_1$ and $B_1$ are represented by white and black squares.
• Figure 2: A very simple example of the graph $G^\times$ (right), given a graph $G$ (left). The set $\partial$ is in this case the single red vertex.
• Figure 3: An example of a graph $G$, where we take $\partial=\overline{\partial}$ to be the edges on the $x$-axis, and real Kasteleyn weights $\xi$ which alternate in sign along $\overline{\partial}$ as $\xi_{w,b} = (-1)^{\mathds{1}\{w>b\}}$, where the ordering on $\partial$ increases clockwise. (We only write the $-1$ phases in the diagram for clarity; the rest are 1).
• Figure 4: The vertices of $G^\times$ in a loop, with all the edges of $G^\times$ joining them.
• Figure 5: The vertices and edges of $G^\times$ in a loop.

Theorems & Definitions (34)

• Theorem 1.1
• Proposition 2.1
• Remark 2.2
• Remark 2.3
• Corollary 2.4
• Lemma 2.5
• proof
• proof : Proof of Proposition \ref{['kast:prop:kast']}
• Definition 3.1: Approximating sequence
• Remark 3.2