Conformally invariant boundary arcs in double dimers

TL;DR

This paper studies boundary arcs in double dimer models on planar domains, showing their conformal invariance in the scaling limit and their relation to Gaussian free fields and the arc loop ensemble.

Contribution

It introduces two variants of the double dimer model with boundary arcs and demonstrates their conformal invariance and connection to Gaussian free fields.

Findings

Boundary arcs converge to conformally invariant quantities

Statistics of arcs match those of the arc loop ensemble in the scaling limit

Results support conjectures relating double dimers, GFF, and ALE

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).