Connection probabilities in the double-dimer model -- the case of two connectivity patterns

TL;DR

This paper calculates connection probabilities in the double-dimer model with specific boundary conditions using Grassmannian representation, confirming consistency with conformal field theory predictions and previous results.

Contribution

It introduces a Grassmannian approach to compute connection probabilities in the double-dimer model with boundary conditions, linking discrete models to conformal invariance results.

Findings

Connection probabilities match those predicted by conformal field theory.

Results are consistent with previous work by Kenyon-Wilson.

The approach confirms the conformal invariance of the double-dimer model.

Abstract

We apply the Grassmannian representation of the dimer model, an equivalent approach to Kasteleyn's solution to the close-packed dimer problem, to calculate the connection probabilities for the double-dimer model with wired/free/wired/free boundary conditions, on a rectangular subdomain of the square lattice with four marked boundary points at the corners. Using some series identities related to Schwarz-Christoffel transformations, we show that the continuum of the result is consistent with the corresponding one in the upper half-plane (previously obtained by Kenyon-Wilson), which is in turn identical to the connection probabilities for 4SLE$_4$ emanating from the boundary, or equivalently, to a conditioned version of CLE$_4$ with wired/free/wired/free boundary conditions in the context of conformal loop ensembles.