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

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 4SLE4 emanating from the boundary, or equivalently, to a conditioned version of CLE4 with wired/free/wired/free boundary conditions in the context of conformal loop ensembles.

Combinatorics of the double-dimer model

“…As discussed, Theorems 1.3.1 and 1.3.2 generalize the combinatorial results of [KW11a, KW09,KW11b]. The main questions of interest in these bodies of work involve asymptotic and probabilistic properties of the double-dimer model, which were further studied in [Ken14,Dub19,GR19]. In [KP16], Kenyon and Pemantle give a connection between the double-dimer model and cluster algebras.…”