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 4SLE emanating from the boundary, or equivalently, to a conditioned version of CLE with wired/free/wired/free boundary conditions in the context of conformal loop ensembles.
Cite
@article{arxiv.1908.07595,
title = {Connection probabilities in the double-dimer model -- the case of two connectivity patterns},
author = {Nahid Ghodratipour and Shahin Rouhani},
journal= {arXiv preprint arXiv:1908.07595},
year = {2019}
}