The expectation value of the number of loops and the left-passage probability in the double-dimer model

Abstract: We study various statistical properties of the double-dimer model, a generalization of the dimer model, on rectangular domains of the square lattice. We take advantage of the Grassmannian representation of the dimer model, first to calculate the probability distribution of the number of nontrivial loops around a cylinder, which is consistent with the previously known result, and then to calculate the expectation value of the number of loops surrounding two faces and the left-passage probability, both in the discrete and the continuum cases. We also briefly explain the calculation of some related observables. As a by-product, we obtain the partition function of the dimer model in the presence of two and four monomers, and a single monomer on the boundary.