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.

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 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. * ghodratipour n@physics.sharif.edu † srouhani@sharif.ir 1 arXiv:1805.03930v2 [cond-mat.stat-mech] 27 Jun 2018 statistical physics, including the O(n) models. These are a large class of critical systems in physics, conjectured to be related to SLE κ and CLE κ concisely expressed by the formula n = −2 cos(4π/κ). The parameter κ is so in significant relation to the central charge c in CFT and as a fact, SLEs (CLEs) with κ ≤ 4 are believed to be the minimal models of CFT with c in [0, 1] [21-23]. One prototypical model where such issues are concerned is the two-dimensional Gaussian Free Field (GFF).The two-dimensional GFF is a central object both in physics and mathematics, related to the Coulomb gas, which falls into the same universality class as other statistical mechanical models such as the XY-model and the two-dimensional sine-Gordon model [17,24,25]. It is the mathematical model of two-dimensional bosonic CFT, and is a two-dimensional analog of the Brownian motion [26]. The notions associated with the Brownian motion, such as the Gaussian measure, the Laplacian, the Green's function and consequently conformal invariance, are also substantial in GFF [25,27], and many stochastic differential equations in two dimensions depend on it as those in one dimension on the Brownian motion [28,29]. We can practically think of GFF as a natural model of random height functions though it is actually a generalized function or a distribution (in sense of Schwartz [26]). It yields a dual interpretation of many loop models, especially the O(n) models, which in SLE/CLE formalism, is very natural in the case of κ = 4 (equivalently n = 2); the zero level line of the Gaussian free field (with appropriate boundary conditions) is a chordal SLE 4 and the collection of level loops of it corresponds to CLE 4 [30,31].Besides, there is a family of conformally invariant loop models in two dimensions, the Brownian Loop Soups (BLSs), which is deeply related to SLE and CLE [20,32]. Each BLS is a Poisson point process of Brownian loops [33], with intensity parameter related to κ via the equation c = (3κ − 8)(6 − κ)/2κ [32], indicating that for low enough intensities the outer boundaries of clusters of Brownian loops are distributed like CLEs for the spectral parameter between 8/3 and 4. This implies that th...