Exact Calculations of Non-Local Quantities in a Loop Model

Abstract

A double dimer model is obtained via superposition of two dimer models. It is essentially a fully packed loop model allowing doubled edges in addition to nontrivial loops. We use quaternion determinants \cite{kenyon2012conformal} to calculate the probability of two edges lying on a loop for various configurations of edges at the boundaries of lattice and in the bulk. This is a non-local quantity in contrast to local correlators like the dimer-dimer correlator. We prove that for the edges on the boundaries, the probability of a single loop passing through them scales like $\frac{1}{r^{2}}$, where $r$ is the distance between the two edges. We also do an exact microscopic calculation for the probability of a single loop passing through two bulk edges separated by a distance $n$ in a column. We obtain this probability to be $\frac{(-1)^{n}}{2}(D_{n}^{2}-D_{n-1}D_{n+1})$ ,where $D_{n}$ is the $n \times n$ Toeplitz matrix generated by $f(\theta)=sgn(\cos{\theta})\frac{\cos{\theta}+i}{\sqrt{cos^{2}(\theta)+1}}$. We use the Basor-Tracy conjecture to determine the asymptotics of the Toeplitz matrices, then this probability is $\sim \frac{0.1223\dots}{n}$ for large $n$. Finally, we propose a modification to the formalism in Ref. \cite{kenyon2012conformal} to allow evaluation of above probabilities on non-bipartite planar graphs. We apply these ideas to a triangular lattice double dimer model. We obtain formal expressions for these probabilities in terms of Green's functions.