The local edge-density conjecture for domino tilings

In the dimer model on an $n \times n$ torus, let $p_a$, $p_b$, $p_c$, and $p_d$ denote the limiting densities of the four edge types in the thermodynamic limit as $n \rightarrow \infty$. For a large planar region, let $f$ be its asymptotic height function, and suppose that on a mesoscopic patch its tilt is $(s,t)=({\partial f}/{\partial x},{\partial f}/{\partial y})$. Local edge-density conjecture. The local densities of $a$-edges, $b$-edges, $c$-edges, and $d$-edges are given by $p_a$, $p_b$, $p_c$, and $p_d$, respectively, in the thermodynamic limit. This conjecture proposes the planar-region interpretation of the torus quantities $p_a,p_b,p_c,p_d$; the paper explicitly states that the corresponding interpretation has not been proved.