Domino Tilings and the Gaussian Free Field

We say a simply connected region \(P_{\varepsilon}\) in \(\mathbb{C}\) is a polyomino if its boundary consists of edges in the \(\varepsilon\mathbb{Z}^2\) lattice. Further, we say the region \(P_{\varepsilon}\) is tileable if we can cover its interior with non-overlapping dominos. Given such a polyomino, there may exist many tilings, and by choosing one uniformly at random we can define a very natural random function on the polyomino, \(h_{\varepsilon}\). Given a bounded, simply connected domain \(U\), we consider a sequence of tileable polyominos \(P_{\varepsilon}\) (\(\varepsilon\rightarrow 0\)) approximating \(U\) from within. Surprisingly, we have that as \(\varepsilon\rightarrow 0\), the associated processes \(h_{\varepsilon}\) tend in distribution to a limiting process on \(U\). And perhaps more surprisingly, we will see that this limiting process is none other than the Gaussian free field (as seen during the Milliman lectures) on the domain \(U\). We will discuss the main ideas behind this result in the talk.