# Domino Tilings and the Gaussian Free Field

Thursday, November 9, 2017 - 2:30pm to 3:20pm
PDL C-401
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.
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