The Local Behavior of Random Lozenge Tilings

Amol Aggarwal, Harvard University

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The statistical behavior of random tilings of large domains has been an intense topic of mathematical research for decades, partly since they highlight a central phenomenon in physics: local behaviors of highly correlated systems can be very sensitive to boundary conditions. Indeed, a salient feature of random tiling models is that the local densities of tiles can differ considerably in different regions of the domain, depending on the shape of the domain. Thus, a question of interest, originally mentioned by Kasteleyn in 1961, is how the shape of the domain affects the local behavior of a random tiling. In this talk, we outline recent work that provides an answer (originally predicted by Cohn-Kenyon-Propp in 2001) to this question for random lozenge tilings of essentially arbitrary domains. The proofs are based on a combination of results from exactly solvable systems with more analytic and probabilistic arguments.