Abstract:

A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. We prove an analogous large deviations principle for dimers in 3D. A lot of the results for dimers in two dimensions use tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. We will show some simulations and explain how to formulate and prove the large deviations principle despite having a much smaller set of tools in 3D (e.g. Hallâ€™s matching theorem or a double dimer swapping operation). Time permitting, we will illustrate some of the key qualitative differences between two and three dimensions.

https://arxiv.org/abs/2304.08468

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