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Résumé |
A dimer tiling of $\mathbb{Z}^d$ 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. In joint work with Nishant
Chandgotia and Scott Sheffield, 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. I will explain
how to formulate the large deviations principle in 3D, show simulations, and try
to give some intuition for why three dimensions is different from two. Time
permitting, I will explain some of the ways that we use a smaller set of tools
(e.g. Hall’s matching theorem or a double dimer swapping operation) in our
arguments. |
