Dimer model in 3D

Dimer model in 3D

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Scott Sheffield, Massachusetts Institute of Technology
Fine Hall 224

The 3D dimer model is more complex than its 2D counterpart.  Classical 2D tools that fail to apply in 3D include Kasteleyn determinants, spanning tree bijections, height function FKG inequalities, amoeba and Ronkin function constructions, and so on. Even the simplest "local move connectedness" results fall apart in dimensions higher than two, although several papers have been written with partial results.

Nonetheless, the 3D dimer model turns out to be surprisingly interesting as a model of a random divergence-free flow.  We develop new tools that enable us to establish a large deviation principle for this random flow which is analogous to the 2D results of Cohn, Kenyon and Propp, with a unique rate function minimizer. We also present several interesting simulations and animations, including for higher dimensional Aztec diamonds and variants. There are many open problems here on which we would welcome assistance. Example: "Is there a finite set of local moves that, for all n, connect the tilings of a 2n by 2n by 2n box?"

The continuum analog of this model is also straightforward to describe. If one starts with a 3-vector-valued white noise, the orthogonal projection onto the space of "curl-free fields" gives the gradient of a Gaussian free field (GFF) while the orthogonal projection onto the space of "divergence-free fields" gives a very closely related object called the Gaussian divergence-free field (GDFF) which itself has many beautiful properties.

This is joint work with Nishant Chandgotia and Catherine Wolfram.