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The 3DÌýdimerÌýmodel is more complex than its 2D counterpart.Ìý Classical 2DÌýtools that fail to apply in 3D include Kasteleyn determinants, spanning treeÌýbijections, height function FKG inequalities, amoeba and Ronkin functionÌýconstructions, and so on. Even the simplest "local move connectedness"Ìýresults fall apart in dimensions higher than two, although several papersÌýhave been written with partial results.

Nonetheless, the 3DÌýdimerÌýmodel turns out to be surprisingly interesting asÌýa model of a random divergence-free flow.Ìý We develop new tools that enableÌýus to establish a large deviation principle for this random flow which isÌýanalogous to the 2D results of Cohn, Kenyon and Propp, with a unique rateÌýfunction minimizer. We also present several interestingÌýsimulations andÌýanimations, including for higher dimensional Aztec diamonds and variants.ÌýThere are many open problems here on which we would welcome assistance.ÌýExample: "Is there a finite set of local moves that, for all n, connect theÌýtilings of a 2n by 2n by 2n box?"

The continuum analog of this model is also straightforward to describe. IfÌýone starts with a 3-vector-valued white noise, the orthogonal projectionÌýonto the space of "curl-free fields" gives the gradientÌýof a Gaussian freeÌýfield (GFF) while the orthogonalÌýprojection onto the space ofÌý"divergence-free fields" gives a very closely related object called theÌýGaussian divergence-free field (GDFF) which itself has many beautifulÌýproperties.

This is joint work with Nishant Chandgotia and Catherine Wolfram.