A 3d representation for lozenge tilings of a hexagon

Alexei Borodin and Matvey Borodin

Random lozenge tilings of a hexagon is, along with random domino tilings of the Aztec diamond, one of the most studied models of statistical physics. At last scale it was first investigated by Cohn-Larsen-Propp in [2], and we refer to lectures by Kenyon [3] and references therein for detailed information about random lozenge tilings of planar domains.

A lozenge tiling of a planar domain has a rather obvious three-dimensional interpretation, unlike domino tilings, where the definition of a height function that adds the third dimension is nontrivial. Here is a sample of the uniformly distributed lozenge tilings of a \( 30\times 30\times 30 \) hexagon:

30x30x30 by borodin on Sketchfab

There many other asymptotically interesting distributions on lozenge tilings of a hexagon. Here is a sample from a distribution with weights proportional to \( q^{volume} \), where \(q\) is a number between 0 and 1:

q-weighted boxed plane partitions 60x60x60 by borodin on Sketchfab

Certain more complicated but still very analyzable distributions on lozenge tilings of a hexagon that are related to classical orthogonal polynomials, were introduced by Borodin-Gorin-Rains in [1]. Here is a sample from a so-called \(q\)-Racah distribution that is reminiscent of a waterfall:

Waterfall 100x100x100 by borodin on Sketchfab

The three samples above were generated using a perfect sampling algorithm of [1] implemented by Vadim Gorin as a stand alone application available via his research webpage.

Three-dimensional interpretation is also available for lozenge tilings of planar domains that are not simply connected (have holes). Here is a sample of the uniformly distributed tilings of a hexagon with a rhombic hole in the middle. The distribution is further conditioned in such a way that this hole, which could be seen as "table" in three dimensions, is positioned at a prescribed height from the "floor" of the picture.

Hexagon with a hole 100x100x100 by borodin on Sketchfab

This sample was generated by Leo Petrov via Glauber dynamics, and we are very grateful to him for providing it. Many more beautiful simulations are available on his gallery page.

Periodic weighting of the three types of lozenges are known to generate a "gaseous phase" that can be observed as two asymptotically (but not perfectly) flat regions in the picture below. This is a sample of lozenge tilings of a hexagon with a 2x3 periodic weighting of the lozenges generated by Christophe Charlier.

2x3 periodic hexagon 150x150x150 by borodin on Sketchfab


1. A. Borodin, V. Gorin, E. Rains, \(q\)-distributions on boxed plane partitions. Selecta Math. 16 (2010), 731-789. arXiv:0905.0679

2. H. Cohn, M. Larsen, J. Propp, The shape of a typical boxed plane partition. New York J. Math. (1998), 137-165. arXiv:math/9801059

3. R. Kenyon, Lectures on dimers Statistical mechanics, 191-230, IAS/Park City Math. Ser., 16, Amer. Math. Soc., Providence, RI, 2009. arXiv:0910.3129