Random tilings

Random tilings are simplified mathematical models of stochastic growth.

Our group focuses on their asymptotic behavior, including limit shapes and fluctuations, using methods from probability theory, combinatorics and complex analysis. 

All figures below were generated using either the Matlab program [here] or the Python program [here]. We thank Lennart Hübner and Max van Horssen for kindly allowing us to use their fast Python implementation.

Lozenge tilings of hexagon

The points seen in the above picture (right) form a determinantal point process. 

Along a fixed vertical line, these points exhibit a natural repulsion, and can be viewed as a discrete analogue of the eigenvalues of Hermitian random matrices. 

One question that our group has studied concerns gap probabilities, that is, the probability that a given interval (the gap) along a fixed vertical line contains no paths .

This problem can be reformulated as a purely combinatorial question: counting the number of lozenge tilings of a suitably reduced hexagon.

The correspondence between non-intersecting paths constrained to avoid a given interval and lozenge tilings of a reduced hexagonal domain is illustrated in the figures below. 

Unlike the full hexagon, there is no known closed-form formula for the number of lozenge tilings of a reduced hexagon.

In particular, no analogue of MacMahon’s formula is available in this setting, which makes the analysis of such constrained tilings challenging. 

The next figures illustrate a system of non-intersecting paths constrained to avoid three disjoint intervals (left), together with the corresponding lozenge tilings of a reduced hexagon (right). 

The following figures show examples of lozenge tilings of reduced hexagons for larger sizes. 

We conclude this section by noting that other random tiling models have been studied, where different probability measures are used. The three figures below show examples of tilings generated randomly according to such models. 

The above figures reveal several interesting phenomena. 

For example, the middle figure shows some frozen regions tiled with two types of lozenges alternating in a stair-like fashion.

The right figure exhibits a new phase, beyond liquid and frozen, which is known as the gas phase.

Domino tilings of Aztec diamonds

The above figures show that a phenomenon similar to that in the hexagon occurs: when the Aztec diamond is large, most tilings feature four frozen regions near the corners, while the central region (approaching a circular shape in the limit) contains all types of dominos. 

The boundary between the liquid and frozen regions is known as the arctic curve. 

Our group has also studied gap probabilities in this model. As illustrated in the figures below, a system of non-intersecting paths constrained not to intersect a semi-infinite interval (right) corresponds to a domino tiling of the Aztec diamond featuring a particular frozen region near the north corner (left). 

The next figures represents three large reduced Aztec diamonds. 

By tuning the size of the removed corner, we observe different types of phenomena. 

The correspondence between constrained non-intersecting paths and domino tilings becomes less clear when the gap is not a single semi-infinite interval. 

In contrast with the previous examples, and also in contrast with lozenge tilings of the hexagon, such gaps in the Aztec diamond do not correspond to a specific region that must freeze. 

Finally, we note that, beyond the uniform measure, other probability measures on the set of all possible domino tilings of the Aztec diamond have been studied. 

The following figures show typical tilings in the so-called doubly periodic Aztec diamond. 

The coloring has been adjusted to improve visualization. The central region is a so-called gas region, distinct from the liquid and frozen regions.