A variational principle for domino tilings

By Henry Cohn, Richard Kenyon, James Gary Propp. 2000

362 citations recorded by the source corpus.

1.1. Description of results. A domino is a 1 x 2 (or 2 x 1) rectangle, and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps. In 1961, Kasteleyn [Kal] found a formula for the number of domino tilings of an m x n rectangle (with mn even), as shown in Figure 1 for rm = n = 68. Temperley and Fisher [TF] used a differenit method and arrived at the same result at almost exactly the same time. Both lines of calculation showed that the logarithm of number of tilings, divided by the number of dominos in a tiling (that is, mn/2), converges to 2G/7r 0.58 (here G is Catalan's constant). On the other hand, in 1992 Elkies et al. [EKLP] studied domino tilings of regions they called Aztec diamonds (Figure 2 shows an Aztec diamond of order 48), and showed that the logarithm of the number of tilings, divided by the number of dominos, converges to the smaller number (log 2)/2 0.35. Thus, even though the region in Figure 1 has slightly smaller area than the region in Figure 2, the former has far more domino tilings. For regions with other shapes, neither of these asymptotic formulas may apply. In the present paper we consider simply-connected regions of arbitrary shape. We give an exact formula for the limiting value of the logarithm of the number of tilings per unit area, as a function of the shape of the boundary of the region, as the size of the region goes to infinity. In particular, we show that computation of this limit is intimately linked with an understanding of long-range variations in the local statistics of random domino tilings. Such variations can be seen by comparing Figures 1 and 2. Each of the two tilings is random in the sense that the algorithm [PWI that was used to create it generates each of the possible tilings of the region being tiled with the same probability. Hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

Subfields: Materials Chemistry, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d191a021e30aa5b347ae715859135e32b713a66d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Materials Chemistry

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Lectures on Dimers (2009, background): “It requires a bit of work [5] to show that Theorem 7.”
Random Skew Plane Partitions and the Pearcey Process (2005, result): “The results of [ 6 ] imply the following form of the law of large numbers: scaled by r in all directions, the surface of our random skew 3D partition converges to a nonrandom surface — the limit shape.”
Stochastic six-vertex model (2014, background): “completely dierent methods to prove this theorem. Note that in a related context of random height function arising from tilings and dimer models, the variational approach is much more developed, see [CKP], [KOS], [KO]. We will revisit the idea of variational problems later from a point of view of in”
The Birth of a Random Matrix (2006, background): “This phenomenon is clearly visible in the simulation in Figure 2 and may be proven on very general grounds, see [6].”
Representations of classical Lie groups and quantized free convolution (2013, methodology): “as a great interest in random lozenge tilings of planar domains and their asymptotic properties as the size of the domain goes to innity in the last 15 years, with many fascinating results, see e.g. [CKP], [OR], [PS], [KOS], [KO], [Ke], [BF], [BG], [BGR], [Pe1], [GP], [Mk] and many others. In partic”
The Cayley Trick and Triangulations of Products of Simplices (2003, background): “The property that the logarithm of the number of tilings is proportional to the area for dilations of a given shape is well-known in the context of usual lozenge tilings, as follows from the following result of Cohn, Kenyon and Propp [4] (they consider mostly the case of domino tilings, i.e., perfec”
Minimizers of convex functionals arising in random surfaces (2008, background): “P which stay O(") away from a given function ’ dened on In connection with problems that arise in statistical mechanics, it is interesting to investigate the limiting behavior as "! 0. Cohn, Kenyon and Propp [ CKP ] proved that as "! 0 the random surfaces almost surely converge uniformly to a nonran”
Brownian Gibbs property for Airy line ensembles (2011, result): “These results are a generalization of [17] where similar results for G = Z2 with constant edge weights were obtained.”
Rigidity of Polyhedral Surfaces, III (2010, methodology): “However, our proof of Rivin’s theorem is different from that in [14] and we use the variational principle established by Cohen, Kenyon and Propp [6].”
Dimers and amoebae (2003, background): “A version of this statement is proved in [2] in the case G = Z(2).”