Triangular tiling

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Abstract: 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

Abstract: We introduce a family of planar regions, called Aztec diamonds, and study tilings of these regions by dominoes. Our main result is that the Aztec diamond of order n has exactly 2n(n+1)/2 domino tilings. In this, the first half of a two-part paper, we give two proofs of this formula. The first proof exploits a connection between domino tilings and the alternating-sign matrices of Mills, Robbins, and Rumsey. In particular, a domino tiling of an Aztec diamond corresponds to a compatible pair of alternating-sign matrices. The second proof of our formula uses monotone triangles, which constitute another form taken by alternating-sign matricess by assigning each monotone triangle a suitable weight, we can count domino tilings of an Aztec diamond.

Abstract: In this article we study domino tilings of a family of finite regions called Aztec diamonds. Every such tiling determines a partition of the Aztec diamond into five sub-regions; in the four outer sub-regions, every tile lines up with nearby tiles, while in the fifth, central sub-region, differently-oriented tiles co-exist side by side. We show that when n is sufficiently large, the shape of the central sub-region becomes arbitrarily close to a perfect circle of radius n/sqrt(2) for all but a negligible proportion of the tilings. Our proof uses techniques from the theory of interacting particle systems. In particular, we prove and make use of a classification of the stationary behaviors of a totally asymmetric one-dimensional exclusion process in discrete time.