Square tiling

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