Tiling pictures of the plane with dominoes

AbstractWe consider the problem of tiling with dominoes pictures of the plane, in theoretical and algorithmic aspects. For generalities and other tiling problems, see for example Refs. Beauquier et al. (1995), Conway and Lagarias (1990), Kannan and Soroker (1992), Kenyon (1992), and Beauquier (1991). The pictures which are considered here may have holes, but uniquely balanced holes, that is every hole, if chessboard-like coloured, has an equal number of black squares and of white ones. We give an algorithmic characterization of tilable pictures and a canonical decomposition into ‘strongly’ tilable subpictures. The given algorithm is linear as far the considered pictures have a finite number of (balanced) holes. In the same hypothesis there is a good parallel algorithm (in class NC). Graphical extension of the used method (heights' method) is applied to a class of bipartite planar graphs. The particular case of without holes pictures is developed in Fournier (1996).As far as I know, the results in this paper are new, except the notions and the theorem in Section 2, which are substantially present in Thurston (1990)

The arctic circle boundary and the Airy process

We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp's conjecture concerning the structure of the tiling at the center of the Aztec diamond.Comment: Published at http://dx.doi.org/10.1214/009117904000000937 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org