Lecture 8: October 1

Consider a simply connected region S of the two-dimensional Cartesian lattice (e.g. a 2n× 2n chessboard). A tiling of S is a covering of all its squares by non-overlapping dominoes, each of which occupies two adjacent squares of the lattice. We want to generate u.a.r. a tiling of S. Such problems arise in statistical physics, where the tilings correspond to configurations of a so-called dimer system on S. Various physical properties of the system are related to the expected value, over the uniform distribution, of some function defined over configurations, such as the number of horizontal dominoes or the correlation between the orientation of dominoes at two given squares. The Markov chain most commonly used to generate u.a.r. a tiling of S picks a 2 × 2 square in S u.a.r. and rotates it if it contains exactly two dominoes. (See, e.g., Figure 8.1.) Thurston [Th90] proved that the set of all tilings is connected by such moves (and we shall see this below).