For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the al...

. We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond 's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance to domino tilings is demonstrated elsewhere), and to combine these estimates with some exponential sum bounds to deduce our final result. This approach generalizes straightforwardly to the case in which the probability distribution on the set of tilings incorporates bias favoring horizontal over vertical tiles or vice versa. We also prove a fairly general large deviation estimate for domino tilings of sim...

Abstract. We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for the measure of maximal entropy µ on the space of tilings of the plane with dominos. We construct a measure ν on the set of lozenge tilings of the plane, show that its entropy is the topological entropy, and compute explicitly the ν-measures of cylinder sets. As applications of these results, we prove that the translation action is strongly mixing for µ and ν, and compute the rate of convergence to mixing (the correlation between distant events). For the measure ν we compute the variance of the height function. Resumé. Soit µ la mesure d’entropie maximale sur l’espace X des pavages du plan par des dominos. On calcule explicitement la mesure des sous-ensembles cylindriques de X. De même, on construit une mesure ν d’entropie maximale sur l’espace X ′ des pavages du plan par losanges, et on calcule explicitement la mesure des sous-ensembles cylindriques. Comme application on calcule, pour µ et ν, les correlations d’évenements distants, ainsi que la ν-variance de la fonction “hauteur ” sur X ′. 1.

Abstract. We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entropy integral. We associate an entropy to every sort of local behavior domino tilings can exhibit, and prove that almost all tilings lie within ε (for an appropriate metric) of the unique entropy-maximizing solution. This gives a solution to the dimer problem with fully general boundary conditions, thereby resolving an issue first raised by Kasteleyn. Our methods also apply to dimer models on other grids and their associated tiling models, such as tilings of the plane by three orientations of unit lozenges. The effect of boundary conditions is, however, not entirely trivial and will be discussed in more detail in a subsequent paper. P. W. Kasteleyn, 1961 1.

Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux. 1.

this paper we deal with the two-dimensional lattice dimer model, or domino tiling model (a domino tiling is a tiling with 2 \Theta 1 and 1 \Theta 2 rectangles). We prove that in the limit as the lattice spacing ffl tends to zero, certain macroscopic properties of the tiling are conformally invariant. The height function h on a domino tiling is an integer-valued function on the vertices in a tiling. It is defined below in section 2.2; see also [4, 19]. One can think of a domino tiling of U as a map h from U

We introduce a concept for random tilings which, comprising the conventional one, is also applicable to tiling ensembles without height representation. In particular, we focus on the random tiling entropy as a function of the tile densities. In this context, and under rather mild assumptions, we prove a generalization of the first random tiling hypothesis which connects the maximum of the entropy with the symmetry of the ensemble. Explicit examples are obtained through the re-interpretation of several exactly solvable models. This also leads to a counterexample to the analogue of the second random tiling hypothesis about the form of the entropy function near its maximum.

We study the four-state antiferromagnetic Potts model on the triangular lattice. We show that the model has six types of defects which diuse and annihilate according to certain conservation laws consistent with their having a vector-valued topological charge. Using the properties of these defects, we deduce a 2 + 2-dimensional height representation for the model and hence show that the model is equivalent to the three-state Potts antiferromagnet on the Kagome lattice and to bond-coloring models on the triangular and honeycomb lattices. We also calculate critical exponents for the ground state ensemble of the model. We nd that the exponents governing the spin{spin correlation function and spin uctuations violate the Fisher scaling law because of constraints on path length which increase the eective wavelength of the spin operator on the height lattice. We conrm our predictions by extensive Monte Carlo simulations of the model using the Wang-SwendsenKoteck y cluster a...

robably do not correspond to any physical properties of the substances that the dimer models were originally introduced to model. However, insofar as the study of exactly solved models has taken on a second life as a chapter of pure mathematics, these results may be of interest to researchers. Moreover, the concept of an "effective field" (described below) could turn out to be useful in more realistic applications of statistical mechanics to non-homogeneous structures. Figure 1 shows a tiling of a certain shape (an "Aztec diamond of order 3") by 1-by-2 tiles ("dominoes"), and an equivalent dimer packing of the associated dual graph. Aztec diamonds were introduced in a paper of Elkies et al., 6 where it was shown that the Aztec diamond of order n has exactly 2 n(n+1)=2 tilings by dominoes. One of the proofs given there used a method called "shuffling," which gave a bijection between tilings of the diamond of ord

Several classic tilings, including rhombuses and dominoes, possess height functions which allow us to 1) prove ergodicity and polynomial mixing times for Markov chains based on local moves, 2) use coupling from the past to sample perfectly random tilings, 3) map the statistics of random tilings at large scales to physical models of random surfaces, and and 4) are related to the “arctic circle ” phenomenon. However, few examples are known for which this approach works in three or more dimensions. Here we show that the rhombus tiling can be generalized to ndimensional tiles for any n ¡ 3. For each n, we show that a certain local move is ergodic, and conjecture that it has a mixing time of O ¢ L n £ 2 logL ¤ on regions of size L. For n ¥ 3, the tiles are rhombohedra, and the local move consists of switching between two tilings of a rhombic dodecahedron. We use coupling from the past to sample random tilings of a large rhombic dodecahedron, and show that arctic regions exist in which the tiling is frozen into a fixed state. However, unlike the two-dimensional case in which the arctic region is an inscribed circle, here it seems to be octahedral. In addition, height fluctuations between the boundary of the region and the center appear to be constant rather than growing logarithmically. We conjecture that this is because the physics of the model is in a “smooth” phase where it is rigid at large scales, rather than a “rough ” phase in which it is elastic.