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Domino statistics of the twoperiodic Aztec diamond
, 2014
"... Abstract. Random domino tilings of the Aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a tw ..."
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Abstract. Random domino tilings of the Aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a twoperiodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. To analyze this model, we use entries of the inverse Kasteleyn matrix which give the probability of any configuration of dominoes. A formula for these entries, for this particular model, was derived by Chhita and Young (2014). In this paper, we find a major simplication of this formula expressing entries of the inverse Kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics. In a part of the Aztec diamond we use this formula to show that the entries of the inverse Kasteleyn matrix converge to the known entries of the fullplane inverse Kasteleyn matrices for the different phases. We also study the detailed asymptotics of the covariance between dominoes at both the ‘solidliquid ’ and ‘liquidgas ’ boundaries. Finally we provide a potential candidate for a combinatorial description of
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"... We describe random generation algorithms for a large class of random combinatorial objects called Schur processes, which are sequences of random (integer) partitions subject to certain interlacing conditions. This class contains several fundamental combinatorial objects as special cases, such as ..."
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We describe random generation algorithms for a large class of random combinatorial objects called Schur processes, which are sequences of random (integer) partitions subject to certain interlacing conditions. This class contains several fundamental combinatorial objects as special cases, such as plane partitions, tilings of Aztec diamonds, pyramid partitions and more generally steep domino tilings of the plane. Our algorithm, which is of polynomial complexity, is both exact (i.e. the output follows exactly the target probability law, which is either Boltzmann or uniform in our case), and entropy optimal (i.e. it reads a minimal number of random bits as an input). It can be viewed as a (far reaching) common generalization of the RSK algorithm for plane partitions and of the domino shuffling algorithm for domino tilings of the Aztec diamond. At a technical level, it relies on unified bijective proofs of the different types of Cauchy identities for Schur functions, and on an adaptation of Fomin’s growth diagram description of the RSK algorithm to that setting. Simulations performed with this algorithm suggest previously unobserved phenomena in the limit shapes for some tiling models. 1