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109
Generating Random Spanning Trees More Quickly than the Cover Time
 PROCEEDINGS OF THE TWENTYEIGHTH ANNUAL ACM SYMPOSIUM ON THE THEORY OF COMPUTING
, 1996
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Lectures on random planar curves and SchrammLoewner evolutions
, 2003
"... The goal of these lectures is to review some of the mathematical results that have been derived in the last years on conformal invariance, scaling limits and properties of some twodimensional random curves. The (distinguished) audience of the SaintFlour summer school consists mainly of probabilist ..."
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Cited by 144 (7 self)
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The goal of these lectures is to review some of the mathematical results that have been derived in the last years on conformal invariance, scaling limits and properties of some twodimensional random curves. The (distinguished) audience of the SaintFlour summer school consists mainly of probabilists and I therefore assume knowledge in stochastic calculus (Itô’s formula etc.), but no special background in basic complex analysis. These lecture notes are neither a book nor a compilation of research papers. While preparing them, I realized that it was hopeless to present all the recent results on this subject, or even to give the complete detailed proofs of a selected portion of them. Maybe this will disappoint part of the audience but the main goal of these lectures will be to try to transmit some ideas and heuristics. As a reader/part of an audience, I often think that omitting details is dangerous, and that ideas are sometimes better understood when the complete proofs are given, but in the present case, partly because the technicalities often use complex analysis tools that the audience might not be so
Correlation function of Schur process with application to local geometry of a random 3dimensional Young Diagram
, 2001
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Nonintersecting paths, random tilings and random matrices
 Probab. Theory Related Fields
, 2002
"... Abstract. We investigate certain measures induced by families of nonintersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abchexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the s ..."
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Cited by 124 (11 self)
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Abstract. We investigate certain measures induced by families of nonintersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abchexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as the eigenvalue measures in random matrix theory like GUE, which can in fact can be obtained from nonintersecting Brownian motions. The derivations of the measures are based on the KarlinMcGregor or LindströmGesselViennot method. We use the measures to show some asymptotic results for the models. 1.
A variational principle for domino tilings
"... 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 entrop ..."
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Cited by 102 (16 self)
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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 entropymaximizing 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.
Local Statistics For Random Domino Tilings Of The Aztec Diamond
 Duke Math. J
, 1996
"... . 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 diam ..."
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Cited by 99 (12 self)
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. 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...
Local statistics of lattice dimers
, 1997
"... 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 ..."
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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 νmeasures 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 sousensembles 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 sousensembles cylindriques. Comme application on calcule, pour µ et ν, les correlations d’évenements distants, ainsi que la νvariance de la fonction “hauteur ” sur X ′. 1.
The arctic circle boundary and the Airy process
 Ann. Prob
, 2005
"... Abstract. 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 struc ..."
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Cited by 88 (6 self)
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Abstract. 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. 1. Introduction and
Conformal Invariance of Domino Tiling
 Ann. Probab
, 1999
"... this paper we deal with the twodimensional 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 ..."
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Cited by 71 (12 self)
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this paper we deal with the twodimensional 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 integervalued 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