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91
Discrete orthogonal polynomial ensembles and the Plancherel measure
, 2001
"... We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble i ..."
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Cited by 194 (10 self)
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We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble is related to a twodimensional directed growth model, and the Charlier ensemble is related to the lengths of weakly increasing subsequences in random words. The Krawtchouk ensemble occurs in connection with zigzag paths in random domino tilings of the Aztec diamond, and also in a certain simplified directed firstpassage percolation model. We use the Charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of Tracy and Widom. As a limit of the Meixner ensemble or the Charlier ensemble we obtain the Plancherel measure on partitions, and using this we prove a conjecture of Baik, Deift and Johansson that under the Plancherel measure, the distribution of the lengths of the first k rows in the partition, appropriately scaled, converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the Gaussian Unitary Ensemble. In this problem a certain discrete kernel, which we call the discrete Bessel kernel, plays an important role.
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 131 (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.
Mixing times of lozenge tiling and card shuffling Markov chains
, 1997
"... Abstract. We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, ..."
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Cited by 104 (1 self)
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Abstract. We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an ℓ×ℓ region we bound the mixing time by O(ℓ 4 log ℓ), which improves on the previous bound of O(ℓ 7), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and SaloffCoste by lower bounding the mixing time of various cardshuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a pathcoupling analysis of Bubley and Dyer, we obtain an O(n 3 log n) upper bound on the mixing time of the KarzanovKhachiyan Markov chain for linear extensions. 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 (15 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.
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 90 (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
Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results
 INT. MATH. RES. NOT
, 2003
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Height Fluctuations in the Honeycomb Dimer Model
 Comm. Math. Phys
"... We study a model of random crystalline surfaces arising in the dimer model on the honeycomb lattice. For a fixed “wire frame ” boundary condition, as the lattice spacing ǫ → 0, Cohn, Kenyon and Propp [3] showed the almost sure convergence of a random surface to a nonrandom limit shape Σ0. We show h ..."
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Cited by 35 (2 self)
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We study a model of random crystalline surfaces arising in the dimer model on the honeycomb lattice. For a fixed “wire frame ” boundary condition, as the lattice spacing ǫ → 0, Cohn, Kenyon and Propp [3] showed the almost sure convergence of a random surface to a nonrandom limit shape Σ0. We show here that when Σ0 has no facets, for a large family of boundary conditions approximating the wire frame, the largescale surface fluctations (height fluctuations) about Σ0 converge as ǫ → 0 to a Gaussian free field for an appropriate conformal structure determined by Σ0. We also show that the local statistics of the fluctuations near a given point x are given by the unique ergodic Gibbs measure (on plane configurations) whose slope is the slope of the tangent plane of Σ0 at x. 1
Universality for mathematical and physical systems
, 2006
"... Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all ..."
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Cited by 34 (0 self)
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Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner’s model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting. 1.
The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis
"... Abstract. We compute the number of rhombus tilings of a hexagon with sides N, M, N, N, M, N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of length M. 1. ..."
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Cited by 24 (8 self)
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Abstract. We compute the number of rhombus tilings of a hexagon with sides N, M, N, N, M, N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of length M. 1.
Toeplitz determinants, random growth and determinantal processes
 in Proceedings of the International Congress of Mathematicians
, 2002
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