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377
Shape fluctuations and random matrices
, 1999
"... We study a certain random growth model in two dimensions closely related to the onedimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the TracyWidom largest eigenvalue distribution for the Gaussian Uni ..."
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Cited by 247 (10 self)
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We study a certain random growth model in two dimensions closely related to the onedimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the TracyWidom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE).
On the distribution of the largest eigenvalue in principal components analysis
 Ann. Statist
, 2001
"... Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribu ..."
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Cited by 216 (2 self)
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Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts. 1. Introduction. The
Longest increasing subsequences: from patience sorting to the BaikDeiftJohansson theorem
 Bull. Amer. Math. Soc. (N.S
, 1999
"... Abstract. We describe a simple oneperson card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of BaikDeiftJoha ..."
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Cited by 146 (2 self)
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Abstract. We describe a simple oneperson card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of BaikDeiftJohansson which yields limiting probability laws via hard analysis of Toeplitz determinants. 1.
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 146 (8 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.
Asymptotics of Plancherel measures for symmetric groups
 J. Amer. Math. Soc
, 2000
"... 1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a representation π ∈ G ∧ the weight (dim π) 2 /G. For the symmetric group S(n), the set S(n) ∧ is the set o ..."
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Cited by 141 (33 self)
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1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a representation π ∈ G ∧ the weight (dim π) 2 /G. For the symmetric group S(n), the set S(n) ∧ is the set of partitions λ of the number
Eigenvalues, invariant factors, highest weights, and Schubert calculus
 Bull. Amer. Math. Soc. (N.S
"... Abstract. We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Gra ..."
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Cited by 121 (3 self)
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Abstract. We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices. Contents 1. Eigenvalues of sums of Hermitian and real symmetric matrices 2. Invariant factors 3. Highest weights 4. Schubert calculus
Universality at the edge of the spectrum in Wigner random matrices
, 1999
"... We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n → +∞. As a corollary, we show that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions ..."
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Cited by 107 (8 self)
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We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n → +∞. As a corollary, we show that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases.
Scale invariance of the PNG droplet and the Airy process
 J. Stat. Phys
"... We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process, A(y). The Airy process is stationary, it has continuous sample paths, its single “time ” (fixed y) distribution is the T ..."
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Cited by 93 (13 self)
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We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process, A(y). The Airy process is stationary, it has continuous sample paths, its single “time ” (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y−2. Roughly the Airy process describes the last line of Dyson’s Brownian motion model for random matrices. Our construction uses a multi–layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation. 1 The PNG droplet The polynuclear growth (PNG) model is a simplified model for layer by layer growth [1, 2]. Initially one has a perfectly flat crystal in contact with its supersaturated vapor. Once in a while a supercritical seed is formed, which then spreads laterally by further attachment of particles at its perimeter sites. Such islands coalesce if they are in the same layer and further islands may be nucleated upon already existing ones. The PNG model ignores the lateral lattice
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 81 (8 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.