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74
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 189 (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.
Infinite wedge and random partitions
 Selecta Mathematica (new series
"... The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example, ..."
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Cited by 94 (6 self)
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The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example,
Limit theorems for height fluctuations in a class of discrete space and time growth models
 J. Statist. Phys
, 2001
"... We introduce a class of onedimensional discrete spacediscrete time stochastic growth models described by a height function h t(x) with corner initialization. We prove, with one exception, that the limiting distribution function of h t(x) (suitably centered and normalized) equals a Fredholm determi ..."
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Cited by 82 (9 self)
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We introduce a class of onedimensional discrete spacediscrete time stochastic growth models described by a height function h t(x) with corner initialization. We prove, with one exception, that the limiting distribution function of h t(x) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large x and large t the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual RSK algorithm, Gessel's theorem, the Borodin Okounkov identity and a novel, rigorous saddle point analysis. In the fixed x, large t regime, we find a Brownian motion representation. This model is equilvalent to the Seppalainen Johansson model. Hence some of our results are not new, but the proofs are.
Random words, Toeplitz determinants and integrable systems
 I
, 2001
"... Abstract. It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject to an overall constraint that the ..."
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Cited by 38 (7 self)
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Abstract. It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject to an overall constraint that the eigenvalues lie in a hyperplane. 1.
Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices
, 2005
"... ..."
Optimal tail estimates for directed last passage site percolation with geometric random variables
, 2001
"... with geometric random variables ..."
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Discrete gap probabilities and discrete Painlevé equations
 DUKE MATH J
, 2003
"... We prove that Fredholm determinants of the form det(1 − Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2F1kernel to {s, s + 1,...}, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations. These Fredholm ..."
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Cited by 29 (6 self)
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We prove that Fredholm determinants of the form det(1 − Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2F1kernel to {s, s + 1,...}, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations. These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed according to a Poissonized Plancherel measure and a zmeasure, or as normalized Toeplitz determinants with symbols eη(ζ +ζ −1) and (1 + ξζ)
Generalized Riffle Shuffles and Quasisymmetric Functions
, 2001
"... Let x i be a probability distribution on a totally ordered set I, i.e., the probability of i 2 I is x i . (Hence x i 0 and x i = 1.) Fix n 2 P = f1; 2; : : :g, and define a random permutation w 2 S n as follows. For each 1 j n, choose independently an integer i j (from the distribution x i ). ..."
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Cited by 27 (0 self)
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Let x i be a probability distribution on a totally ordered set I, i.e., the probability of i 2 I is x i . (Hence x i 0 and x i = 1.) Fix n 2 P = f1; 2; : : :g, and define a random permutation w 2 S n as follows. For each 1 j n, choose independently an integer i j (from the distribution x i ). Then standardize the sequence i = i 1 \Delta \Delta \Delta i n in the sense of [34, p. 322], i.e., let ff 1 ! \Delta \Delta \Delta ! ff k be the elements of I actually appearing in i, and let a i be the number of ff i 's in i. Replace the ff 1 's in i by 1; 2; : : : ; a 1 from lefttoright, then the ff 2 's in i by a 1 + 1; a 1 + 2; : : : ; a 1 + a 2 from lefttoright, etc. For instance, if I = P and i = 311431, then w = 412653. This defines a probability distribution on the symmetric group S n , which we call the QSdistribution (because of the close connection with quasisymmetric functions explained below). If we need to be explicit about the parameters x = (x i ) i2I , t
On a Toeplitz determinant identity of Borodin and Okounkov
 Integral Equations Operator Theory
"... In this note we give two other proofs of an identity of A. Borodin and A. Okounkov which expresses a Toeplitz determinant in terms of the Fredholm determinant of a product of two Hankel operators. The second of these proofs yields a generalization of the identity to the case of block Toeplitz determ ..."
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Cited by 25 (6 self)
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In this note we give two other proofs of an identity of A. Borodin and A. Okounkov which expresses a Toeplitz determinant in terms of the Fredholm determinant of a product of two Hankel operators. The second of these proofs yields a generalization of the identity to the case of block Toeplitz determinants. The authors of the title proved in [2] an elegant identity expressing a Toeplitz determinant in terms of the Fredholm determinant of an infinite matrix which (although not described as such) is the product of two Hankel matrices. The proof used combinatorial theory, in particular a theorem of Gessel expressing a Toeplitz determinant as a sum over partitions of products of Schur functions. The purpose of this note is to give two other proofs of the identity. The first uses an identity of the second author [4] for the quotient of Toeplitz determinants in which the same product of Hankel matrices appears and the second, which is more direct and extends the identity to the case of block Toeplitz determinants, consists of carrying the first author’s collaborative proof [1] of the strong Szegö limit theorem one step further. We begin with the statement of the identity of [2], changing notation slightly. If φ is a function on the unit circle with Fourier coefficients φk then Tn(φ) denotes the Toeplitz matrix (φi−j)i,j=0,···,n−1 and Dn(φ) its determinant. Under general conditions φ has a representation φ = φ+ φ − where φ+ (resp. φ−) extends to a nonzero analytic function in the interior (resp. exterior) of the circle. We assume that φ has geometric mean 1, and normalize φ ± so that φ+(0) = φ−(∞) = 1. Define the infinite matrices Un and Vn acting on ℓ2 (Z+), where Z + = {0, 1, · · ·}, by Un(i, j) = (φ−/φ+)n+i+j+1, and the matrix Kn acting on ℓ 2 ({n, n + 1, · · ·}) by Vn(i, j) = (φ+/φ−)−n−i−j−1 Kn(i, j) = (φ−/φ+)i+k (φ+/φ−)−k−j. k=1 Notice that Kn becomes UnVn under the obvious identification of ℓ 2 ({n, n + 1, · · ·}) with ℓ 2 (Z +). It is easy to check that, aside from a factor (−1) i+j which does not affect its Fredholm determinant, the entries of Kn are the same as given by the integral formula (2.2) of [2]. The formula of Borodin and Okounkov is