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55
On the distribution of the length of the longest increasing subsequence of random permutations
 J. Amer. Math. Soc
, 1999
"... Let SN be the group of permutations of 1, 2,...,N. If π ∈ SN,wesaythat π(i1),...,π(ik) is an increasing subsequence in π if i1
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Cited by 347 (28 self)
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Let SN be the group of permutations of 1, 2,...,N. If π ∈ SN,wesaythat π(i1),...,π(ik) is an increasing subsequence in π if i1 <i2 <·· · <ikand π(i1) < π(i2) < ···<π(ik). Let lN (π) be the length of the longest increasing subsequence. For example, if N =5andπis the permutation 5 1 3 2 4 (in oneline notation:
Double scaling limit in the random matrix model: the RiemannHilbert approach
"... Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1. ..."
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Cited by 40 (7 self)
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Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1.
Generic Behavior of the Density of States in Random Matrix Theory and Equilibrium Problems in the Presence of Real Analytic External Fields
, 2000
"... The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example in random matrix theory: the limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1. it is pos ..."
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Cited by 36 (14 self)
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The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example in random matrix theory: the limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1. it is positive on the interior of a finite number of intervals, 2. it vanishes like a square root at endpoints, and 3. outside the support, there is strict inequality in the EulerLagrange variational conditions. If these conditions hold, then the limiting local eigenvalue statistics is loosely described by a "bulk" in which there is universal behavior involving the sine kernel, and "edge effects" in which there is a universal behavior involving the Airy kernel. Through techniques from potential theory and integrable systems, we show that this "regular" behavior is generic for equilibrium measures associated with real analytic external fields. In particular, we show that for any oneparameter family of external fields V=c the equilibrium measure exhibits this regular behavior, except for an at most countable number of values of c. We discuss applications of our results to random matrices, orthogonal polynomials and integrable systems.
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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Asymptotics of the partition function for random matrices via RiemannHilbert techniques, and applications to graphical enumeration
 Internat. Math. Research Notices
, 2003
"... Abstract. We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed RiemannHilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N ..."
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Cited by 33 (6 self)
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Abstract. We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed RiemannHilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.
Random vicious walks and random matrices
 Comm. Pure Appl. Math
"... A lock step walk is a onedimensional integer lattice walk in discrete time. Suppose that initially there are infinitely many walkers on the nonnegative even integer sites. At each moment of time, every walker moves either to its left or to its right with equal probability. The only constraint is t ..."
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Cited by 24 (4 self)
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A lock step walk is a onedimensional integer lattice walk in discrete time. Suppose that initially there are infinitely many walkers on the nonnegative even integer sites. At each moment of time, every walker moves either to its left or to its right with equal probability. The only constraint is that no two walkers can occupy the same site at the same time. Hence we describe the walk as vicious. It is proved that as time tends to infinity, a certain limiting conditional distribution of the displacement of the leftmost walker is identical to the limiting distribution of the (scaled) largest eigenvalue of a random GOE matrix (GOE TracyWidom distribution). The proof is based on the bijection between path configurations and semistandard Young tableaux established recently by Guttmann, Owczarek and Viennot. The distribution of semistandard Young tableaux is analyzed using the Hankel determinant expression for the probability obtained from the work of Rains and the author. The asymptotics of the Hankel determinant are then obtained by applying the DeiftZhou steepestdescent method to the RiemannHilbert problem for the related orthogonal polynomials. 1
Orthogonal polynomial ensembles in probability theory
 Prob. Surv
, 2005
"... Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary ..."
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Cited by 24 (1 self)
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Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other wellknown ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, noncolliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther
First order asymptotics of matrix integrals; a rigorous approach towards the understanding of matrix models
, 2002
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