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266
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:
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 240 (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).
General Orthogonal Polynomials
 in “Encyclopedia of Mathematics and its Applications,” 43
, 1992
"... Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed. ..."
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Cited by 59 (6 self)
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Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
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.
The Asymptotic Zero Distribution of Orthogonal Polynomials With Varying Recurrence Coefficients
"... this paper to ll this gap. To state our theorem we use the notation lim n=N!t X n;N = X 4 Kuijlaars and Van Assche to denote the property that in the doubly indexed sequence X n;N we have lim j!1 X n j ;N j = X whenever n j and N j are two sequences of natural numbers such that N j ! 1 and n j ..."
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Cited by 40 (9 self)
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this paper to ll this gap. To state our theorem we use the notation lim n=N!t X n;N = X 4 Kuijlaars and Van Assche to denote the property that in the doubly indexed sequence X n;N we have lim j!1 X n j ;N j = X whenever n j and N j are two sequences of natural numbers such that N j ! 1 and n j =N j ! t as j !1. For example, the convergence in Proposition 1.3 may be expressed by lim n=N!t (p n;N ) = w;t : We will use this notation throughout the rest of the paper. Our main result is the following. Theorem 1.4 Let for each N 2 N, two sequences fa n;N g 1 n=1 , a n;N > 0 and fb n;N g 1 n=0 of recurrence coecients be given, together with orthogonal polynomials p n;N generated by the recurrence xp n;N (x) = a n+1;N p n+1;N (x) + b n;N p n;N (x) + a n;N p n 1;N (x); n 0; (1.6) and the initial conditions p 0;N 1 and p 1;N 0. Suppose that there exist two continuous functions a : (0; 1) ! [0; 1), b : (0; 1) ! R, such that lim n=N!t a n;N = a(t); lim n=N!t b n;N = b(t) (1.7) whenever t > 0. Dene the functions (t) := b(t) 2a(t); (t) := b(t) + 2a(t); t > 0: (1.8) Then we have for every t > 0, lim n=N!t (p n;N ) = 1 t Z t 0 ! [(s);(s)] ds: (1.9) Here ! [;] is the measure given by (1.4) if < . If = , then ! [;] is the Dirac point mass at . Remark 1.5 The measure on the righthand side of (1.9) is the average of the equilibrium measures of the varying intervals [(s); (s)] for 0 < s < t. Its support is given by " inf 0<s<t (s); sup 0<s<t (s) # : (1.10) In particular, the support is always an interval. The support is unbounded if or are unbounded near 0. J. Approx. Theory 99 (1999), 167197. 5 Remark 1.6 Theorem 1.4 has an obvious extension to polynomials that are orthogonal with respect to a discrete measure supp...
Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation
"... Acknowledgements viii ..."
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.
Discretizing manifolds via minimum energy points
 Notices of the AMS
, 2004
"... There are a variety of needs for the discretization of a manifold—statistical sampling, quadrature rules, starting points for Newton’s method, computeraided design, interpolation schemes, finite element tessellations—to name but a few. So let us assume we are given a ddimensional manifold A in the ..."
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Cited by 34 (13 self)
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There are a variety of needs for the discretization of a manifold—statistical sampling, quadrature rules, starting points for Newton’s method, computeraided design, interpolation schemes, finite element tessellations—to name but a few. So let us assume we are given a ddimensional manifold A in the Euclidean space R d′ and wish to determine, say, 5000 points that “represent A”. How can we go about this if A is described by some geometric property or by some parametrization of the unit cube U d: = [0, 1] d in R d? Naturally, we must be guided by the particular application in mind. For an historical perspective as well as a brief motivational journey, let’s look at the simple case when A is the interval [−1, 1] ⊂ R. One obvious choice for N points that discretize A is the set of equally spaced points xk,N = −1 +
Constrained Energy Problems with Applications to Orthogonal Polynomials of a Discrete Variable
 J. Anal. Math
, 1997
"... Given a positive measure oe with koek ? 1 we write ¯ 2 M oe if ¯ is a probability measure and oe \Gamma ¯ is a positive measure. Under some general assumptions on the constraining measure oe and a weight function w we prove existence and uniqueness of a measure oe w that minimizes the weighted ..."
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Cited by 33 (4 self)
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Given a positive measure oe with koek ? 1 we write ¯ 2 M oe if ¯ is a probability measure and oe \Gamma ¯ is a positive measure. Under some general assumptions on the constraining measure oe and a weight function w we prove existence and uniqueness of a measure oe w that minimizes the weighted logarithmic energy over the class M oe . We also obtain a characterization theorem, a saturation result and a balayage representation for the measure oe w . As applications of our results we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials. 1 Introduction In this paper we shall investigate constrained energy problems in the presence of an external field. Before defining the problem, we briefly recall the classical and the weighted energy problems of potential theory. In so doing, we introduce the terminology that will be needed for statin...