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68
On the distribution of the length of the second row of a Young diagram under Plancherel measure
 Geom. Funct. Anal
, 1999
"... We investigate the probability distribution of the length of the second row of a Young diagram of size N equipped with Plancherel measure. We obtain an expression for the generating function of the distribution in terms of a derivative of an associated Fredholm determinant, which can then be used to ..."
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Cited by 53 (8 self)
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We investigate the probability distribution of the length of the second row of a Young diagram of size N equipped with Plancherel measure. We obtain an expression for the generating function of the distribution in terms of a derivative of an associated Fredholm determinant, which can then be used to show that as N ## the distribution converges to the TracyWidom distribution [TW] for the second largest eigenvalue of a random GUE matrix. This paper is a sequel to [BDJ], where we showed that as N # # the distribution of the length of the first row of a Young diagram, or equivalently, the length of the longest increasing subsequence of a random permutation, converges to the TracyWidom distribution [TW] for the largest eigenvalue of a random GUE matrix. 1
The RiemannHilbert approach to strong asymptotics for orthogonal polynomials on [1, 1]
"... We consider polynomials that are orthogonal on [1, 1] with respect to a modified Jacobi weight (1  x) # (1 + x) # h(x), with #, # > 1 and h real analytic and stricly positive on [1, 1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [ ..."
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Cited by 44 (23 self)
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We consider polynomials that are orthogonal on [1, 1] with respect to a modified Jacobi weight (1  x) # (1 + x) # h(x), with #, # > 1 and h real analytic and stricly positive on [1, 1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [1, 1], for the recurrence coe#cients and for the leading coe#cients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants. For the asymptotic analysis we use the steepest descent technique for RiemannHilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szego function associated with the weight and for the local analysis around the endpoints 1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions. 1 Supported by FWO research project G.0176.02 and by INTAS project 00272 2 Supported by NSF grant #DMS9970328 3 Supported by FWO research project G.0184.01 and by INTAS project 00272 4 Research Assistant of the Fund for Scientific Research  Flanders (Belgium) 1 1
Application of the τfunction theory of Painlevé equations to random matrices
 PV, PIII, the LUE, JUE and CUE
, 2002
"... Okamoto has obtained a sequence of τfunctions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be reexpressed as multidim ..."
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Cited by 42 (16 self)
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Okamoto has obtained a sequence of τfunctions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be reexpressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter N, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVI theory. We show that the Hamiltonian also satisfies an equation related to the discrete PV equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general PV transcendent in σ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter a a nonnegative
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.
Universal Results for Correlations of Characteristic Polynomials
 RiemannHilbert Approach. Commun. Math. Phys
, 2003
"... Abstract We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same ..."
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Cited by 26 (6 self)
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Abstract We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the RiemannHilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via DeiftZhou steepestdescent/stationary phase method for RiemannHilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of β = 2 symmetry class. 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
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 21 (5 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)
Universality in random matrix theory for orthogonal and symplectic ensembles
, 2004
"... We give a proof of the Universality Conjecture for orthogonal (β = 1) and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e −V (x) where V is a polynomial, V (x) = κ2mx 2m + · · ·, κ2m> 0. For such weights the associated equilibrium measure is su ..."
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Cited by 18 (5 self)
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We give a proof of the Universality Conjecture for orthogonal (β = 1) and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e −V (x) where V is a polynomial, V (x) = κ2mx 2m + · · ·, κ2m> 0. For such weights the associated equilibrium measure is supported on a single interval. The precise statement of our results is given in Theorem 1.1 below. An announcement of our results can be found in [DG]. For a proof of the Universality Conjecture for unitary ensembles (β = 2), for the same class of weights, see [DKMVZ2]. Our starting point is Widom’s representation [W] of the orthogonal and symplectic correlation kernels in terms of the kernel arising in the unitary case (β = 2) plus a correction term which is constructed out of derivatives and integrals of orthonormal polynomials (OP’s) {pj}j≥0 with respect to the weight w(x). The calculations in [W] in turn depend on the earlier work of Tracy and Widom [TW2]. It turns out (see [W] and also Theorems 2.1 and 2.2 below) that only the OP’s in the range j = N + O(1), N → ∞, contribute to the correction term. In controlling this correction term, and hence proving Universality for β = 1 and 4, the uniform Plancherel–Rotach type asymptotics for the OP’s found in [DKMVZ2] play an important role, but there are significant new analytical difficulties that must be overcome which are not present in the case β = 2. We note that we do not use skew orthogonal polynomials. In later work we plan to consider weights of the form e −NV (x) for polynomial V, where the equilibrium measure may be supported on a finite number of intervals.
Riemann–Hilbert analysis for Laguerre polynomials with large negative parameter
 Comput. Meth. Funct. Theory
"... Abstract. We study the asymptotic behavior of Laguerre polynomials L (αn) n (nz) as n → ∞, where αn is a sequence of negative parameters such that −αn/n tends to a limit A> 1 as n → ∞. These polynomials satisfy a nonhermitian orthogonality on certain contours in the complex plane. This fact allows ..."
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Cited by 17 (11 self)
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Abstract. We study the asymptotic behavior of Laguerre polynomials L (αn) n (nz) as n → ∞, where αn is a sequence of negative parameters such that −αn/n tends to a limit A> 1 as n → ∞. These polynomials satisfy a nonhermitian orthogonality on certain contours in the complex plane. This fact allows the formulation of a Riemann–Hilbert problem whose solution is given in terms of these Laguerre polynomials. The asymptotic analysis of the Riemann–Hilbert problem is carried out by the steepest descent method of Deift and Zhou, in the same spirit as done by Deift et al. for the case of orthogonal polynomials on the real line. A main feature of the present paper is the choice of the correct contour. Keywords: Riemann–Hilbert problems, generalized Laguerre polynomials, strong asymptotics, steepest descent method.