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35
Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices
, 2005
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Averages of characteristic polynomials in Random Matrix Theory
, 2004
"... We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with G ..."
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Cited by 35 (6 self)
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We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with Gaussian weights. Classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulas by a simple computation. We employ a discrete approximation method: the problem is solved for discrete analogues of random matrix ensembles originating from representation theory, and then a limit transition is performed. Exact pfaffian/determinantal formulas for the discrete averages are proved using standard tools of linear algebra; no application of orthogonal or skeworthogonal polynomials is needed.
Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups UN
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Strong Asymptotics of LaguerreType Orthogonal Polynomials and Applications in Random Matrix Theory
, 2005
"... We consider polynomials orthogonal on [0, ∞) with respect to Laguerretype weights w(x) = x α e −Q(x) , where α> −1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine PlancherelRotach type asymptotics in the entire complex plane fo ..."
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Cited by 29 (3 self)
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We consider polynomials orthogonal on [0, ∞) with respect to Laguerretype weights w(x) = x α e −Q(x) , where α> −1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine PlancherelRotach type asymptotics in the entire complex plane for the orthonormal polynomials with respect to w, as well as asymptotics of the corresponding recurrence coefficients and of the leading coefficients of the orthonormal polynomials. As an application we will use these asymptotics to prove universality results in random matrix theory. We will prove our results by using the characterization of orthogonal polynomials via a 2 × 2 matrix valued RiemannHilbert problem, due to Fokas, Its and Kitaev, together with an application of the DeiftZhou steepest descent method to analyze the RiemannHilbert problem asymptotically.
On the average of characteristic polynomials from classical groups
 COMM. MATH. PHYS
, 2005
"... We provide an elementary and selfcontained derivation of formulae for products and ratios of characteristic polynomials from classical groups using classical results due to Weyl and Littlewood. ..."
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Cited by 28 (1 self)
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We provide an elementary and selfcontained derivation of formulae for products and ratios of characteristic polynomials from classical groups using classical results due to Weyl and Littlewood.
Autocorrelation of ratios of Lfunctions
 COMM. NUMBER THEORY AND PHYSICS
, 2007
"... We give a new heuristic for all of the main terms in the quotient of products of Lfunctions averaged over a family. These conjectures generalize the recent conjectures for mean values of Lfunctions. Comparison is made to the analogous quantities for the characteristic polynomials of matrices ave ..."
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Cited by 21 (3 self)
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We give a new heuristic for all of the main terms in the quotient of products of Lfunctions averaged over a family. These conjectures generalize the recent conjectures for mean values of Lfunctions. Comparison is made to the analogous quantities for the characteristic polynomials of matrices averaged over a classical compact group.
Averages of ratios of characteristic polynomials for the compact classical groups
"... Averages of ratios of characteristic polynomials for the compact classical groups are evaluated in terms of determinants whose dimensions are independent of the matrix rank. These formulas are shown to be equivalent to expressions for the same averages obtained in a previous study, which was motivat ..."
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Cited by 18 (4 self)
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Averages of ratios of characteristic polynomials for the compact classical groups are evaluated in terms of determinants whose dimensions are independent of the matrix rank. These formulas are shown to be equivalent to expressions for the same averages obtained in a previous study, which was motivated by applications to analytic number theory. Our approach uses classical methods of random matrix theory, in particular determinants and orthogonal polynomials, and can be considered more elementary than the method of Howe pairs used in the previous study.
Painléve IV and degenerate Gaussian unitary ensembles
 J. Phys. A: Math. Gen
, 2006
"... We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a ..."
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Cited by 13 (4 self)
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We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painlevé IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable.
On the largest eigenvalue of a Hermitian random matrix model with spiked external source I. Rank one case
"... Abstract Consider a Hermitian matrix model under an external potential with spiked external source. When the external source is of rank one, we compute the limiting distribution of the largest eigenvalue for general, regular, analytic potential for all values of the external source. There is a tran ..."
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Cited by 13 (4 self)
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Abstract Consider a Hermitian matrix model under an external potential with spiked external source. When the external source is of rank one, we compute the limiting distribution of the largest eigenvalue for general, regular, analytic potential for all values of the external source. There is a transitional phenomenon, which is universal for convex potentials. However, for nonconvex potentials, new types of transition may occur. The higher rank external source is analyzed in the subsequent paper.
ON THE SECONDORDER CORRELATION FUNCTION OF THE CHARACTERISTIC POLYNOMIAL OF A REAL SYMMETRIC WIGNER MATRIX
, 2007
"... We consider the asymptotic behaviour of the secondorder correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble, obtained by Brézin and Hikami [BH2], essentially ..."
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Cited by 11 (4 self)
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We consider the asymptotic behaviour of the secondorder correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble, obtained by Brézin and Hikami [BH2], essentially continues to hold for a general real symmetric Wigner matrix. To obtain this result, we adapt the approach by Götze and Kösters [GK], who proved the analogous result for the Hermitian case.