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24
Linear functionals of eigenvalues of random matrices
 Trans. Amer. Math. Soc
, 2001
"... Abstract. Let Mn be a random n × n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n →∞. By Fourier analysis, this result ..."
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Cited by 97 (7 self)
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Abstract. Let Mn be a random n × n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n →∞. By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of Mn. For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of Mn are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices. 1.
The efficient evaluation of the hypergeometric function of a matrix argument
 MATH. COMP
, 2005
"... We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix. ..."
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Cited by 79 (17 self)
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We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix.
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 62 (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
Products and ratios of characteristic polynomials of random Hermitian matrices
 J. Math. Phys
, 2003
"... We present new and streamlined proofs of various formulae for products and ratios of characteristic polynomials of random Hermitian matrices that have appeared recently in the literature. 1 1 ..."
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Cited by 35 (5 self)
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We present new and streamlined proofs of various formulae for products and ratios of characteristic polynomials of random Hermitian matrices that have appeared recently in the literature. 1 1
Large N Expansion for Normal and Complex Matrix Ensembles
 in ”Frontiers in Number Theory, Physics, and Geometry I
, 2006
"... We present the first two leading terms of the 1/N (genus) expansion of the free energy for ensembles of normal and complex random matrices. The results are expressed through the support of eigenvalues (assumed to be a connected domain in the complex plane). In particular, the subleading (genus1) te ..."
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Cited by 20 (2 self)
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We present the first two leading terms of the 1/N (genus) expansion of the free energy for ensembles of normal and complex random matrices. The results are expressed through the support of eigenvalues (assumed to be a connected domain in the complex plane). In particular, the subleading (genus1) term is given by the regularized determinant of the Laplace operator in the complementary domain with the Dirichlet boundary conditions. An explicit expression of the genus expansion through harmonic moments of the domain gives some new representations of the mathematical objects related to the Dirichlet boundary problem, conformal analysis and spectral geometry.
Convergence analysis of Krylov subspace iterations with methods from potential theory
 SIAM Review
"... Abstract. Krylov subspace iterations are among the bestknown and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on t ..."
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Cited by 16 (2 self)
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Abstract. Krylov subspace iterations are among the bestknown and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix. This leads to an extremal problem in polynomial approximation theory: how small can a monic polynomial of a given degree be on the spectrum? This survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices. It is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure. Then depending on the number of iterations, the Lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region, which is characterized by means of a constrained equilibrium problem from potential theory. The same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems. Key words. Krylov subspace iterations, Ritz values, eigenvalue distribution, equilibrium measure, contrained equilibrium, potential theory AMS subject classifications. 15A18, 31A05, 31A15, 65F15 1. Introduction. Krylov
Unitary Correlations and the Fejér Kernel
 Mathematical Physics, Analysis, and Geometry
, 2002
"... Let M be a unitary matrix with eigenvalues t j , and let f be a function on the unit circle. Define X f (M) = P f(t j ). We derive exact and asymptotic formulae for the covariance of X f and X g with respect to the measures j(M)j 2 dM where dM is Haar measure and an irreducible character. The ..."
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Cited by 4 (1 self)
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Let M be a unitary matrix with eigenvalues t j , and let f be a function on the unit circle. Define X f (M) = P f(t j ). We derive exact and asymptotic formulae for the covariance of X f and X g with respect to the measures j(M)j 2 dM where dM is Haar measure and an irreducible character. The asymptotic results include an analysis of the Fej'er kernel which may be of independent interest.