Universality of random matrices in the microscopic limit and the Dirac operator spectrum

Akemann, Gernot; Damgaard, Poul H.; Magnea, U.; Nishigaki, Shinsuke M.

doi:10.1016/s0550-3213(96)00713-4

Cited by 186 publications

(298 citation statements)

References 20 publications

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“…The generalised ensemble follows easily, by inserting this result into eq. (4.15) 19) which is also exact for finite and infinite N . This very fact implies that we have full control of the large-N limit.…”

Section: Generalised Universal First Eigenvalue Distribution At the Hmentioning

confidence: 67%

“…It is governed by the Bessel-law [15,4,16,17,18] labelled by β and M − N being finite. The robustness or universality under polynomial deformations of the Gaussian weight function has also been proven [19,20]. This perturbation destroys the independence of the uncorrelated degrees of freedom of X, without changing the microscopic correlation functions.…”

Section: Introductionmentioning

confidence: 95%

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Power law deformation of Wishart–Laguerre ensembles of random matrices

Akemann¹,

Vivo²

2008

J. Stat. Mech.

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We introduce a one-parameter deformation of the Wishart-Laguerre or chiral ensembles of positive definite random matrices with Dyson index β = 1, 2 and 4. Our generalised model has a fat-tailed distribution while preserving the invariance under orthogonal, unitary or symplectic transformations. The spectral properties are derived analytically for finite matrix size N × M for all three β, in terms of the orthogonal polynomials of the standard Wishart-Laguerre ensembles. For large-N in a certain double scaling limit we obtain a generalised Marčenko-Pastur distribution on the macroscopic scale, and a generalised Bessel-law at the hard edge which is shown to be universal. Both macroscopic and microscopic correlations exhibit power-law tails, where the microscopic limit depends on β and the difference M − N . In the limit where our parameter governing the power-law goes to infinity we recover the correlations of the Wishart-Laguerre ensembles. To illustrate these findings the generalised Marčenko-Pastur distribution is shown to be in very good agreement with empirical data from financial covariance matrices.

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Section: Generalised Universal First Eigenvalue Distribution At the Hmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 95%

Power law deformation of Wishart–Laguerre ensembles of random matrices

Akemann¹,

Vivo²

2008

J. Stat. Mech.

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“…What is perhaps more astonishing, the same results can be obtained from universality classes of large-N (chiral) Random Matrix Theories [4,6,7,8]. To be precise, using the relation to effective field theory it has been shown how to derive the microscopic spectral density ρ s (ζ) [5] and spectral two-point function [9] for the symmetry breaking class of QCD.…”

Section: Introductionmentioning

confidence: 71%

Distributions of Dirac operator eigenvalues

Akemann

Damgaard²

2004

Physics Letters B

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The distribution of individual Dirac eigenvalues is derived by relating them to the density and higher eigenvalue correlation functions. The relations are general and hold for any gauge theory coupled to fermions under certain conditions which are stated. As a special case, we give examples of the lowest-lying eigenvalue distributions for QCD-like gauge theories without making use of earlier results based on the relation to Random Matrix Theory.

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“…It is consistent with our direct measurements. We have thus provided yet more independent confirmation of the formation of a condensate in the confined phase, while simultaneously testing the predictions about the microscopic spectral density of the Dirac operator [4,5]. …”

mentioning

confidence: 74%

“…The formation of a chiral condensate ψ ψ would be a signal for such spontaneous symmetry breaking. If correct, it would open up the possibility of comparing the detailed universality predictions [4,5] for the Dirac operator spectrum around eigenvalues near λ = 0 with lattice Monte Carlo data, as has recently been done in the case of four-dimensional lattice QCD [6]. The first step in such a program is to establish the existence of the chiral condensate ψ ψ .…”

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confidence: 99%

A quark-antiquark condensate in three-dimensional QCD

Damgaard¹,

Heller²,

Krasnitz³

et al. 1998

Physics Letters B

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Three-dimensional lattice QCD is studied by Monte Carlo simulations within the quenched approximation. At zero temperature a quark-antiquark condensate is observed in the limit of vanishing quark masses. The condensate vanishes continuously at the finite-temperature deconfinement phase transition of the theory. A natural interpretation of this phenomenon in the full theory with dynamical quarks is in terms of the spontaneous flavor symmetry breakingIn addition, the spectrum of low-lying Dirac operator eigenvalues is computed and found to be consistent with a flat distribution at zero temperature, in agreement with analytical predictions.NBI-HE-98-06 FSU-SCRI-98-28 UALG/TP/98-4 hep-lat/9803012

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Universality of random matrices in the microscopic limit and the Dirac operator spectrum

Cited by 186 publications

References 20 publications

Power law deformation of Wishart–Laguerre ensembles of random matrices

Power law deformation of Wishart–Laguerre ensembles of random matrices

Distributions of Dirac operator eigenvalues

A quark-antiquark condensate in three-dimensional QCD

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