New critical matrix models and generalized universality

Akemann, Gernot; Vernizzi, Graziano

doi:10.1016/s0550-3213(02)00173-6

Cited by 12 publications

(9 citation statements)

References 54 publications

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“…If the spectral density vanishes, it still possible to derive sum rules for the inverse Dirac eigenvalues [84,85,86,87]. It could be that such sum rules are easier obtained for lattice simulation of random Dirac fermions than for lattice simulations of the Schwinger model with two massless flavors [18].…”

Section: A Spectral Dualitymentioning

confidence: 99%

Sum rules for the Dirac spectrum of the Schwinger model

Shifrin

Verbaarschot

2006

Phys. Rev. D

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The inverse eigenvalues of the Dirac operator in the Schwinger model satisfy the same Leutwyler-Smilga sum rules as in the case of QCD with one flavor. In this paper we give a microscopic derivation of these sum rules in the sector of arbitrary topological charge. We show that the sum rules can be obtained from the clustering property of the scalar correlation functions. This argument also holds for other theories with a mass gap and broken chiral symmetry such as QCD with one flavor.For QCD with several flavors a modified clustering property is derived from the low energy chiral Lagrangian. We also derive sum rules for a fixed external gauge field.Both types of sum rules are obtained from the bosonized version of the Schwinger model.In the sector of topological charge ν the sum rules are consistent with a shift of the Dirac spectrum away from zero by ν/2 average level spacings. This shift is also required to obtain a nonzero chiral condensate in the massless limit. Finally, we discuss the Dirac spectrum for a closely related two-dimensional theory with a gauge field action that is quadratic in the gauge field and has applications to the quantum Hall effect and d-wave super-conductors.

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Section: A Spectral Dualitymentioning

confidence: 99%

Sum rules for the Dirac spectrum of the Schwinger model

Shifrin

Verbaarschot

2006

Phys. Rev. D

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“…The condition (1.7) then says that the mean eigenvalue density ψ should be strictly positive there. If the origin belongs to the interior of the support of µ V but the mean eigenvalue density vanishes there, then the potential is called multicritical, see [3,4,21]. This case will not be treated in this paper.…”

Section: Introductionmentioning

confidence: 99%

Universality for Eigenvalue Correlations at the Origin of the Spectrum

Kuijlaars

Vanlessen

2003

Communications in Mathematical Physics

131

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We establish universality of local eigenvalue correlations in unitary random matrix ensembles 1 Zn | det M | 2α e −ntr V (M) dM near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x| 2α e −nV (x) have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local eigenvalue correlations have universal behavior described in terms of Bessel functions. We extend this to a much wider class of confining potentials V . Our approach is based on the steepest descent method of Deift and Zhou for the asymptotic analysis of Riemann-Hilbert problems. This method was used by Deift et al. to establish universality in the bulk of the spectrum. A main part of the present work is devoted to the analysis of a local Riemann-Hilbert problem near the origin.

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“…Then restricting oneself to the class of random matrix measures invariant with respect to the change of basis (here the unitary invariant ensembles) the quantity of interest can be expressed as a determinant containing kernels made of polynomials orthogonal with respect to that measure [12]. In this way studying universality amounts mainly to investigating asymptotics of the orthogonal polynomials in various regimes, and is well established by now both on physical [13,14,15,16,17,18,19] as well as mathematical [20,21,22,23] level of rigor.…”

Section: Introductionmentioning

confidence: 99%

Universal random matrix correlations of ratios of characteristic polynomials at the spectral edges

Akemann

Fyodorov

2003

Nuclear Physics B

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It has been shown recently by Fyodorov and Strahov [math-ph/0204051] that Cauchy transforms of orthogonal polynomials appear naturally in general correlation functions containing ratios of characteristic polynomials of random N×N Hermitian matrices. Our main goal is to investigate the issue of universality of large N asymptotics for those Cauchy transforms for a wide class of weight functions. Our analysis covers three different scaling regimes: the “hard edge”, the “bulk” and the “soft edge” of the spectrum, thus extending the earlier results known for the bulk. The principal tool is to show that for finite matrix size N the auxiliary “wave functions” associated with the Cauchy transforms obey the same second order differential equation as those associated with the orthogonal polynomials themselves

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New critical matrix models and generalized universality

Cited by 12 publications

References 54 publications

Sum rules for the Dirac spectrum of the Schwinger model

Sum rules for the Dirac spectrum of the Schwinger model

Universality for Eigenvalue Correlations at the Origin of the Spectrum

Universal random matrix correlations of ratios of characteristic polynomials at the spectral edges

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