Universality for Eigenvalue Correlations at the Origin of the Spectrum

Kuijlaars, Arno B. J.; Vanlessen, M.

doi:10.1007/s00220-003-0960-z

Cited by 75 publications

(143 citation statements)

References 43 publications

(113 reference statements)

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“…It follows from (2.24)-(2.26) that W α takes the form 27) for matrices P α , Q α which are independent of z. Moreover, it is easily verified that X α exhibits the following symmetry, 28) which yields the relation…”

Section: Recursive Construction Of ψmentioning

confidence: 95%

“…The convergence is uniform for u, v in compact subsets of R. If t = 0, it was proved in [27] that for u, v in bounded subsets of (0, ∞) that 22) where J ν is the Bessel function of order ν (see [29] for a reference on Bessel functions and other special functions which appear throughout the paper). The convergence is uniform for u, v in compact subsets of (0, ∞).…”

Section: Asymptotics For the Correlation Kernel Nearmentioning

confidence: 99%

“…The limits are uniform for u, v in compact subsets of R. 27) and the functions Φ j , j = 1, 2, will be defined in Section 2 for τ < 0 and in Section 3 for τ > 0.…”

mentioning

confidence: 99%

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Random Matrices with Merging Singularities and the Painlevé V Equation

Claeys

Fahs²

2016

SIGMA

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Abstract. We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the formwhere M is an n × n Hermitian matrix, α > −1/2 and t ∈ R, in double scaling limits where n → ∞ and simultaneously t → 0. If t is proportional to 1/n 2 , a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of α, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel.

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Section: Recursive Construction Of ψmentioning

confidence: 95%

Section: Asymptotics For the Correlation Kernel Nearmentioning

confidence: 99%

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Random Matrices with Merging Singularities and the Painlevé V Equation

Claeys

Fahs²

2016

SIGMA

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“…Then, ν 0 is of the form [9], 24) with χ [a,b] the indicator function of the set [a, b], and with h 0 real analytic and satisfying,…”

Section: Assumptions 11mentioning

confidence: 99%

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

Claeys

Vanlessen

2007

Commun. Math. Phys.

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115

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We consider unitary random matrix ensembles Z −1 n,s,t e −n tr Vs,t(M) dM on the space of Hermitian n × n matrices M , where the confining potential V s,t is such that the limiting mean density of eigenvalues (as n → ∞ and s, t → 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P 2 I equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the wellknown connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P 2 I equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights e −nVs,t on R. The special solution of the P 2 I equation pops up in the n −2/7 -term of the asymptotics.

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“…This can be done by showing that the microscopic spectral density is insensitive to deformations of the probability distribution. The interested reader is referred to the extensive literature on this subject [85,86,87,88,89,90,91,92].…”

Section: The Dirac Spectrum In Qcdmentioning

confidence: 99%

The supersymmetric method in random matrix theory and applications to QCD

Verbaarschot¹

2004

AIP Conference Proceedings

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Abstract.The supersymmetric method is a powerful method for the nonperturbative evaluation of quenched averages in disordered systems. Among others, this method has been applied to the statistical theory of S-matrix fluctuations, the theory of universal conductance fluctuations and the microscopic spectral density of the QCD Dirac operator.We start this series of lectures with a general review of Random Matrix Theory and the statistical theory of spectra. An elementary introduction of the supersymmetric method in Random Matrix Theory is given in the second and third lecture. We will show that a Random Matrix Theory can be rewritten as an integral over a supermanifold. This integral will be worked out in detail for the Gaussian Unitary Ensemble that describes level correlations in systems with broken time reversal invariance. We especially emphasize the role of symmetries.As a second example of the application of the supersymemtric method we discuss the calculation of the microscopic spectral density of the QCD Dirac operator. This is the eigenvalue density near zero on the scale of the average level spacing which is known to be given by chiral Random Matrix Theory. Also in this case we use symmetry considerations to rewrite the generating function for the resolvent as an integral over a supermanifold.The main topic of the second last lecture is the recent developments on the relation between the supersymmetric partition function and integrable hierarchies (in our case the Toda lattice hierarchy). We will show that this relation is an efficient way to calculate superintegrals. Several examples that were given in previous lectures will be worked out by means of this new method. Finally, we will discuss the quenched QCD Dirac spectrum at nonzero chemical potential. Because of the nonhermiticity of the Dirac operator the usual supersymmetric method has not been successful in this case. However, we will show that the supersymmetric partition function can be evaluated by means of the replica limit of the Toda lattice equation.

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Universality for Eigenvalue Correlations at the Origin of the Spectrum

Cited by 75 publications

References 43 publications

Random Matrices with Merging Singularities and the Painlevé V Equation

Random Matrices with Merging Singularities and the Painlevé V Equation

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

The supersymmetric method in random matrix theory and applications to QCD

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