Asymptotic correlations at the spectrum edge of random matrices

Nagao, Taro; Forrester, Peter J.

doi:10.1016/0550-3213(94)00545-p

Cited by 99 publications

(115 citation statements)

References 19 publications

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“…The corresponding results for β = 1, 4 are given in terms of a Pfaffian of a matrix kernel [18], and for a discussion of a relation between the three universal kernels we refer to [20]. A feature we observe for all three β is that for α ≤ O(1) the oscillations of the Bessel density are completely smoothed out, apart from the first peak.…”

Section: Generalised Universal Bessel-lawmentioning

confidence: 79%

“…The corresponding microscopic densities of the WL ensembles are again universal [20]. Computed initially in [17,18] the obtained expressions can be simplified. They can be expressed through the β = 2 density eq.…”

Section: Generalised Universal Bessel-lawmentioning

confidence: 99%

“…(2.17) holds for β = 1 and 4 when replacing the determinant by a Pfaffian, Pf[κ N (λ i , λ j )], where κ N is a 2 × 2 matrix kernel. For a Gaussian weight its skew orthogonal polynomials are explicitly known as well in terms of Laguerre polynomials [17,18].…”

Section: Definition Of the Model And Finite-n Solution For General Pomentioning

confidence: 99%

“…(2.17) in terms of the norms h i of orthogonal polynomials 18) or their ratios r i ≡ h i+1 h i . An identical result holds for β = 1 and 4 in terms of the skew orthogonal norms [13].…”

Section: Definition Of the Model And Finite-n Solution For General Pomentioning

confidence: 99%

“…On the microscopic domain much effort has been spent on the so-called hard edge at the origin for the positive definite eigenvalues, which is absent in the WD class. 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].…”

Section: Introductionmentioning

confidence: 99%

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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 Bessel-lawmentioning

confidence: 79%

Section: Generalised Universal Bessel-lawmentioning

confidence: 99%

Section: Definition Of the Model And Finite-n Solution For General Pomentioning

confidence: 99%

“…(2.17) in terms of the norms h i of orthogonal polynomials 18) or their ratios r i ≡ h i+1 h i . An identical result holds for β = 1 and 4 in terms of the skew orthogonal norms [13].…”

Section: Definition Of the Model And Finite-n Solution For General Pomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Power law deformation of Wishart–Laguerre ensembles of random matrices

Akemann¹,

Vivo²

2008

J. Stat. Mech.

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The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

Blackstone

Charlier

Lenells

2023

Comm Pure Appl Math

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The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider the probability where and is any non‐negative integer. We obtain asymptotics for this probability as the size of the intervals becomes large, up to and including the oscillations of order 1. In these asymptotics, the most intricate term is a one‐dimensional integral along a linear flow on a g‐dimensional torus, whose integrand involves ratios of Riemann θ‐functions associated to a genus g Riemann surface. We simplify this integral in two generic cases: (a) If the flow is ergodic, we compute the leading term in the asymptotics of this integral explicitly using Birkhoff's ergodic theorem. (b) If the linear flow has certain “good Diophantine properties”, we obtain improved estimates on the error term in the asymptotics of this integral. In the case when the flow is both ergodic and has “good Diophantine properties” (which is always the case for , and “almost always” the case for ), these results can be combined, yielding particularly precise and explicit large gap asymptotics.

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Universal Behavior in Dirac Spectra

Verbaarschot

NATO Science Series: B:

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Asymptotic correlations at the spectrum edge of random matrices

Cited by 99 publications

References 19 publications

Power law deformation of Wishart–Laguerre ensembles of random matrices

Power law deformation of Wishart–Laguerre ensembles of random matrices

The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

Universal Behavior in Dirac Spectra

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