Distribution of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>k</mml:mi><mml:mi mathvariant="normal">th</mml:mi></mml:math>smallest Dirac operator eigenvalue

Damgaard, Poul H.; Nishigaki, Shinsuke M.

doi:10.1103/physrevd.63.045012

Cited by 127 publications

(195 citation statements)

References 37 publications

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“…It is known that for standard WL the maxima of the Bessel density correspond to the location of individual eigenvalues [37], as we will see in the next subsection. On the other hand the microscopic density of the WL ensembles in the bulk is completely flat, equalling 1 π in our normalisation.…”

Section: Generalised Universal Bessel-lawmentioning

confidence: 99%

“…In WL the gap probability E(s) and the first eigenvalue distribution p(s) are explicitly known and universal in the microscopic large-N limit for all ν at β = 2, for odd values of ν and 0 at β = 1, and for ν = 0 at β = 4. This has been shown by various authors independently [16,38,39,37]. In some cases only finite-N results are know in terms of a hypergeometric function of a matrix valued argument [40,41], from which limits are difficult to extract.…”

Section: Generalised Universal First Eigenvalue Distribution At the Hmentioning

confidence: 99%

“…In some cases only finite-N results are know in terms of a hypergeometric function of a matrix valued argument [40,41], from which limits are difficult to extract. Although p(s) follows from E(s), the most compact universal formulas are known directly for p(s) for all three β [37]. There, also the second and higher eigenvalue distributions are given, which we will not consider here.…”

Section: Generalised Universal First Eigenvalue Distribution At the Hmentioning

confidence: 99%

“…Here the first eigenvalue distribution of the WL ensembles is only known explicitly for odd values of ν in the large-N limit [37], given by a Pfaffian with indices running over half integers:…”

Section: Generalised Universal First Eigenvalue Distribution At the Hmentioning

confidence: 99%

“…Next we give the first eigenvalue distribution for general ν. Here we directly use the most compact universal expression [37] for ℘ (2) ν (y) in WL, without making the detour over E (2) (y) [39],…”

Section: Generalised Universal First Eigenvalue Distribution At the Hmentioning

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: 99%

Section: Generalised Universal First Eigenvalue Distribution At the Hmentioning

confidence: 99%

Section: Generalised Universal First Eigenvalue Distribution At the Hmentioning

confidence: 99%

Section: Generalised Universal First Eigenvalue Distribution At the Hmentioning

confidence: 99%

Section: Generalised Universal First Eigenvalue Distribution At the Hmentioning

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 large N limit of four dimensional Yang-Mills field coupled to adjoint fermions on a single site lattice

Hietanen

Narayanan

2010

J. High Energ. Phys.

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We consider the large N limit of four dimensional SU (N ) Yang-Mills field coupled to adjoint fermions on a single site lattice. We use perturbative techniques to show that the Z 4 N center-symmetries are broken with naïve fermions but they are not broken with overlap fermions. We use numerical techniques to support this result. Furthermore, we present evidence for a non-zero chiral condensate for one and two Majorana flavors at one value of the lattice gauge coupling.

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Individual eigenvalue distributions for the Wilson Dirac operator

Akemann¹,

Ipsen²

2012

J. High Energ. Phys.

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We derive the distributions of individual eigenvalues for the Hermitian Wilson Dirac Operator D 5 as well as for real eigenvalues of the Wilson Dirac Operator D W . The framework we provide is valid in the epsilon regime of chiral perturbation theory for any number of flavours N f and for non-zero low energy constants W 6,7,8 . It is given as a perturbative expansion in terms of the k-point spectral density correlation functions and integrals thereof, which in some cases reduces to a Fredholm Pfaffian. For the real eigenvalues of D W at fixed chirality ν this expansion truncates after at most ν terms for small lattice spacing a. Explicit examples for the distribution of the first and second eigenvalue are given in the microscopic domain as a truncated expansion of the Fredholm Pfaffian for quenched D 5 , where all k-point densities are explicitly known from random matrix theory. For the real eigenvalues of quenched D W at small a we illustrate our method by the finite expansion of the corresponding Fredholm determinant of size ν.

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Distribution of thekthsmallest Dirac operator eigenvalue

Cited by 127 publications

References 37 publications

Power law deformation of Wishart–Laguerre ensembles of random matrices

Power law deformation of Wishart–Laguerre ensembles of random matrices

The large N limit of four dimensional Yang-Mills field coupled to adjoint fermions on a single site lattice

Individual eigenvalue distributions for the Wilson Dirac operator

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