Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

Akemann, Gernot; Kieburg, Mario; Phillips, M. J.

doi:10.1088/1751-8113/43/37/375207

Cited by 33 publications

(69 citation statements)

References 39 publications

(151 reference statements)

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“…(4.3) gives a double sum, which in general cannot be reduced to a single sum. Note that the definition of such skeworthogonal polynomials is not unique for β = 4 and β = 1 [24,36]. In our choice they have parity according to their degree,…”

Section: The Interpolating Airy Kernel For β =mentioning

confidence: 99%

The Interpolating Airy Kernels for the $$\beta =1$$ β = 1 and $$\beta =4$$ β = 4 Elliptic Ginibre Ensembles

Akemann¹,

Phillips²

2014

J Stat Phys

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We consider two families of non-Hermitian Gaussian random matrices, namely the elliptical Ginibre ensembles of asymmetric N -by-N matrices with Dyson index β = 1 (real elements) and with β = 4 (quaternion-real elements). Both ensembles have already been solved for finite N using the method of skew-orthogonal polynomials, given for these particular ensembles in terms of Hermite polynomials in the complex plane. In this paper we investigate the microscopic weakly non-Hermitian large-N limit of each ensemble in the vicinity of the largest or smallest real eigenvalue. Specifically, we derive the limiting matrix-kernels for each case, from which all the eigenvalue correlation functions can be determined. We call these new kernels the "interpolating" Airy kernels, since we can recover -as opposing limiting cases -not only the well-known Airy kernels for the Hermitian ensembles, but also the complementary error function and Poisson kernels for the maximally non-Hermitian ensembles at the edge of the spectrum. Together with the known interpolating Airy kernel for β = 2, which we rederive here as well, this completes the analysis of all three elliptical Ginibre ensembles in the microscopic scaling limit at the spectral edge.

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Section: The Interpolating Airy Kernel For β =mentioning

confidence: 99%

The Interpolating Airy Kernels for the $$\beta =1$$ β = 1 and $$\beta =4$$ β = 4 Elliptic Ginibre Ensembles

Akemann¹,

Phillips²

2014

J Stat Phys

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“…The additional Pfaffian determinant containing part of the weight function F (x; m) is a feature shared by the partition function for the non-Hermitian Wilson Dirac operator eq. (3.14) [23], as well as by other non-Hermitian RMT, see [38]. The problem of computing individual eigenvalue distributions in terms of densities is now specified.…”

Section: The Jpdf and Quenched K-point Densities Of Dmentioning

confidence: 99%

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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“…All complex (and real for β = 1) eigenvalue correlation functions as well as the partition functions are known in the presence of these terms, see [5,12,6,13,7,14,15,28]. For simplicity we will mainly restrict ourselves to the so-called quenched case N f = 0 in the following, although N f = 0 is particularly interesting for the QCD application as it may lead to a nonpositive overall weight, the so-called sign problem.…”

Section: Wishart Picturementioning

confidence: 99%

“…Second, more source terms were added to these parameter dependent two-matrix models, by inserting an arbitrary but fixed number of characteristic polynomials (as additional determinants of the Dirac operator) into the measure. All complex eigenvalue correlation functions were computed in this more general setting in [5,12] for β = 2, in [6,13] for β = 4 and in [14,15] for β = 1. We also mention that the β = 1 symmetry class at maximal non-Hermiticity appears in the superconducting phase of QCD with two colours [16].…”

Section: Introductionmentioning

confidence: 99%

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Akemann¹

2011

Acta Phys. Pol. B

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We briefly review the solution of three ensembles of non-Hermitian random matrices generalizing the Wishart-Laguerre (also called chiral) ensembles. These generalizations are realized as Gaussian two-matrix models, where the complex eigenvalues of the product of the two independent rectangular matrices are sought, with the matrix elements of both matrices being either real, complex or quaternion real. We also present the more general case depending on a non-Hermiticity parameter, that allows us to interpolate between the corresponding three Hermitian Wishart ensembles with real eigenvalues and the maximally non-Hermitian case. All three symmetry classes are explicitly solved for finite matrix size N × M for all complex eigenvalue correlations functions (and real or mixed correlations for real matrix elements). These are given in terms of the corresponding kernels built from orthogonal or skew-orthogonal Laguerre polynomials in the complex plane. We then present the corresponding three Bessel kernels in the complex plane in the microscopic large-N scaling limit at the origin, both at weak and strong non-Hermiticity with M − N ≥ 0 fixed.

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Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

Cited by 33 publications

References 39 publications

The Interpolating Airy Kernels for the $$\beta =1$$ β = 1 and $$\beta =4$$ β = 4 Elliptic Ginibre Ensembles

The Interpolating Airy Kernels for the $$\beta =1$$ β = 1 and $$\beta =4$$ β = 4 Elliptic Ginibre Ensembles

Individual eigenvalue distributions for the Wilson Dirac operator

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