The Asymptotic Expansion of the Meijer G-Function

Fields, Jerry L.

doi:10.2307/2005104

Cited by 23 publications

(22 citation statements)

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“…Using (4.7), it is immediately seen that the local scaling regime near the origin gives (1.13) announced in Theorem 2. Compared to the same result for the complex density (4.6), the real density has the advantage that the 15) as obtained in [23,48,10]. We remark that it has been argued by Beenakker and co-workers [8] that the statistical state implied by (4.15) is realised by the level crossings of so-called Majorana zero modes for a disordered semiconducting wire at a Josephson junction, in a weak magnetic field.…”

Section: Real Eigenvaluessupporting

confidence: 68%

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Forrester

Ipsen

2016

Linear Algebra and its Applications

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Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly k real eigenvalues as a determinant with entries involving particular Meijer G-functions. We also compute the explicit form of the Pfaffian correlation kernel for the correlation between real eigenvalues, and the correlation between complex eigenvalues. The simplest example of these -the eigenvalue density of the real eigenvalues -gives by integration the expected number of real eigenvalues. Our ability to perform these calculations relies on the construction of certain skew-orthogonal polynomials in the complex plane, the computation of which is carried out using their relationship to particular random matrix averages.

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Section: Real Eigenvaluessupporting

confidence: 68%

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Forrester

Ipsen

2016

Linear Algebra and its Applications

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“…Our computation is very similar to that. Actually, this special case has been studied by Barnes [7], and the readers can also find it in the Fields' work [14]. Lemma 3.3.…”

Section: )mentioning

confidence: 98%

Universality for Products of Random Matrices I: Ginibre and Truncated Unitary Cases

Liu

Wang

2015

Int Math Res Notices

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Recently, the joint probability density functions of complex eigenvalues for products of independent complex Ginibre matrices have been explicitly derived as determinantal point processes. We express truncated series coming from the correlation kernels as multivariate integrals with singularity and investigate saddle point method for such a type of integrals. As an application, we prove that the eigenvalue correlation functions have the same scaling limits as those of the single complex Ginibre ensemble, both in the bulk and at the edge of the spectrum. We also prove that the similar results hold true for products of independent truncated unitary matrices.

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“…with first parameter in the bottom row zero are examples of the generalized hypergeometric function p F q [16,Eq. (2.26)].…”

Section: (X)mentioning

confidence: 99%

Eigenvalue statistics for product complex Wishart matrices

Forrester¹

2014

J. Phys. A: Math. Theor.

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The eigenvalue statistics for complex N × N Wishart matrices X † r,s X r,s , where X r,s is equal to the product of r complex Gaussian matrices, and the inverse of s complex Gaussian matrices, are considered. In the case r = s the exact form of the global density is computed. The averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function s+1 F r . For finite N the eigenvalue probability density function is computed, and is shown to be an example of a biorthogonal ensemble. A double contour integral form of the corresponding correlation kernel is derived, which allows the hard edge scaled limit to be computed. The limiting kernel is given in terms of certain Meijer G-functions, and is identical to that found in the recent work of Kuijlaars and Zhang in the case s = 0. Properties of the kernel and corresponding correlation functions are discussed.

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The Asymptotic Expansion of the Meijer G-Function

Cited by 23 publications

References 0 publications

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Universality for Products of Random Matrices I: Ginibre and Truncated Unitary Cases

Eigenvalue statistics for product complex Wishart matrices

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