The asymptotic expansion of the Meijer 𝐺-function

Fields, Jerry L.

doi:10.1090/s0025-5718-1972-0361202-7

Cited by 11 publications

(17 citation statements)

References 3 publications

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“…In order to make our paper self contained we compute here the leading order asymptotic behaviour of the weights w n (z) for |z| → ∞ using the saddle point method. The result can also be found in Theorem 2 in reference [27]. We use the multidimensional representation of the weights (3.14):…”

Section: B Asymptotic Of the Weight Functionmentioning

confidence: 94%

Universal microscopic correlation functions for products of independent Ginibre matrices

Akemann

Burda

2012

J. Phys. A: Math. Theor.

114

278

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We consider the product of n complex non-Hermitian, independent random matrices, each of size N × N with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n = 1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n > 1 and generalise the known Bessel-law in the complex plane for n = 2 to a new hypergeometric kernel 0 F n−1 .

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Section: B Asymptotic Of the Weight Functionmentioning

confidence: 94%

Universal microscopic correlation functions for products of independent Ginibre matrices

Akemann

Burda

2012

J. Phys. A: Math. Theor.

114

278

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“…where G is Meijer-G function [67], we finally derive the spin susceptibility of the n-doped phosphorene as…”

Section: B Rkky Interactionmentioning

confidence: 99%

“…In order to calculate the long-range behavior of the RKKY interaction, we do need to expand the Meijer-G functions [67] and use its asymptotic expansion. In the both electron and hole doped phosphorene along the armchair direction (x), the asymptotic term of the spin susceptibility reads as…”

Section: B Rkky Interactionmentioning

confidence: 99%

Strongly anisotropic RKKY interaction in monolayer black phosphorus

Zare

Parhizgar

Asgari

2018

Journal of Magnetism and Magnetic Materials

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We theoretically study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in twodimensional black phosphorus, phosphorene. The RKKY interaction enhances significantly for the low levels of hole doping owing to the nearly valence flat band. Remarkably, for the hole-doped phosphorene, the highest RKKY interaction occurs when two impurities are located along the zigzag direction and it tends to a minimum value with changing the direction from the zigzag to the armchair direction. We show that the interaction is highly anisotropic and the magnetic ground-state of two magnetic adatoms can be tuned by changing the rotational configuration of impurities. Owing to the anisotropic band dispersion, the oscillatory behavior with respect to the angle of the rotation is well-described by sin(2kFR), where the Fermi wavelength kF changes in different directions. We also find that the tail of the RKKY oscillations falls off as 1/R 2 at large distances. arXiv:1712.08905v1 [cond-mat.mes-hall]

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“…In particular it is different from the well-known large argument expansion, cf. [49]. The expression (3.10) can be further simplified for large t 1 since the mean value m ab −→ m b , cf.…”

Section: Large T Limitmentioning

confidence: 99%

Universal distribution of Lyapunov exponents for products of Ginibre matrices

Akemann¹,

Burda²,

Kieburg³

2014

J. Phys. A: Math. Theor.

102

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Abstract. Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random N × N matrices also called Ginibre ensemble we rederive the Lyapunov exponents for an infinite product. We show that for a large number t of product matrices the distribution of each Lyapunov exponent is normal and compute its t-dependent variance as well as corrections in a large-t expansion. Originally Lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution. Surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order. We discuss a general mechanism for 2 × 2 matrices why the singular values and the radii of complex eigenvalues collapse onto the same value in the large-t limit. Thereby we rederive Newman's triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits t → ∞ and N → ∞ on the global and the local scale. As a mathematical byproduct we show that a particular asymptotic expansion of a Meijer G-function with large index leads to a Gaussian.

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The asymptotic expansion of the Meijer 𝐺-function

Abstract: Gamma function identities are integrated to expand the Meijer G-function in a basic set of functions, each of which is simply characterized asymptotically.

Cited by 11 publications

References 3 publications

Universal microscopic correlation functions for products of independent Ginibre matrices

Universal microscopic correlation functions for products of independent Ginibre matrices

Strongly anisotropic RKKY interaction in monolayer black phosphorus

Universal distribution of Lyapunov exponents for products of Ginibre matrices

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