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Krasovsky, I. V.

doi:10.1155/s1073792804140221

Cited by 71 publications

(7 citation statements)

References 28 publications

(39 reference statements)

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“…in the interior of the support of ρ(x), we show that the asymptotic series progresses in powers of 1/N , and we obtain the explicit form of the 1/N correction. We find for the LUE that the coefficient of the 1/N term consists of a component which is oscillatory in N as well a component which is non-oscillatory in N , whereas for the GUE it consists of only an oscillatory component, as shown in (8). For the soft edge we will see that the asymptotic series progresses in powers of N −1/3 , and we obtain explicit expressions for the coefficients of the N −1/3 and N −2/3 terms, which again involve Airy functions.…”

Section: Introductionmentioning

confidence: 60%

“…For recent advances in asymptotic questions related to these interpretations, complementary to the present study, see Refs. 6,7,8 As background to the present study we note that aspects of the large N form of ρ N (x) first arose in studies of field theories related to Hermitian matrix models 9 . There, for the GUE the large N asymptotic expansion of the moments m N (p) := Ω x p ρ N (x) dx (p = 1, 2, .…”

Section: Introductionmentioning

confidence: 96%

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Asymptotic corrections to the eigenvalue density of the GUE and LUE

Garoni

Forrester

Frankel

2005

Journal of Mathematical Physics

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We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N by N matrices, both in the bulk of the spectrum and near the spectral edge. This is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density, to which we apply the saddle point method. The main correction to the bulk density is oscillatory in N and depends on the distribution function of the limiting density, while the corrections to the Airy kernel at the soft edge are again expressed in terms of the Airy function and its first derivative. We demonstrate numerically that these expansions are very accurate. A matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk.Comment: 14 pages, 4 figure

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Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 96%

Asymptotic corrections to the eigenvalue density of the GUE and LUE

Garoni

Forrester

Frankel

2005

Journal of Mathematical Physics

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“…For T e = 1, (but ξ still infinite) the behavior is known exactly [45,56,57] from an analogy with random matrix theory. The decay at large times is mainly Gaussian with power law and higher order corrections but is well approximated by the Wigner distribution…”

Section: Infinite Overlapmentioning

confidence: 99%

Waiting time distribution for trains of quantized electron pulses

Albert

Devillard

2014

Phys. Rev. B

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We consider a sequence of quantized Lorentzian pulses of non-interacting electrons impinging on a quantum point contact (QPC) and study the waiting time distribution (WTD), for any transmission and any number of pulses. As the degree of overlap between the electronic wave functions is tuned, the WTD reveals how the correlations between particles are modified. In the weak overlap regime, the WTD is made of several equidistant peaks, separated by the same period as the incoming pulses, contained in an almost exponentially decaying envelope. In the other limit, the WTD of a single quantum channel subjected to a constant voltage is recovered. In both cases, the WTD stresses the difference between the fluctuations induced by the scatterer and the ones encoded in the incoming quantum state. A clear cross-over between these two situations is studied with numerical and analytical calculations based on scattering theory.Comment: 12 pages, 4 figure

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“…Our motivation here is to extend similar considerations done for Hermitian RMT. These include the so-called Widom-Dyson constant to the third order, and the first computation in [11] was made rigorous only very recently [12].…”

Section: Introductionmentioning

confidence: 99%

Gap probabilities in non-Hermitian random matrix theory

Akemann

Phillips

Shifrin

2009

Journal of Mathematical Physics

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We compute the gap probability that a circle of\ud radius r around the origin contains exactly k complex eigenvalues. Four different ensembles of random matrices are considered: the Ginibre ensembles and their chiral complex counterparts, with both complex (beta=2) or quaternion real (beta=4) matrix elements. For general non-Gaussian weights we give a Fredholm determinant or Pfaffian representation respectively, depending on the non-Hermiticity parameter. At maximal non-Hermiticity, that is for rotationally invariant weights, the product of Fredholm eigenvalues for beta=4 follows from beta=2 by skipping every second factor, in contrast to the known relation for Hermitian ensembles. On additionally choosing Gaussian weights we give new explicit expressions for the Fredholm eigenvalues in the chiral case, in terms of Bessel-K and incomplete Bessel-I functions. This compares to known results for the Ginibre ensembles in terms of incomplete exponentials. Furthermore we present an asymptotic expansion of the logarithm of the gap probability for large argument r at large N in all four ensembles, up to including the third order linear term. We can provide strict upper and lower bounds and present numerical evidence for its conjectured values, depending on the number of exact zero eigenvalues in the chiral ensembles. For the Ginibre ensemble at beta=2 exact results were previously derived by Forrester

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Cited by 71 publications

References 28 publications

Asymptotic corrections to the eigenvalue density of the GUE and LUE

Asymptotic corrections to the eigenvalue density of the GUE and LUE

Waiting time distribution for trains of quantized electron pulses

Gap probabilities in non-Hermitian random matrix theory

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