A Random Matrix Decimation Procedure Relating β = 2/(r + 1) to β =  2(r + 1)

Forrester, Peter J.

doi:10.1007/s00220-008-0616-0

Cited by 23 publications

(18 citation statements)

References 17 publications

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“…Thus, this investigation based on the Selberg integral reveals parameters for which the validity of (4.9) may be expected. One can in fact proceed further and prove, using a generalization of the Dixon-Anderson integral, that for these parameters (4.9) is indeed valid [60].…”

Section: The Importance Of the Selberg Integral 519mentioning

confidence: 96%

The importance of the Selberg integral

Forrester

Warnaar

2008

Bull. Amer. Math. Soc.

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292

272

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Abstract. It has been remarked that a fair measure of the impact of Atle Selberg's work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an n-dimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after its initial publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the Macdonald conjectures. It further initiated the study of q-analogues, which in turn enriched the Macdonald conjectures. We review these developments and proceed to exhibit the sustained prominence of the Selberg integral as evidenced by its central role in random matrix theory, Calogero-Sutherland quantum many-body systems, Knizhnik-Zamolodchikov equations, and multivariable orthogonal polynomial theory.

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Section: The Importance Of the Selberg Integral 519mentioning

confidence: 96%

The importance of the Selberg integral

Forrester

Warnaar

2008

Bull. Amer. Math. Soc.

Self Cite

292

272

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“…Note that r (1) = r. For the Gaussian β ensemble, it can be shown rigorously that the distribution of r (n) for β = 2/(n + 1) is equivalent to the distribution of r (1) for β = 2(n + 1) [41]. Evidence for a broader class of interrelations involving β = 1, 2, 4 has been provided recently in Ref.…”

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confidence: 90%

Random Matrix Ensemble for the Level Statistics of Many-Body Localization

2019

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We numerically study the level statistics of the Gaussian β ensemble. These statistics generalize Wigner-Dyson level statistics from the discrete set of Dyson indices β = 1, 2, 4 to the continuous range 0 < β < ∞. The Gaussian β ensemble covers Poissonian level statistics for β → 0, and provides a smooth interpolation between Poissonian and Wigner-Dyson level statistics. We establish the physical relevance of the level statistics of the Gaussian β ensemble by showing near-perfect agreement with the level statistics of a paradigmatic model in studies on many-body localization over the entire crossover range from the thermal to the many-body localized phase. In addition, we show similar agreement for a related Hamiltonian with broken time-reversal symmetry.

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“…(22) and (24) are well defined for any half-integer n. It is useful to generalize it, because the evaluation of the moments for β = 1 requires moments for β = 4 computed at half-integer n.…”

Section: Remark 26mentioning

confidence: 99%

“…Similar dualities have appeared in the literature before. 19,24,34 Lemma 6.1: Let n be an even integer. We have the following dualities:…”

Section: A a Duality Between β = 1 And β =mentioning

confidence: 99%

Moments of the transmission eigenvalues, proper delay times, and random matrix theory. I

Mezzadri

Simm

2011

Journal of Mathematical Physics

111

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We develop a method to compute the moments of the eigenvalue densities of matrices in the Gaussian, Laguerre and Jacobi ensembles for all the symmetry classes β ∈ {1, 2, 4} and finite matrix dimension n. The moments of the Jacobi ensembles have a physical interpretation as the moments of the transmission eigenvalues of an electron through a quantum dot with chaotic dynamics. For the Laguerre ensemble we also evaluate the finite n negative moments. Physically, they correspond to the moments of the proper delay times, which are the eigenvalues of the Wigner-Smith matrix. Our formulae are well suited to an asymptotic analysis as n → ∞. Appendix B. The Generating Function f j (s) References

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A Random Matrix Decimation Procedure Relating β = 2/(r + 1) to β = 2(r + 1)

Cited by 23 publications

References 17 publications

The importance of the Selberg integral

The importance of the Selberg integral

Random Matrix Ensemble for the Level Statistics of Many-Body Localization

Moments of the transmission eigenvalues, proper delay times, and random matrix theory. I

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