Transmission eigenvalue densities and moments in chaotic cavities from random matrix theory

Vivo, Pierpaolo; Vivo, Edoardo

doi:10.1088/1751-8121/41/23/239801

Cited by 15 publications

(24 citation statements)

References 1 publication

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“…Since experiments can now be performed in quantum dots with a number of channels arbitrarily small, 46 recently there has been an increasing interest in computing finite n formulae. 32,44,47,55 Some of these integrals have never been computed before, others are already available in the literature. 27,30,35,44,55 In particular, most of the formulae for β = 1 and β = 4 are original.…”

Section: A Backgroundmentioning

confidence: 99%

“…34 Explicit formulae were then obtained by Novaes 44 and by Vivo and Vivo. 55 In a similar setting Bai et al 5 computed the leading order term of the asymptotic expansion for β = 1.…”

Section: A the Moments Of Transmission Eigenvalues And Of The Propermentioning

confidence: 99%

“…A 4-th order recurrence relation for the exact moments at β = 2 were first reported Ledoux [34]. Explicit formulae were then obtained by Novaes [44] and by Vivo and Vivo [54]. Bai et al [5] computed the leading order term of the asymptotic expansion for β = 1.…”

Section: The Moments Of Transmission Eigenvalues and Of The Proper De...mentioning

confidence: 99%

“…Remark 5.2. The complexity of the expression (90) is mainly due to formula (54), which relates Jacobi polynomials of different weights. The Laguerre ensemble is slightly simpler, because the associated coefficients (61) and normalizations h j are more concise.…”

Section: Symplectic Ensemblesmentioning

confidence: 99%

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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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Section: A Backgroundmentioning

confidence: 99%

“…34 Explicit formulae were then obtained by Novaes 44 and by Vivo and Vivo. 55 In a similar setting Bai et al 5 computed the leading order term of the asymptotic expansion for β = 1.…”

Section: A the Moments Of Transmission Eigenvalues And Of The Propermentioning

confidence: 99%

Section: The Moments Of Transmission Eigenvalues and Of The Proper De...mentioning

confidence: 99%

Section: Symplectic Ensemblesmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Mean and variance of G and other quantities have been known for a long time when N, M are large [27,40,41], and recently also for finite N, M [41,42,43,44].…”

Section: Landauer Conductance In Open Cavitiesmentioning

confidence: 99%

Largest Schmidt eigenvalue of random pure states and conductance distribution in chaotic cavities

Vivo¹

2011

J. Stat. Mech.

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A strategy to evaluate the distribution of the largest Schmidt eigenvalue for entangled random pure states of bipartite systems is proposed. We point out that the multiple integral defining the sought quantity for a bipartition of sizes N, M is formally identical (upon simple algebraic manipulations) to the one providing the probability density of Landauer conductance in open chaotic cavities supporting N and M electronic channels in the two leads. Known results about the latter can then be straightforwardly employed in the former problem for both systems with broken (β = 2) and preserved (β = 1) time reversal symmetry. The analytical results, yielding a continuous but not everywhere analytic distribution, are in excellent agreement with numerical simulations.

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Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes

2016

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We study three instances of log-correlated processes on the interval: the logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the Gaussian log-correlated potential in presence of edge charges, and the Fractional Brownian motion with Hurst index H → 0 (fBM0). In previous collaborations we obtained the probability distribution function (PDF) of the value of the global minimum (equivalently maximum) for the first two processes, using the freezing-duality conjecture (FDC). Here we study the PDF of the position of the maximum x m through its moments. Using replica, this requires calculating moments of the density of eigenvalues in the β-Jacobi ensemble. Using Jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary β > 0 and positive integer n in terms of sums over partitions. For positive moments, this expression agrees with a very recent independent derivation by Mezzadri and Reynolds. We check our results against a contour integral formula derived recently by Borodin and Gorin (presented in the Appendix 1 from these authors). The duality necessary for the FDC to work is proved, and on our expressions, found to correspond to exchange of partitions with their dual. Performing the limit n → 0 and to negative Dyson index β → −2, we obtain the moments of x m and give explicit expressions for the lowest ones. Numerical checks for the GUE polynomials, performed independently by N. Simm, indicate encouraging agreement. Some results are also obtained for moments in Laguerre, Hermite-Gaussian, as well as circular and related ensembles. The correlations of the position and the value of the field at the minimum are also analyzed.With Appendix 1 written

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Transmission eigenvalue densities and moments in chaotic cavities from random matrix theory

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Cited by 15 publications

References 1 publication

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

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

Largest Schmidt eigenvalue of random pure states and conductance distribution in chaotic cavities

Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes

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