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Tau-Function Theory of Chaotic Quantum Transport with β = 1, 2, 4

Communications in Mathematical Physics volume 324, pages 465–513 (2013)Cite this article

Abstract

We study the cumulants and their generating functions of the probability distributions of the conductance, shot noise and Wigner delay time in ballistic quantum dots. Our approach is based on the integrable theory of certain matrix integrals and applies to all the symmetry classes \({\beta \in \{1, 2, 4\}}\) of Random Matrix Theory. We compute the weak localization corrections to the mixed cumulants of the conductance and shot noise for β = 1, 4, thus proving a number of conjectures of Khoruzhenko et al. (in Phys Rev B 80:(12)125301, 2009). We derive differential equations that characterize the cumulant generating functions for all \({\beta \in \{1, 2, 4 \} }\). Furthermore, when β = 2 we show that the cumulant generating function of the Wigner delay time can be expressed in terms of the Painlevé III′ transcendant. This allows us to study properties of the cumulants of the Wigner delay time in the asymptotic limit \({n \to \infty}\). Finally, for all the symmetry classes and for any number of open channels, we derive a set of recurrence relations that are very efficient for computing cumulants at all orders.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    F. Mezzadri

  2. School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK

    N. J. Simm

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  1. F. Mezzadri
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Correspondence to F. Mezzadri.

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Communicated by P. Deift

Research partially supported by EPSRC, grant no: EP/G019843/1.

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Mezzadri, F., Simm, N.J. Tau-Function Theory of Chaotic Quantum Transport with β = 1, 2, 4. Commun. Math. Phys. 324, 465–513 (2013). https://doi.org/10.1007/s00220-013-1813-z

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Keywords

  • Recurrence Relation
  • Quantum Channel
  • Shot Noise
  • Moment Generate Function
  • Symmetry Class