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Random-matrix theory of quantum transport
Open accessAbstract
This is a review of the statistical properties of the scattering matrix of a mesoscopic system. Two geometries are contrasted: A quantum dot and a disordered wire. The quantum dot is a confined region with a chaotic classical dynamics, which is coupled to two electron reservoirs via point contacts. The disordered wire also connects two reservoirs, either directly or via a point contact or tunnel barrier. One of the two reservoirs may be in the superconducting state, in which case conduction involves Andreev reflection at the interface with the superconductor. In the case of the quantum dot, the distribution of the scattering matrix is given by either Dyson's circular ensemble for ballistic point contacts or the Poisson kernel for point contacts containing a tunnel barrier. In the case of the disordered wire, the distribution of the scattering matrix is obtained from the Dorokhov-Mello-Pereyra-Kumar equation, which is a one-dimensional scaling equation. The equivalence is discussed with the nonlinear $\ensuremath{\sigma}$ model, which is a supersymmetric field theory of localization. The distribution of scattering matrices is applied to a variety of physical phenomena, including universal conductance fluctuations, weak localization, Coulomb blockade, sub-Poissonian shot noise, reflectionless tunneling into a superconductor, and giant conductance oscillations in a Josephson junction.
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“The simplest form of multiple linear regression, which is in essence fitting a plane through the origin to a cloud of points, is not sloppy.”
From Sloppy-model universality class and the Vandermonde matrix. (2006) · cites this paper as [13]
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“The Hessian matrix for this type of model is the sample covariance matrix of the data points and is known as a Wishart matrix [13].”
From Sloppy-model universality class and the Vandermonde matrix. (2006) · cites this paper as [13]
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Metadata
- Authors
- Beenakker, C. W. J.
- Affiliations
- Leiden University (NL)
- Corresponding author
- Beenakker, C. W. J.
- ORCID
- Beenakker, C. W. J.
- Venue
- Reviews of Modern Physics
- Publisher
- American Physical Society (APS)
- Published
- 1997-07-01
- Volume
- 69
- Issue
- 3
- Pages
- 731-808
- ISSN
- 0034-6861 (ISSN-L)
- 1539-0756
- OpenAlex
- W2092583264
- Cited by
- 2,551
- References
- 347
- Citations by year
- 2026: 58 · 2025: 92 · 2024: 79 · 2023: 75 · 2022: 98 · 2021: 111 · 2020: 116 · 2019: 100 · 2018: 76 · 2017: 98 · 2016: 91 · 2015: 88 · 2014: 99 · 2013: 110 · 2012: 83
- FWCI
- 68.17
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Reviews of Modern Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Leiden Repository (Leiden University) (submittedVersion)
- Data Archiving and Networked Services (DANS) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- Stichting voor Fundamenteel Onderzoek der Materie
- Nederlandse Organisatie voor Wetenschappelijk Onderzoek
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Quantum and electron transport phenomena
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Keywords
- random matrix theory, quantum transport, scattering matrix, mesoscopic systems, quantum dots, disordered wires, point contacts, tunnel barrier, superconducting junctions, Andreev reflection, Poisson kernel, nonlinear sigma model, supersymmetry, localization, universal conductance fluctuations, weak localization, Coulomb blockade
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