# Spectral correlations of individual quantum graphs

## Abstract

We investigate the spectral properties of chaotic quantum graphs. We demonstrate that the energy-average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear {sigma}-model action. This proves that spectral correlations of individual quantum graphs behave according to the predictions of Wigner-Dyson random matrix theory. We explore the stability of the universal random matrix behavior with regard to perturbations, and discuss the crossover between different types of symmetries.

- Authors:

- Institut fuer Theoretische Physik, Freie Universitaet Berlin, Arnimallee 14, 14195 Berlin (Germany) and Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100 (Israel)
- (Germany)

- Publication Date:

- OSTI Identifier:
- 20709843

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; Journal Volume: 72; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevE.72.056215; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; CHAOS THEORY; CORRELATIONS; DISTURBANCES; GRAPH THEORY; NONLINEAR PROBLEMS; RANDOMNESS; SIGMA MODEL; SUPERSYMMETRY

### Citation Formats

```
Gnutzmann, Sven, Altland, Alexander, and Institut fuer Theoretische Physik, Universitaet zu Koeln, Zuelpicher Str. 77, 50937 Cologne.
```*Spectral correlations of individual quantum graphs*. United States: N. p., 2005.
Web. doi:10.1103/PhysRevE.72.056215.

```
Gnutzmann, Sven, Altland, Alexander, & Institut fuer Theoretische Physik, Universitaet zu Koeln, Zuelpicher Str. 77, 50937 Cologne.
```*Spectral correlations of individual quantum graphs*. United States. doi:10.1103/PhysRevE.72.056215.

```
Gnutzmann, Sven, Altland, Alexander, and Institut fuer Theoretische Physik, Universitaet zu Koeln, Zuelpicher Str. 77, 50937 Cologne. Tue .
"Spectral correlations of individual quantum graphs". United States.
doi:10.1103/PhysRevE.72.056215.
```

```
@article{osti_20709843,
```

title = {Spectral correlations of individual quantum graphs},

author = {Gnutzmann, Sven and Altland, Alexander and Institut fuer Theoretische Physik, Universitaet zu Koeln, Zuelpicher Str. 77, 50937 Cologne},

abstractNote = {We investigate the spectral properties of chaotic quantum graphs. We demonstrate that the energy-average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear {sigma}-model action. This proves that spectral correlations of individual quantum graphs behave according to the predictions of Wigner-Dyson random matrix theory. We explore the stability of the universal random matrix behavior with regard to perturbations, and discuss the crossover between different types of symmetries.},

doi = {10.1103/PhysRevE.72.056215},

journal = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},

number = 5,

volume = 72,

place = {United States},

year = {Tue Nov 01 00:00:00 EST 2005},

month = {Tue Nov 01 00:00:00 EST 2005}

}

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