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Random Matrix Theory

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Random Matrix Theory (RMT) is a method of studying the statistical behavior of large complex systems, by defining an ensemble which considersall possible laws of interactions within the system. The important question addressed by random matrix theory is: Given a random matrix ensemble whatare the probability laws which govern its eigenvalues or eigenvectors. This question is pertinent to many areas in physics and mathematics, for instancestatistical behavior of compound nucleus, conductivity in disordered metals, behavior of chaotic systems or zeros of the Riemann zeta function. Thesuccess of random matrices lies in the universality regime of the eigenvalue statistics. There is compelling evidence that when the size of the matrix isvery large then the eigenvalue distribution tends, in a certain sense, towards a limiting distribution. This only depends on the symmetryproperties of the matrix and is independent of the initial probability law imposed on the...

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Notes

  1. 1.

    This was alreadyknown to Wigner and von Neuman [51].

Abbreviations

Random matrices :

Large matrices with randomly distributed elements obeying the given probability laws and symmetry classes.

Orthogonal ensembles :

Real symmetric random matrix ensembles which are invariant under all orthogonal transformations. Majority of practical systems are described by these ensembles.

Unitary ensembles :

Hermitian random matrix ensembles which are invariant under all unitary transformations.

Symplectic ensembles :

Hermitian self-dual random matrix ensembles which are invariant under all symplectic transformations.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, UK

    Güler Ergün

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  1. Güler Ergün
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Ergün, G. (2009). Random Matrix Theory . In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_443

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