Random matrix

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In probability theory and statistics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.

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[edit] Motivation

In the case of disordered physical systems (such as amorphous materials), the corresponding matrices are randomised. Essentially, the physics of these systems can be captured in simplified forms by studying ordered counterparts (such as crystals), and disordered counterparts. In the latter case, the mathematical properties of matrices with elements drawn randomly from statistical distributions (i.e. random matrices) determine the physical properties. The eigenvectors and eigenvalues of random matrices are often of particular interest.

The random behaviour (of the spectral measure, norm, etc. - see below) often disappears in the limit (that is, the limit is deterministic). This phenomenon is a particular case of self-averaging.

[edit] Spectral theory of random matrices

Many remarkable proofs and a great deal of empirical evidence have been published on random matrix theory. One of the most famous results is the so-called Wigner law which states that the spectral measure (known as the density of states) of a random symmetric N \times N matrix with Gaussian-distributed elements tends to the semicircle distribution as N \rightarrow \infty. The Wigner law actually holds in much more general cases, for example for symmetric matrices with i.i.d. entries under mild assumptions on their distribution.

A more general theory was developed by Dan-Virgil Voiculescu in the 1980's to treat the spectral measure of several random matrices under assumptions on their joint moments; it is called Free probability.

[edit] Applications

[edit] More random matrix topics

  • Types of random matrices:
  • Differential equations for gap distributions and transition probabilities
  • Relations to integrable systems and isomonodromic deformations
  • Growth processes; applications to fluid dynamics and crystal growth

[edit] See also

[edit] External links

[edit] References

  • M.L.Mehta, Random matrices, Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. xviii+688 pp