Random matrix

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.

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.

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 × N matrix with Gaussian-distributed elements tends to the semicircle distribution as N → ∞. 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 1980s to treat the spectral measure of several random matrices under assumptions on their joint moments; it is called free probability.

Applications

• Applications to random tilings, random words, random partitions
• Applications to L-functions, including support for the Hilbert-Pólya conjecture.
• Applications to multivariate statistics
• Applications to nuclear physics, including the Gaussian unitary ensemble, the Gaussian symplectic ensemble, and the Gaussian orthogonal ensemble. The spectra and cross sections nuclei measured in laboratories show that the dynamics of the nucleus is exceedingly complex. Evidence points at a chaotic behaviour similar to that seen on hyperbolic manifolds; random matrix theory attempts to model the gross properties of the nuclear spectra (distribution of resonances, spectral line widths) through ensembles of random matrices.
• Applications to signal processing and wireless communications
• Applications to quantum chaos and mesoscopic physics
• Applications to number theory
• Applications to operator algebras [1], and Free probability [2].
• Applications to models of quantum gravity in two dimensions
• Current research suggests it could have applications in improving web search engines [3]

More random matrix topics

• Types of random matrices:
  • Matrix models
  • Random matrices with independent entries

• Methods:
  • Matrix Riemann–Hilbert methods, applications to large-N asymptotics
• Integral operators in random matrix theory

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

See also

• Determinantal ensembles
• Brownian ensembles
• Dyson processes. Airy, Bessel, sine and Laguerre processes
• Free probability
• Wishart distribution

References

• Diaconis, Persi (2003), "Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture", American Mathematical Society. Bulletin. New Series, 40 (2): 155–178, doi:10.1090/S0273-0979-03-00975-3, MR1962294

• Diaconis, Persi (2005), "What is ... a random matrix?", Notices of the American Mathematical Society, 52 (11): 1348–1349, ISSN 0002-9920, MR2183871
• M.L.Mehta, Random matrices, Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. xviii+688 pp

External links

• Random Matrix at MathWorld
• RMTool A MATLAB based Random Matrix Calculator
• Random Matrix Theories in Quantum Physics: Common Concepts
• Developments in Random Matrix Theory
• Random-matrix physics: spectrum and strength fluctuations
• Random matrix theory survey
• Alcatel-Lucent Chair on Flexible Radio Applications of Random Matrix theory to Wireless Communications.