Random matrix theory
|Yan Fyodorov (2011), Scholarpedia, 6(3):9886.||doi:10.4249/scholarpedia.9886||revision #90306 [link to/cite this article]|
Random Matrix Theory (frequently abbreviated as RMT) is an active research area of modern Mathematics with input from Mathematical and Theoretical Physics, Mathematical Analysis and Probability, and with numerous applications, most importantly in Theoretical Physics, Number Theory, and Combinatorics, and further in Statistics, Financial Mathematics, Biology and Engineering & Telecommunications.
The main goal of the Random Matrix Theory is to provide understanding of the diverse properties (most notably, statistics of matrix eigenvalues) of matrices with entries drawn randomly from various probability distributions traditionally referred to as the random matrix ensembles. Three classical random matrix ensembles are the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic Ensemble (GSE). They are composed respectively of real symmetric, complex Hermitian and complex self-adjoint quaternion matrices with independent, normally distributed mean-zero entries whose variances are adjusted to ensure the invariance of their joint probability density with respect to Orthogonal (respectively, Unitary or Symplectic) similarity transformations. Such invariance is also shared by the corresponding Lebesgue measures. If one keeps the requirement of invariance of the joint probability density of all entries but relaxes the property of entries being independent, one arrives at a broader classes of Invariant non-Gaussian Ensembles (Orthogonal, Unitary, or Symplectic). Note that being self-adjoint the matrices of those types have all their eigenvalues real.
Equally important are the Circular Ensembles composed of complex unitary matrices sharing the same invariance properties of the measure as their Gaussian counterparts, but with eigenvalues confined to the unit circle in the complex plane rather than to the real line. Those ensembles are known as the Circular Orthogonal Ensemble (COE), the Circular Unitary Ensemble (CUE) and the Circular Symplectic Ensemble (CSE). Finally deserves mentioning the so-called Ginibre Ensemble of matrices with independent, identically and normally distributed real, complex, or quaternion real entries, and no further constraints imposed. The corresponding eigenvalues are scattered in the complex plane.
Brief History and Background Information
Although origins of RMT could be traced back to works by Wishart (1928)  and James (1954-1964)  in the field of Statistics (see also Hua's works in multivariate harmonic analysis ) the real start of the field is usually attributed to highly influential papers by Eugene Wigner in 1950's  motivated by applications in Nuclear Physics. Wigner suggested that fluctuations in positions of compound nuclei resonances can be described in terms of statistical properties of eigenvalues of very large real symmetric matrices with independent, identically distributed entries. The rational behind such a proposal was the idea that in the situation when it is hardly possible to understand in detail individual spectra associated with any given nucleus composed of many strongly interacting quantum particles, it may be reasonable to look at the corresponding systems as "black boxes" and adopt a kind of statistical description, not unlike thermodynamics approach to classical matter. A system in Quantum Mechanics can be characterized by a self-adjoint linear operator in Hilbert space, its Hamiltonian, which we may think informally of as a matrix of infinitely many dimensions. This suggests that input to such theory should be very general properties of the underlying generic Hamiltonians, most importantly such global features as the Hermiticity, the time-inversion invariance as well as other symmetries Hamiltonians may obey from general principles. Wigner hoped that the output of the model will be universal, that is independent on the detail and common to majority of systems sharing the corresponding symmetri