the eigenvalues of the random matrix. The point process captures the statistical properties of eigenvalues in the vicinity of . For the Gaussian ensembles, the limit of is known; thus, for GUE it is a determinantal point process with the kernel (the sine kernel). The universality principle postulates that the limit of as should depend only on the symmetry class of the random matrix (and neither on the specific model of random matrices nor on ). Rigorous proofs of universality are known for invariant matrix ensembles and Wigner matrices. Edge statistics Correlation functions The joint probability density of the eigenvalues of random Hermitian matrices , with partition functions of the form where and is the standard Lebesgue measure on the space of Hermitian matrices, is given by The -point correlation functions (or marginal distributions) are defined as which are skew symmetric functions of their variables. In particular, the one-point correlation function, or density of states, is Its integral over a Borel set gives the expected number of eigenvalues contained in : The following result expresses these correlation functions as determinants of the matrices formed from evaluating the appropriate integral kernel at the pairs of points appearing within the correlator. Theorem [Dyson-Mehta] For any , the -point correlation function can be written as a determinant where is the th Christoffel-Darboux kernel associated to , written in terms of the quasipolynomials where is a complete sequence of monic polynomials, of the degrees indicated, satisfying the orthogonilty conditions Other classes of random matrices Wishart matrices Wishart matrices are n × n random matrices of the form , where X is an n × m random matrix (m ≥ n) with independent entries, and X* is its conjugate transpose. In the important special case considered by Wishart, the entries of X are identically distributed Gaussian random variables (either real or complex). The limit of the empirical spectral measure of Wishart matrices was found by Vladimir Marchenko and Leonid Pastur. Random unitary matrices Non-Hermitian random matrices References Books Survey articles Historic works Footnotes External links Algebra of random variables Mathematical physics