by the Gaussian measure with density on the space of Hermitian matrices . Here is a normalization constant, chosen so that the integral of the density is equal to one. The term unitary refers to the fact that the distribution is invariant under unitary conjugation. The Gaussian unitary ensemble models Hamiltonians lacking time-reversal symmetry. The Gaussian orthogonal ensemble is described by the Gaussian measure with density on the space of n × n real symmetric matrices H = (Hij). Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry. The Gaussian symplectic ensemble is described by the Gaussian measure with density on the space of n × n Hermitian quaternionic matrices, e.g. symmetric square matrices composed of quaternions, . Its distribution is invariant under conjugation by the symplectic group, and it models Hamiltonians with time-reversal symmetry but no rotational symmetry. The Gaussian ensembles GOE, GUE and GSE are often denoted by their Dyson index, β = 1 for GOE, β = 2 for GUE, and β = 4 for GSE. This index counts the number of real components per matrix element. The ensembles as defined here have Gaussian distributed matrix elements with mean ⟨Hij⟩ = 0, and two-point correlations given by from which all higher correlations follow by Isserlis' theorem. The joint probability density for the eigenvalues of GUE/GOE/GSE is given by where Zβ,n is a normalization constant which can be explicitly computed, see Selberg integral. In the case of GUE (β = 2), the formula (1) describes a determinantal point process. Eigenvalues repel as the joint probability density has a zero (of th order) for coinciding eigenvalues . For the distribution of the largest eigenvalue for GOE, GUE and Wishart matrices of finite dimensions, see. Distribution of level spacings From the ordered sequence of eigenvalues , one defines the normalized spacings , where is the mean spacing. The probability distribution of spacings is approximately given by, for the orthogonal ensemble GOE , for the unitary ensemble GUE , and for the symplectic ensemble GSE . The numerical constants are such that is normalized: and the mean spacing is, for . Generalizations Wigner matrices are random Hermitian matrices such that the entries above the main diagonal are independent random variables with zero mean and have identical second moments. Invariant matrix ensembles are random Hermitian matrices with density on the space of real symmetric/Hermitian/quaternionic Hermitian matrices, which