the Gaussian ensembles GOE, GUE, and GSE
/ GOE = gee-oh-EE; Wigner = VIG-ner /
The Gaussian ensembles are the most symmetric, most exactly solvable random matrices — the hydrogen atom of random matrix theory. They were introduced by Wigner and Dyson to model the spectra of heavy atomic nuclei, whose energy levels were too complicated to compute but seemed to obey universal statistical laws. The three ensembles, GOE, GUE, and GSE, correspond to the three classical symmetry classes a quantum system can have (real, complex, or quaternionic) and are indexed by a parameter beta = 1, 2, 4 measuring the number of real degrees of freedom per off-diagonal entry.
Each ensemble is defined by an invariant probability density on matrices: the law has density proportional to exp(-(beta/4) N trace(H^2)) (with a conventional normalisation), which is the same as saying the entries are independent Gaussians with the symmetry constraint imposed. The GOE consists of real symmetric matrices invariant under orthogonal conjugation H -> O^T H O; the GUE consists of complex Hermitian matrices invariant under unitary conjugation H -> U^* H U; the GSE consists of quaternionic self-dual matrices invariant under symplectic conjugation. Because the density depends only on the eigenvalues through trace(H^2) = sum of lambda_i^2, one can integrate out the eigenvectors and write the joint eigenvalue density explicitly: it is proportional to (product over i<j of |lambda_i - lambda_j|^beta) times exp(-(beta/4) N sum lambda_i^2). That factor product |lambda_i - lambda_j|^beta is the Vandermonde repulsion, and the exponent beta is exactly 1, 2, or 4.
The Gaussian ensembles matter because they are exactly solvable: the GUE in particular has determinantal eigenvalue correlations (its kernel is built from Hermite polynomials), which let one compute the bulk sine kernel, the edge Airy kernel, and the Tracy-Widom law in closed form. They serve as the reference point against which universality is stated: the deep theorems of the field say that a generic Wigner matrix has the same local eigenvalue statistics as the corresponding Gaussian ensemble of the same symmetry class. A caveat: the eigenvalue eigenvalue-density of all three ensembles is the same semicircle in the limit; the beta dependence shows up not in the bulk density but in the strength of level repulsion and in the precise local statistics.
For the GUE (beta = 2) one can write the bulk eigenvalue correlations exactly using Hermite polynomials. Take two eigenvalues near the centre of the spectrum: the probability of finding them at small separation s (in rescaled units) is proportional to 1 - (sin(pi s)/(pi s))^2, vanishing like s^2 as s -> 0. That clean closed form is exactly the level repulsion the GUE is famous for, and it is what real nuclear energy-level data was found to obey.
The GUE's exact solvability lets one read level repulsion straight off a closed formula.
GOE/GUE/GSE differ in the repulsion exponent beta = 1, 2, 4 and hence in fine local statistics, but all three share the same limiting semicircle bulk density. Do not confuse the symmetry class (which fixes beta) with the bulk law (which is universal across the three).