Random Matrix - Gaussian Ensembles

Gaussian Ensembles

The most studied random matrix ensembles are the Gaussian ensembles.

The Gaussian unitary ensemble GUE(n) is described by the Gaussian measure with density

on the space of n × n Hermitian matrices H = (H_ij)n
i,j=1. Here Z_GUE(n) = 2n/2 πn2/2 is a normalisation 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 GOE(n) is described by the Gaussian measure with density

on the space of n × n real symmetric matrices H = (H_ij)n
i,j=1. Its distribution is invariant under orthogonal conjugation, and it models Hamiltonians with time-reversal symmetry.

The Gaussian symplectic ensemble GSE(n) is described by the Gaussian measure with density

on the space of n × n quaternionic Hermitian matrices H = (H_ij)n
i,j=1. Its distribution is invariant under conjugation by the symplectic group, and it models Hamiltonians with time-reversal symmetry but no rotational symmetry.

The joint probability density for the eigenvalues λ₁,λ₂,...,λ_n of GUE/GOE/GSE is given by

where β = 1 for GOE, β = 2 for GUE, and β = 4 for GSE; Z_β,n is a normalisation constant which can be explicitly computed, see Selberg integral. In the case of GUE (β = 2), the formula (1) describes a determinantal point process.