In random matrix theory, the Gaussian ensembles are specific probability distributions over self-adjoint matrices whose entries are independently sampled from the gaussian distribution. They are among the most-commonly studied matrix ensembles, fundamental to both mathematics and physics. The three main examples are the Gaussian orthogonal (GOE), unitary (GUE), and symplectic (GSE) ensembles. These are classified by the Dyson index β, which takes values 1, 2, and 4 respectively, counting the number of real components per matrix element (1 for real elements, 2 for complex elements, 4 for quaternions). The index can be extended to take any real positive value.

The gaussian ensembles are also called the Wigner ensembles, or the Hermite ensembles.

There are many conventions for defining the Gaussian ensembles. In this article, we specify exactly one of them.

In all definitions, the Gaussian ensemble have zero expectation.

When referring to the main reference works, it is necessary to translate the formulas from them, since each convention leads to different constant scaling factors for the formulas.

There are equivalent definitions for the GβE(N) ensembles, given below.

For all ${\displaystyle \beta =1,2,4}$ cases, the GβE(N) ensemble is defined by how it is sampled:

For all ${\displaystyle \beta =1,2,4}$ cases, the GβE(N) ensemble is defined with density function $${\displaystyle \rho (W_{N})={\frac {1}{Z}}e^{-{\frac {\beta }{4}}\sum _{i=1}^{N}W_{N,ii}^{2}-{\frac {\beta }{2}}\sum _{1\leq i<j\leq N}|W_{N,ij}|^{2}}={\frac {1}{Z}}e^{-{\frac {\beta }{4}}\mathrm {Tr} W_{N}^{2}}}$$where the partition function is ${\displaystyle Z=2^{{\frac {1}{2}}N}\left({\frac {2\pi }{\beta }}\right)^{{\frac {1}{2}}N+{\frac {1}{4}}\beta N(N-1)}}$.