Gaussian ensemble
random matrix with gaussian entries
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.
Definitions
Conventions
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.
β
{\displaystyle \beta }
: a positive real number.
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