Gaussian Orthogonal Ensemble

The Gaussian orthogonal ensemble (GOE) is a probability distribution on real symmetric matrices with independent centered normal distribution entries on and above the diagonal. In the unscaled convention, entries strictly above the diagonal have variance 1 and diagonal entries have variance 2. Entries below the diagonal are determined by symmetry. This is a special case of a Wigner matrix (Tao 2012).

Its law is invariant under conjugation by an orthogonal matrix. Dividing a sampled matrix by gives the usual Wigner's semicircle law normalization. The rescaled bulk eigenvalue limit is the sine process (He et al. 2026).