The Gaussian Orthogonal Ensemble (GOE) consists of real symmetric matrices with independent Gaussian entries. Wigner's semicircle law: eigenvalue density converges to ρ(λ) = (2/π)√(1−λ²/4) as N→∞. Level spacing obeys the Wigner surmise P(s) ∝ s·exp(−πs²/4) — eigenvalues repel! Compare to Poisson (no repulsion) for random uncorrelated levels.
Eigenvalue repulsion: P(s)~s for GOE vs P(s)~1 for Poisson (random). Repulsion ↔ quantum chaos.