Wigner Semicircle Law (1955): For an N×N GOE (Gaussian Orthogonal Ensemble) matrix
H = (A + Aᵀ)/√(2N) where Aᵢⱼ ~ N(0,1), the empirical spectral distribution converges to the semicircle
ρ(λ) = (1/2π)√(4−λ²) for |λ| ≤ 2. Level spacing statistics follow the Wigner surmise P(s) = (π/2)s·e^{−πs²/4},
showing level repulsion — unlike Poisson statistics of uncorrelated systems where P(s) = e^{−s}.
This universality connects to quantum chaos: energy levels of chaotic quantum systems have GOE statistics.
Proved rigorously (Tao-Vu 2012) for general Wigner matrices via four-moment theorem.