Wigner Semicircle Law

Wigner's semicircle law (1958): for an N×N symmetric random matrix with i.i.d. Gaussian entries (variance σ²/N), the empirical spectral density converges as N→∞ to ρ(x) = (2πσ²)⁻¹ √(4σ² − x²) on [−2σ, 2σ] — a perfect semicircle. The GOE (Gaussian Orthogonal Ensemble) uses real entries; GUE uses complex Hermitian matrices. Level repulsion: eigenvalues avoid each other (unlike Poisson diagonal matrices where they cluster). This universality — the bulk statistics are ensemble-independent — underpins connections to nuclear physics, quantum chaos, and the Riemann zeta zeros.