Wigner Semicircle Law

Eigenvalue density of large random symmetric matrices converges to ρ(x) = (2/πR²)√(R²−x²)

Matrix size N

Histogram bins

N—

Semicircle radius R—

λ_max observed—

λ_min observed—

GOE: H = (A + Aᵀ)/(2√N)
where Aᵢⱼ ~ N(0,1)

R = 2σ = 2 as N → ∞

Wigner's semicircle law (1955): take an N×N symmetric random matrix with i.i.d. Gaussian entries, scaled by 1/√N. As N→∞, the empirical distribution of eigenvalues converges to ρ(x) = (2/πR²)√(R²−x²) supported on [−R, R], regardless of the entry distribution (universality). This is a cornerstone of random matrix theory (RMT), with connections to nuclear physics, quantum chaos, number theory (GUE and Riemann zeros), and wireless communications.