For a random symmetric N×N matrix with i.i.d. Gaussian entries (GUE/GOE), the empirical eigenvalue distribution converges to Wigner's semicircle: ρ(x) = (2/πR²)√(R²−x²) as N→∞. This is a universal law — independent of the entry distribution.
Wigner (1955): eigenvalue density of random matrices converges universally to the semicircle. The same law governs nuclear energy levels, network spectra, and quantum chaos. Radius R=2√N (for unit variance entries).