Random Matrix Level Repulsion (GOE)

Wigner-Dyson statistics: For GOE (β=1), the nearest-neighbor spacing distribution is P(s) ≈ (π/2)s·exp(−πs²/4) — the Wigner surmise. The key feature: P(0) = 0, i.e., level repulsion. Eigenvalues avoid each other like particles with a logarithmic repulsion from the Jacobian of the eigenvalue measure. Poisson statistics (P(s)=e^{-s}) apply to integrable/non-interacting systems. The crossover GOE↔Poisson occurs as disorder breaks time-reversal symmetry or drives Anderson localization. GUE (β=2) applies with broken TRS (magnetic field); GSE (β=4) to spin-1/2 systems with TRS.