PARAMETERS
Total spacings: 0
Mean spacing: —
Wigner ratio r: —
Wigner surmise — exact for 2×2 GOE, excellent approximation for large N:
P(s) = (π/2)s·exp(−πs²/4)
Level repulsion: P(s→0) ~ s (eigenvalues avoid each other). Poisson statistics P(s) = e^{-s} shows no repulsion — uncorrelated levels.
The GUE Wigner surmise: P(s) = (32/π²)s²·exp(−4s²/π) — stronger repulsion (β=2 vs GOE β=1).
Random matrices describe quantum chaotic systems (BGS conjecture), heavy nucleus energy levels, Riemann zeros, and random graph spectra.