Random Matrix Level Repulsion — GOE/GUE/Poisson

Level spacing statistics reveal universality classes. GOE (real symmetric) gives Wigner surmise P(s) ~ s·exp(−πs²/4); GUE (complex Hermitian) gives P(s) ~ s²·exp(−4s²/π). Poisson statistics (no repulsion) arise for integrable systems. The nearest-neighbour spacing distribution distinguishes quantum chaos from regularity.

Matrix size N 50

Samples 100

Ensemble GOE

GOE: P(s) ≈ (π/2)s·exp(−πs²/4)
GUE: P(s) ≈ (32/π²)s²·exp(−4s²/π)
Poisson: P(s) = e^(−s)

Wigner surmise (exact for 2×2)