Random matrix theory — Wigner surmise vs Poisson eigenvalue statistics
The Gaussian Orthogonal Ensemble (GOE) consists of real symmetric matrices with Gaussian entries. After unfolding the spectrum, eigenvalue spacing follows the Wigner surmise P(s)=(πs/2)e^{−πs²/4} — exhibiting level repulsion P(s)→0 as s→0. Poisson statistics P(s)=e^{−s} arise for integrable/random systems with no repulsion. Spectral rigidity Δ₃(L) quantifies long-range correlations: GOE grows logarithmically while Poisson grows linearly.