Random Matrix β-Ensembles

GOE (β=1, real symmetric), GUE (β=2, complex Hermitian), GSE (β=4, quaternion self-dual). The Wigner surmise P(s) = A·sᵝ·exp(−B·s²) governs level repulsion. Compare against Poisson (no correlations) from uncorrelated levels.

Matrix size N80

Ensemble β

Samples (accumulated)5

Left: Eigenvalue density (semicircle law) | Center: Level spacing histogram vs Wigner surmise | Right: Eigenvalue scatter