Wigner surmise · level repulsion · random matrix universality
Samples: 0
Spacings: 0
⟨s⟩: —
⟨s²⟩: —
Random Matrix Theory (Wigner, Dyson, Mehta) — Random matrices from the Gaussian
Unitary Ensemble (GUE, β=2) have correlated eigenvalues that repel each other.
The nearest-neighbor spacing distribution is approximated by the Wigner surmise:
• GUE: p(s) = (32/π²) s² exp(−4s²/π) — quadratic repulsion
• GOE: p(s) = (π/2) s exp(−πs²/4) — linear repulsion
• GSE: p(s) = (2¹⁸/3⁶π³) s⁴ exp(−64s²/9π) — quartic repulsion
• Poisson: p(s) = exp(−s) — no repulsion (integrable systems)
The spacing exponent β=1,2,4 equals the dimension of number fields (real, complex,
quaternion). GUE describes time-reversal-broken quantum chaos. The right panel shows
the empirical eigenvalue density (semicircle law): ρ(λ) = √(2N−λ²)/(2π) for GUE.