Random Matrix Theory
Wigner semicircle law & GOE/GUE level spacing statistics
Ensemble Parameters
Ensemble:
GOE — real symmetric
GUE — complex Hermitian
Wishart (Marchenko-Pastur)
Poisson (uncorrelated)
Matrix size N:
80
Samples (accumulated):
1
Histogram bins:
40
New Sample
Accumulate
Wigner semicircle:
ρ(λ) = (2/πR²)√(R²-λ²)
R = 2σ√N
Level spacing
s = λ_{n+1}-λ_n (unfolded):
GOE: P(s) ≈ (πs/2)exp(-πs²/4) (Wigner surmise)
GUE: P(s) ≈ (32s²/π²)exp(-4s²/π)
Poisson: P(s) = exp(-s)
Level repulsion: GOE β=1, GUE β=2.