Eigenvalues of random Hermitian matrices repel each other — GUE spacing follows Wigner surmise
Empirical histogram
Wigner surmise (GOE/GUE)
Poisson (exp)
Level repulsion: eigenvalues of random matrices from GOE (β=1) or GUE (β=2) repel each other.
Nearest-neighbor spacing p(s) follows Wigner surmise: p(s) ∝ sβ exp(−c_β s²) — linear repulsion for GOE, quadratic for GUE.
In contrast, uncorrelated (Poisson) levels have p(s) = exp(−s): no repulsion, level crossings are generic.
GOE arises for time-reversal symmetric systems; GUE for broken time-reversal (magnetic fields).