Dyson's Coulomb Gas — Random Matrix Continuum Limit

Dyson's Coulomb Gas (1962): Eigenvalues of random matrices behave like 2D charges on a line repelling logarithmically: H = −β Σᵢ﹤ⱼ log|xᵢ−xⱼ| + β/2 Σᵢ xᵢ².
Wigner semicircle: For N→∞ GUE/GOE, eigenvalue density → ρ(x) = (1/2π)√(4−x²) on [−2,2].
Level repulsion: P(s) ~ s^β for small spacing s. GOE β=1 (orthogonal), GUE β=2 (unitary), GSE β=4 (symplectic).
Wigner surmise: P(s) = (π/2)s·exp(−πs²/4) for GOE — eigenvalues avoid each other (quantum chaos signature).