Random Matrix Theory

Wigner's miracle: For a random symmetric matrix with i.i.d. entries (variance σ²), the eigenvalue density converges to the semicircle law: ρ(x) = √(4Nσ² − x²)/(2πNσ²). More remarkably, eigenvalues repel each other — the nearest-neighbor spacing follows P(s) ≈ (π/2)s·exp(−πs²/4) (Wigner surmise), not the Poisson distribution of uncorrelated levels. This same statistics describes heavy nuclei, quantum chaos, and even prime number gaps.