Dyson Brownian Motion — Eigenvalue Repulsion

Dyson Brownian motion is the stochastic process describing how eigenvalues of a random matrix evolve when matrix elements undergo Brownian motion. The SDE is: dλ_i = dB_i/√N + (β/2) Σ_{j≠i} 1/(λ_i−λ_j) dt − λ_i/2 dt. The repulsion term 1/(λ_i−λ_j) reflects level repulsion: eigenvalues behave like a log-gas at inverse temperature β. β=1 (GOE), β=2 (GUE), β=4 (GSE). The stationary distribution gives the Wigner semicircle.