Dyson Brownian Motion: Eigenvalues in Motion

This simulator is the 2D dynamical counterpart to the 3D Wigner semicircle law simulator, computed independently rather than as a flattened render of the same scene. Instead of diagonalizing a static random matrix, it integrates Freeman Dyson's 1962 Brownian-motion construction directly: N points on a line, each pushed by real log-repulsion from every other point (exactly the 2D Coulomb-gas force restricted to a line) plus a confining pull toward zero and ordinary Brownian noise. Because this stochastic process has the GOE/GUE/GSE random-matrix eigenvalue distributions as its exact equilibrium state, simply running it forward in time and pooling snapshots reproduces Wigner's semicircle law — the point of the simulator is to watch that convergence happen live, frame by frame, rather than see it appear fully-formed from a one-shot matrix diagonalization. A second view pools the nearest-neighbour level-spacing ratio to show how the repulsion strength β (GOE, GUE or GSE) sets universal local statistics, and a β=0 "no repulsion" option shows what is lost without it: a plain Gaussian instead of a semicircle.