This is the 2D companion to the 3D Wigner semicircle simulator. Instead of diagonalizing a static random matrix once with a Jacobi solver, it integrates Dyson's 1962 Brownian motion directly — a real stochastic differential equation whose N interacting particles are the eigenvalues, live, frame by frame, with no matrix ever built.
dλᵢ = [ (β/N)·Σⱼ≠ᵢ 1/(λᵢ−λⱼ) − κ·λᵢ ] dτ + √(2dτ/N)·dBᵢ
β = repulsion strength (GOE=1, GUE=2, GSE=4)
κ = β/2 for β>0 (keeps the semicircle edge fixed at ±2 for every β)
κ = 0.5 for the β=0 "Poisson" reference (a confining well is
needed there too, or free particles with no repulsion at all
would just diffuse away forever with no equilibrium)
The log-repulsion term 1/(λᵢ−λⱼ) is exactly the Coulomb repulsion between charges confined to a line — it is what forces the eigenvalues apart and, together with the confining term, is what makes the equilibrium distribution converge to the semicircle law as more time-samples are pooled, for any β>0. Switch to β=0 and the repulsion term vanishes entirely: the particles become independent Ornstein–Uhlenbeck processes and the histogram settles on a plain Gaussian bump instead — the simulator's live proof that repulsion, not confinement alone, is what produces the semicircle.
Semicircle law (β>0, N large): ρ(x) = (1/2π)√(4−x²), x∈[−2,2]
Level-spacing ratio (Atas et al. 2013 surmise):
rᵢ = min(sᵢ,sᵢ₋₁)/max(sᵢ,sᵢ₋₁) ∈ [0,1]
⟨r⟩ ≈ 0.5307 (GOE) 0.5996 (GUE) 0.6744 (GSE) 0.3863 (Poisson)
- Symmetry class β — sets how strongly neighbouring eigenvalues repel. The global semicircle shape is universal across β (only the confinement κ is retuned to keep the edge at ±2); what changes is the local level-spacing statistic, shown in the second view.
- Particle count N — more particles means a smoother, more finely-resolved spectrum; the underlying SDE is re-derived to hold the same [−2,2] support regardless of N.
- Simulation speed — scales the physical time step dτ; the process is ergodic, so speeding it up only changes how fast the running histogram fills in, not the destination it converges to.
- Reseed particles — restarts all N eigenvalues from fresh random positions and clears the pooled histogram, so you can watch the approach to equilibrium from scratch.
- Level spacing view — pools the nearest-neighbour ratio statistic rᵢ from the live dynamics against the β-dependent theoretical surmise curve, the same universality diagnostic used in nuclear physics and quantum chaos.
Real-world relevance: Dyson's construction is the dynamical process for which the GOE/GUE/GSE ensembles are literally the equilibrium state — the same repulsion mechanism shows up in the energy levels of heavy nuclei, zeros of the Riemann zeta function, and eigenvalues of large random neural-network weight matrices.