Random Matrix Theory: The Wigner Semicircle Law

This simulator builds real random symmetric matrices — Gaussian Orthogonal Ensemble by default, or coin-flip ±1 entries to demonstrate universality — and diagonalizes them in the browser with a hand-written Jacobi eigenvalue algorithm, no linear-algebra library involved. As matrix size N grows, the rescaled eigenvalue spectrum λ/√N visibly settles onto Eugene Wigner's 1955 semicircle law, the random-matrix analogue of the Central Limit Theorem. A second view switches to the level-spacing ratio statistic ⟨r⟩, a diagnostic physicists use to tell correlated ("chaotic") spectra from uncorrelated (Poisson) ones — and shows that it lands on the same universal GOE value of about 0.5307 no matter which entry distribution generated the matrix.