The eigenvalues of a random matrix fill Wigner's semicircle

Only the entries' variance matters; these are uniform on $[-\sqrt{3},\sqrt{3}]$, which has variance 1 and looks nothing like a Gaussian. The even moments of the semicircle are the Catalan numbers $1,2,5,\dots$ because $\frac{1}{N}\mathrm{Tr}{H}^{2k}$ counts closed walks that never cross themselves. The tests check the second, fourth and sixth moments (1, 2 and 5), and that no eigenvalue lies beyond $\pm 2.1$.

At $N=200$ the program runs in your browser in about a tenth of a second. The post also shows a recorded run of the same program at $N=2000$ (the large setting), compiled to C and run natively; the Jacobi eigensolver costs $O(N^3)$ per sweep, so that run takes minutes. Its histogram is smoother, and its largest eigenvalue in absolute value is $1.996$.