Take an N×N random symmetric matrix with i.i.d. entries, normalize by 1/√N. As N→∞, the empirical eigenvalue distribution converges almost surely to the semicircle ρ(x) = (2/π)√(1−x²) on [−1,1].
Wigner's theorem (1958) is universal: it holds for any distribution with finite variance, not just Gaussian. This is the random matrix analogue of the central limit theorem. The semicircle arises because of free probability — the free cumulants of a large random matrix are controlled by its variance alone.