Wigner Semicircle Law

Consider the Gaussian distribution, given by the density function f(x) \propto \exp({-x^2/g}). Wigner asked: What if you consider a higher-dimensional (matrix) analogue? That is, x is now replaced by X , a symmetric N \times N matrix, with each entry being a random variable. Now let us consider its eigenvalues. Generically, there will be N of them. Wigner found that if N \gg 1 and g<<1 with Ng fixed, then the eigenvalues are distributed according to a density function that is a perfect semicircle! The size is R \sim \sqrt{N g} (technically speaking, Wigner did this for Hermitian matrices).