Probability

This subject is not so classical as what's gone before. It has its roots in practical statistics (1930 or so), quantum mechanics (1958), and, very recently, in an astonishing variety of other questions, touched upon below. §12.7 et seq. are technically more demanding than any thing we've done before, employing for the most part radically new methods, but it's fascinating and worth the effort. Mehta [1967] is the best general reference.

The Gaussian orthogonal ensemble (GOE)

Quantum-mechanically speaking, the energy levels E of a large collection of n like atoms, without internal degrees of freedom such as angular momentum or spin, may be found by solving Hψ ≡ −Δψ + Vψ = Eψ, in which Δ is Laplace's operator in ℝ3n = ℝ3 × … × ℝ3, one copy for each atom to record its location, and V is the (classical) total potential energy of such a configuration. But, if n is big, the computation is already way out of reach of the biggest machine. What to do? Well, you can stop there, or you can look for something simpler, something you can compute, to see if that helps. Wigner [1958, 1967] proposed the simplest such caricature in connection with the scattering of neutrinos off heavy nuclei, replacing H by a typical, real symmetric, n × n matrix x = (xij :1 ≤ i, j ≤ n) and looking at its spectrum as n ↑ ∞ far away from Nature perhaps, but never mind.