Besides the spacing distribution, there is another thing you can calculate, which is just the general correlation function of any number of levels; the probability for finding, say, five levels with five given separations can be calculated, and I was able to find a very elegant formula for this general correlation function, which turns out to be Pfaffian, which is a very elegant object. It's not a determinant but it's like the square root of a determinant, and with the Pfaffians, this mathematical theory of Pfaffians, you can then deduce essentially everything that these level distributions do. The only difficulty with this whole subject is to find any practical use for it, and Wigner originally designed it to be used in nuclear physics, but it's only useful if you have experimental data to compare it with, and there don't exist, in fact, enough levels of heavy nuclei to get good statistics. I mean, the best case is a certain isotope of erbium which is - I forget the exact atomic weight of this particular isotope, but in any case there's one case in which you have about two hundred well-defined levels, but two hundred isn't enough, really, to do a good statistical analysis. So the subject died essentially for lack of applications. But it was revived more recently because now we have all the virtual reality, now we have these big computers, and we can compute ideal systems of energy levels in enormous quantities, so it's trivial now to find a million levels for a suitably defined system and then compare that with the theory. So this is now a large industry. So the subject has actually found quite a lot of use in dealing with the behaviour of complex systems, simulated on the computer, whether or not it has anything to do with the real world. And there is one very interesting case which is the zeros of the zeta function, which is a purely mathematical problem . You have this famous Riemann zeta function, which is one of the great mysteries of pure mathematics where you have the zeros which are all on a certain line with real part equal to one half, with the imaginary parts of the zeros distributed in this mysterious way up the line. And it turns out - Odlyzko, a mathematician at the Bell Labs, calculated about 10 to the 20th zeros of the zeta function - and, well, it turns out they precisely obey the unitary distribution law, as if they were the Eigen values of a random unitary matrix. And that's an empirical fact; it still is not understood, there is no theoretical proof of that. So it's a very interesting connection between the physics of random systems and this mathematical mystery.

[Q] But to you, working on it was beautiful mathematics.

Yes, it's fun to do and the applications to me were always secondary. The connection with zeta function, in a way, is to me the most exciting part of it.

[Q] Which has possibly stimulated you to go back to looking at that problem?

Yes. And of course there's been a certain amount of progress. Hugh Montgomery actually proved a weaker version of this unitary distribution law. I mean, he was able to prove rigorously that if you take Fourier transform as the distribution within a certain subset of the Fourier space, unitary distribution is exact. He couldn't extend it over the whole space. And at least, as far as it goes, his proof actually covers a substantial region. So it almost certainly is correct, and the mathematicians are still busily working on it. Sarnak here in Princeton is a mathematician who is very much interested in this, so I keep in communication with him.