The eigenvalues of random matrices can have very interesting properties. As one changes the probability distribution one can drive a phase transition between a `disk' phase where the eigenvalues have nonzero probability on a circular region around the origin to a `ring' phase where the density of eigenvalues near the origin is zero. An amazing theorem states that one cannot have phases with more than one ring!

Relevant Publications:

[106.] ``Single Ring Theorem' and the Disk--Annulus Phase Transition,'' J. Feinberg, R.T. Scalettar, and A. Zee, J. Math. Phys. 42, 5718 (2001).

[109.] ``Phase Transitions in Non-Hermitian Matrix Models and the `Single Ring' Theorem,'' J. Feinberg, R.T. Scalettar, and A. Zee, Proceedings of Johns-Hopkins Conference on Non--Perturbative Quantum Field Theory, World Scientific.