4/23/2003

Russell Lyons
Indiana University and Georgia Tech

Eigenvalues of random matrices arise in various areas of physics and mathematics.  The most-studied such probability measures have a determinantal form.  Several people have studied other specific determinantal processes, as well as a general theory.   We shall discuss the general theory of stationary random fields on integer lattices that are defined via minors of multi-dimensional Toeplitz matrices. Explicit examples include combinatorial models, finitely dependent processes, and renewal processes in one dimension. Among the interesting properties of these processes, we focus mainly on whether they have a phase transition analogous to that which occurs in statistical mechanics.  We describe necessary and sufficient conditions for the existence of such a phase transition and give several examples to illustrate the theorem.  This is joint work with Jeff Steif.