Title: Stationary Determinantal Processes


Jeff Steif

Department of Mathematics
Chalmers University of Technology


Abstract:  Given a function f on the d-dimensional torus with values
in the unit interval, there is a 2-state stationary random field on
the d-dimensional integer lattice that is defined via minors of the
d-dimensional Toeplitz matrix of the function f.  The variety of such
systems includes certain combinatorial models, certain finitely
dependent models, and certain 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 on f for the existence of such a phase
transition and give several examples to illustrate the theorem.

[Since these processes are somewhat different than the types of processes
that probabilists tend to look at (at least different than the types
of processes that I usually look at), I will spend a good deal of time
on different examples. In addition, terms like "Toeplitz matrix" and
things like this will be explained and not assumed. This is joint
work with Russ Lyons.