Duke Math. J. 120 (3), 515-575, (1 December 2003) DOI: 10.1215/S0012-7094-03-12032-3
Russell Lyons, Jeffrey E. Steif
KEYWORDS: 82B26, 28D05, 60G10, 82B20, 37A05, 37A25, 37A60, 60G25, 60G60, 60B15
We study a class of stationary processes indexed by ℤd that are defined via minors of d-dimensional (multilevel) Toeplitz matrices. We obtain necessary and sufficient conditions for phase multiplicity (the existence of a phase transition) analogous to that which occurs in statistical mechanics. Phase uniqueness is equivalent to the presence of a strong K-property, a particular strengthening of the usual K (Kolmogorov) property. We show that all of these processes are Bernoulli shifts (isomorphic to independent identically distributed (i.i.d.) processes in the sense of ergodic theory). We obtain estimates of their entropies, and we relate these processes via stochastic domination to product measures.