Stationary Determinantal Processes: Phase Multiplicity, Bernoullicity, Entropy, and Domination

Details Stationary Determinantal Processes: Phase Multiplicity, Bernoullicity, Entropy, and Domination Stationary Determinantal Processes: Phase Multiplicity, Bernoullicity, Entropy, and Domination

2002-04-26

arxiv.org/abs/math/0204324v5

Duke Math. J. 120, no. 3 (2003), 515-575.

Mathematics

Probability

56 pp

Scientific paper

We study a class of stationary processes indexed by $\Z^d$ 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 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.

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