Duke Mathematical Journal

Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination

Russell Lyons and Jeffrey E. Steif

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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.

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Duke Math. J., Volume 120, Number 3 (2003), 515-575.

First available in Project Euclid: 16 April 2004

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Zentralblatt MATH identifier

Primary: 82B26: Phase transitions (general) 28D05: Measure-preserving transformations 60G10: Stationary processes
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 37A05: Measure-preserving transformations 37A25: Ergodicity, mixing, rates of mixing 37A60: Dynamical systems in statistical mechanics [See also 82Cxx] 60G25: Prediction theory [See also 62M20] 60G60: Random fields 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization


Lyons, Russell; Steif, Jeffrey E. Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination. Duke Math. J. 120 (2003), no. 3, 515--575. doi:10.1215/S0012-7094-03-12032-3. https://projecteuclid.org/euclid.dmj/1082137353

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