×

A Markov random field which is \(K\) but not Bernoulli. (English) Zbl 0939.60036

The author constructs a \(Z^2\) Markov random field which is of completely positive entropy but is not isomorphic to a Bernoulli shift. The basis of this example is a discrete time exclusion process which was originally studied by Yaguchi. The author also proves that such random field does not belong to the class of loosely Bernoulli transformations.

MSC:

60G60 Random fields
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Conze, J. P., Entropie d’un groupe abélian de transformations, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 25, 11-30 (1972) · Zbl 0261.28015 · doi:10.1007/BF00533332
[2] Feldman, J., New K-automorphisms and a problem of Kakutani, Israel Journal of Mathematics, 24, 16-38 (1976) · Zbl 0336.28003 · doi:10.1007/BF02761426
[3] Friedman, N. A.; Ornstein, D., On isomorphism of weak Bernoulli transformations, Advances in Mathematics, 5, 365-394 (1970) · Zbl 0203.05801 · doi:10.1016/0001-8708(70)90010-1
[4] F. den Hollander and J. Steif,On K-automorphisms, Bernoulli shifts and Markov random fields, Ergodic Theory and Dynamical Systems, to appear. · Zbl 0949.60501
[5] Ledrappier, F., Un champ markovien peut être d’entropie nulle et mélangeant, Comptes Rendus de l’Académie des Sciences, Paris, 287, A561-A563 (1978) · Zbl 0387.60084
[6] Mejilson, I., Mixing properties of a class of skew-products, Israel Journal of Mathematics, 19, 266-270 (1974) · Zbl 0305.28008 · doi:10.1007/BF02757724
[7] Yaguchi, H., Stationary measures for an exclusion process on one-dimensional lattices with infinitely many hopping sites, Hiroshima Mathematical Journal, 16, 449-475 (1986) · Zbl 0615.60097
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.