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Markov and almost Markov properties in one, two or more directions. (English) Zbl 1475.60202

Summary: In this review-type paper written at the occasion of the Oberwolfach workshop One-sided vs. Two-sided stochastic processes (February 22–29, 2020), we discuss and compare Markov properties and generalisations thereof in more directions, as well as weaker forms of conditional dependence, again either in one or more directions. In particular, we discuss in both contexts various extensions of Markov chains and Markov fields and their properties, such as \(g\)-measures, variable length Markov chains, variable neighbourhood Markov fields, variable neighbourhood (parsimonious) random fields, and generalized Gibbs measures.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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