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Erschienen in:
Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

One-Sided Versus Two-Sided Stochastic Descriptions

verfasst von : Aernout C. D. van Enter

Erschienen in: Statistical Mechanics of Classical and Disordered Systems

Verlag: Springer International Publishing

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Abstract

It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearest-neighbor Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or “Dyson”) models.

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Vorheriges Kapitel Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics
Nächstes Kapitel The Free Energy of the GREM with Random Magnetic Field
Fußnoten
1
We denote \(\mu [f]\) for the expectation \(\mathbb {E}_\mu [f]\) under a measure \(\mu \).
 
2
Expressing that \(\forall \Lambda \in \mathcal {S},\; \forall A \in \mathcal {F}_\Lambda \), \(\rho (A)>0\) implies that \(\gamma _\Lambda (A | \omega ) >0\) for any \(\omega \in \Omega \).
 
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Metadaten
Titel
One-Sided Versus Two-Sided Stochastic Descriptions
verfasst von
Aernout C. D. van Enter
Copyright-Jahr
2019
Buch
Statistical Mechanics of Classical and Disordered Systems

Print ISBN: 978-3-030-29076-4

Electronic ISBN: 978-3-030-29077-1

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-29077-1_2