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International Conference on Statistical Mechanics of Classical and Disordered Systems

StaMeClaDys 2018: Statistical Mechanics of Classical and Disordered Systems pp 21–33Cite as

One-Sided Versus Two-Sided Stochastic Descriptions

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 293))

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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Notes

  1. 1.

    We denote \(\mu [f]\) for the expectation \(\mathbb {E}_\mu [f]\) under a measure \(\mu \).

  2. 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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Acknowledgements

I dedicate this paper with pleasure to Anton Bovier, on the occasion of his 60th birthday. Anton has been a reliable and stimulating guide for me in matters literary, culinary and scientific, for more than 30 years. His many-sided personality and friendship have enriched my life in many respects. I wish him many more years, and look forward to have a share therein. I thank my collaborators and discussion partners on Dyson models, long-range models and g-measures for the pleasure of these collaborations and for all I have learned from them. In alphabetical order I owe: Stein Bethuelsen, Rodrigo Bissacot, Diana Conache, Loren Coquille, Eric Endo, Roberto Fernández, Frank den Hollander, Bruno Kimura, Arnaud Le Ny, Brian Marcus, Pierre Picco, Wioletta Ruszel, Cristian Spitoni, Siamak Taati and Evgeny Verbitskiy. I thank Eric, Siamak and Arnaud for helpful advice on the manuscript. I thank Louis-Pierre Arguin, Veronique Gayrard, Nicola Kistler and Irina Kourkova for inviting me to be present at the CIRM meeting to celebrate Anton, and also to contribute to this volume.

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  1. Bernoulli Institute, University of Groningen, Nijenborgh 9, 9747AG, Groningen, The Netherlands

    Aernout C. D. van Enter

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  1. Aernout C. D. van Enter
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Correspondence to Aernout C. D. van Enter .

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Editors and Affiliations

  1. Aix-Marseille Université CNRS, Institut de Mathématiques de Marseille, Marseille, France

    Véronique Gayrard

  2. Baruch College, CUNY, New York, NY, USA

    Louis-Pierre Arguin

  3. Goethe-Universität Frankfurt, Frankfurt am Main, Hessen, Germany

    Nicola Kistler

  4. Laboratoire de Probabilités, Pierre and Marie Curie University VI, Paris, France

    Irina Kourkova

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van Enter, A.C.D. (2019). One-Sided Versus Two-Sided Stochastic Descriptions. In: Gayrard, V., Arguin, LP., Kistler, N., Kourkova, I. (eds) Statistical Mechanics of Classical and Disordered Systems . StaMeClaDys 2018. Springer Proceedings in Mathematics & Statistics, vol 293. Springer, Cham. https://doi.org/10.1007/978-3-030-29077-1_2

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