Abstract
We study the one-dimensional projection of the extremal Gibbs measures of the two-dimensional Ising model – the “Schonmann projection”. These measures are known to be non-Gibbsian at low temperatures, since their conditional probabilities as a function of the (two-sided) boundary conditions are not continuous. We prove the conjecture that they are g-measures, which means that their conditional probabilities have a continuous dependence on one-sided boundary conditions.
Nous étudions la projection unidimensionnelle des mesures extrémales de Gibbs du modèle d’Ising bidimensionnel – la “projection Schonmann”. Ces mesures sont connues pour être non-Gibbsiennes à basses températures, puisque leurs probabilités conditionnelles en fonction des conditions au bord (bilatérales) ne sont pas continues. Nous prouvons la conjecture que néanmoins ce sont des g-mesures, ce qui signifie que leurs probabilités conditionnelles dépendent de façon continue des conditions au bord unilatérales.
Funding Statement
Part of the work of S.S. has been carried out at Skoltech and at IITP RAS. The support of Russian Science Foundation (projects No. 14-50-00150 and 20-41-09009) is gratefully acknowledged.
Dedication
Dedicated to the memory of our dear friend Dima Ioffe
Acknowledgements
S.S. thanks Serge Pirogov, Yvan Velenik and Sebastien Ott for enlightening discussions. A.v.E. thanks the participants of the Oberwolfach miniworkshop [1], for various discussions as well as collaborations on related issues, and the International Emerging Action “Long-Range” of the CNRS to make his participation in the miniworkshop possible. He also thanks Rodrigo Bissacot, Eric Endo and Roberto Fernández for earlier collaborations and various discussions on the issue of g-measures versus Gibbs measures. Both of the authors are grateful to Giambattista Giacomin for his helpful comments.
Citation
Aernout van Enter. Senya Shlosman. "The Schonmann projection: How Gibbsian is it?." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 2 - 10, February 2024. https://doi.org/10.1214/22-AIHP1266
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