One-Sided Continuity Properties for the Schonmann Projection

We consider the plus-phase of the two-dimensional Ising model below the critical temperature. In 1989 Schonmann proved that the projection of this measure onto a one-dimensional line is not a Gibbs measure. After many years of continued research which have revealed further properties of this measure...

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Bibliographic Details
Published in:Journal of statistical physics 2018-08, Vol.172 (4), p.1147-1163
Main Authors: Bethuelsen, Stein Andreas, Conache, Diana
Format: Article
Language:English
Subjects:
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
COUPLING
CRITICAL TEMPERATURE
ISING MODEL
Mathematical and Computational Physics
ONE-DIMENSIONAL CALCULATIONS
Physical Chemistry
Physics
Physics and Astronomy
Projection
Properties (attributes)
Quantum Physics
SIMULATION
Statistical Physics and Dynamical Systems
Theoretical
Two dimensional models
TWO-DIMENSIONAL CALCULATIONS
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Summary:We consider the plus-phase of the two-dimensional Ising model below the critical temperature. In 1989 Schonmann proved that the projection of this measure onto a one-dimensional line is not a Gibbs measure. After many years of continued research which have revealed further properties of this measure, the question whether or not it is a Gibbs measure in an almost sure sense remains open. In this paper we study the same measure by interpreting it as a temporal process. One of our main results is that the Schonmann projection is almost surely a regular g -measure. That is, it does possess the corresponding one-sided notion of almost Gibbsianness. We further deduce strong one-sided mixing properties which are of independent interest. Our proofs make use of classical coupling techniques and some monotonicity properties which are known to hold for one-sided, but not two-sided conditioning for FKG measures.
ISSN:0022-4715
1572-9613
DOI:10.1007/s10955-018-2092-z