One-Sided Continuity Properties for the Schonmann Projection

• 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
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-018-2092-z

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.