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arXiv · 1802.02059

One-sided continuity properties for the Schonmann projection

Stein Andreas Bethuelsen·Diana Conache
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Abstract

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.

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Stein Andreas Bethuelsen, Diana Conache. 2018-02-06. One-sided continuity properties for the Schonmann projection. https://doi.org/10.1007/s10955-018-2092-z

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