One-sided continuity properties for the Schonmann projection

TL;DR

This paper investigates the Schonmann projection of the 2D Ising model's plus-phase, showing it is almost surely a regular g-measure with strong one-sided mixing properties, using coupling and monotonicity techniques.

Contribution

It proves the Schonmann projection is almost surely a regular g-measure, establishing one-sided Gibbsianness and mixing properties for this measure.

Findings

Schonmann projection is almost surely a regular g-measure.

Establishes strong one-sided mixing properties.

Uses coupling and monotonicity techniques for proof.

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.