A new perspective on the equivalence between Weak and Strong Spatial Mixing in two dimensions

TL;DR

This paper offers a new proof and extension of the equivalence between weak and strong spatial mixing in two-dimensional lattice models, using a percolative perspective on information propagation.

Contribution

It provides a novel proof, extends the class of models where the equivalence holds, and introduces a percolative framework for understanding information flow.

Findings

New proof of weak implying strong mixing in 2D models.

Extension of the equivalence to broader classes of models.

Percolative interpretation of information propagation in lattice systems.

Abstract

Weak mixing in lattice models is informally the property that ``information does not propagate inside a system''. Strong mixing is the property that ``information does not propagate inside and on the boundary of a system''. In dimension two, the boundary of reasonable systems is one dimensional, so information should not be able to propagate there. This led to the conjecture that in 2D, weak mixing implies strong mixing. The question was investigated in several previous works, and proof of this conjecture is available in the case of finite range Gibbsian specifications, and in the case of nearest-neighbour FK percolation (under some restrictions). The present work gives a new proof of these results, extends the family of models for which the implication holds, and, most interestingly, provides a ``percolative picture'' of the information propagation.