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

TL;DR

The paper proves that in two dimensions, exponential weak mixing implies exponential strong mixing under a broad set of hypotheses by linking information propagation to a subcritical inhomogeneous Bernoulli percolation on a coarse-grained lattice. It introduces a detailed patch-exploration and coarse-graining scheme that builds a coupling between boundary-conditioned and unconditioned blocks, reducing dependence to a percolation event surrounding the region of interest. The main theorem then yields concrete strong-mixing conclusions for three major model families—finite-range Gibbsian specifications, FK percolation, and hard-core models—via the percolation-control mechanism, and it also outlines how ratio-mixing results can relax the initial hypotheses. This percolative viewpoint provides a true picture of how information travels through 2D systems and offers a pathway to extend strong mixing results beyond classical finite-range settings. The work thus advances the understanding of 2D mixing phenomena and broadens the reach of strong mixing results in statistical mechanics models.

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.

Figures (8)

• Figure 1: A volume $\Lambda$ (circular dots), and its inner boundary $\partial^{\mathrm{i}}\Lambda$ (orange dots). Crosses are $\mathbb{Z}^2\setminus \Lambda$.
• Figure 2: Left: a star-path (black disks) and the set it surrounds (yellow area, the surrounded sites of $\mathbb{Z}^2$ are crosses). Right: a collection of paths (black disks) with the set they surround (in yellow); blue is separated form green by the paths, red is not separated from blue, and the set of blue-surrounded sites are the crosses inside the left and top right paths.
• Figure 3: Blue blocks are explored first, then orange and finally red sites. Weak mixing is used to show that sampling blue and orange blocks away from the system boundary is done uniformly over the past with high probability. The entropy of the red-orange junctions between blue blocks compensates for the use of finite energy on the red sites.
• Figure 4: Illustration of the partitioning with $r=L=2$, $a=1$. The "elementary cell" of the partitioning is depicted in orange. The sets $E_{ij}^k$ correspond to the rectangular pink boxes, the sets $F_{ij}^k$ to rectangular boxes plus the sites between the rectangular box and the neighbouring square boxes, and the sets $H_{ij}$ are the sets of sites between two square boxes. Grey dots represent sites where $X=\psi(\sigma)$ lives.
• Figure 5: The set $\varrho(\{i,j\},i,1)$ is given by the red sites in $E_{ij}^1$ while the set $\varrho(\{i,j\}, j,1)$ is given by the blue sites in $E_{ij}^1$. The sets $\varrho(i,j,1)$ and $\varrho(j,i,1)$ are respectively given by the red sites in $\mathrm{B}(i)$ and the blue sites in $\mathrm{B}(j)$. Here, $L=3, a=2$. Grey dots represent sites where $X=\psi(\sigma)$ lives.

Theorems & Definitions (37)

• Example 1.1
• Remark 1.1
• Remark 1.2
• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• proof : Proof of Theorem \ref{['thm:weakMix_to_ratioWeakMix']}
• Definition 1
• Lemma 2.1
• proof