Percolation without FKG

Technical Summary: Percolation Without FKG

Problem Statement
The paper addresses a fundamental gap in the theory of two-dimensional statistical mechanics: the proof of the Box-Crossing Property (BXP), also known as the Russo-Seymour-Welsh (RSW) theorem, for models that do not satisfy the Fortuin-Kasteleyn-Ginibre (FKG) condition of positive association.

Standard proofs of the BXP rely heavily on the FKG inequality, which ensures that increasing events are positively correlated. This property is crucial for constructing percolation crossings of long rectangles from elementary blocks and for defining pure phases. While the BXP is well-established for Bernoulli percolation and positively associated models (like the ferromagnetic Ising model), it has remained unproven for models lacking positive association, such as the antiferromagnetic Ising model or certain negatively correlated continuous models. The authors aim to establish the BXP for a class of discrete percolation models that exhibit the domain Markov property and fast decorrelation, specifically targeting the high-temperature (small $\beta$) antiferromagnetic Ising model on a periodic planar triangulation.

Methodology
The authors develop a novel proof strategy that replaces the FKG inequality with a combination of topological surgery, precise control of correlations, and a renormalization scheme. The core components of their methodology include:

1. Topological Surgery and Random Quads: Instead of relying on positive association to glue crossing events, the authors utilize geometric operations on "quads" (quadrilaterals defined by paths on the lattice). They employ lemmas regarding the "shaving" of quads and the concatenation of paths to relate crossing probabilities of different geometries.
2. Oscillation and Decorrelation: The proof introduces a measure of "oscillation" ($\delta(Q) = \pi^+(Q) - \pi^-(Q)$), representing the variation in crossing probabilities under different boundary conditions. The authors assume the model is "well-decorrelated," meaning the influence of boundary conditions decays rapidly (specifically, $\alpha(\ell, L) \leq C L^2 \ell^{-2-\gamma}$). This allows them to control the oscillation and show that crossing probabilities become stable at large scales.
3. Scale Induction (Renormalization): The central argument is a scale induction scheme. The authors define a quantity $\beta(r, R)$ representing the supremum of crossing probabilities for quads crossing an annulus of radii $r$ and $R$. They prove a "last scale" lemma (Lemma 35) showing that if a model is "good" (satisfies a power-law bound on $\beta$) up to a scale $R$, and possesses sufficient decorrelation, it remains "good" up to scale $2R$. This propagation relies on the fact that decorrelation ensures that the probability of a crossing in a large annulus is bounded away from 1 by the probability of a dual circuit, which is controlled by the oscillation estimates.
4. Interpolation Argument: To apply these results to the Ising model, the authors use a continuity argument. They consider a family of models interpolating between critical Bernoulli percolation (where BXP is known) and the target model (antiferromagnetic Ising with small $\beta$). By showing that the "goodness" property is open with respect to the parameter (due to the domain Markov property and continuity of specifications), they extend the BXP from the known case to the target model.

Key Contributions and Results

• Theorem 1 (Ising Model): The paper proves that for the Ising model on the face-centered square lattice, there exists a threshold $\beta_0 > 0$ such that for all inverse temperatures $\beta \in (-\beta_0, \beta_0)$, the Ising measure satisfies the box-crossing property. Consequently, the model does not percolate; almost surely, all clusters of constant spin are finite.
• General Framework (Theorem 43): The authors establish a general theorem for a one-parameter family of models $(P_u)$. If the family satisfies conditions including uniqueness, symmetry, duality, the domain Markov property, continuity, and uniform decorrelation, and if the base model $P_0$ (critical Bernoulli percolation) satisfies the BXP, then there exists a neighborhood of $P_0$ where all models satisfy the BXP and do not percolate.
• Scale Induction (Theorem 37): A rigorous proof is provided that a model which is "good" (satisfies a specific power-law decay of crossing probabilities) up to a certain scale and is well-decorrelated, remains good at all larger scales. This removes the need for FKG by substituting it with quantitative decorrelation estimates.
• Polynomial Decay: The authors demonstrate that for these "good" and well-decorrelated models, the probability of a connection from the origin to distance $R$ decays polynomially ($R^{-c}$), confirming the absence of infinite clusters.

Significance and Claims
The paper claims to provide the first proof of the box-crossing property in the standard statistical mechanics setup (fixed models in infinite volume) that does not utilize the FKG inequality.

• Breaking the FKG Barrier: The work demonstrates that positive association is not strictly necessary for the existence of critical behavior characterized by the BXP. Instead, the combination of the domain Markov property and fast decorrelation is sufficient to propagate estimates across scales.
• Universality Class: The results suggest that the high-temperature Ising model (both ferromagnetic and antiferromagnetic) belongs to the same universality class as Bernoulli percolation regarding the existence of the BXP and the absence of percolation.
• Limitations and Modesty: The authors explicitly state that while the BXP is proven, other consequences (such as the exact values of universal exponents or the tightness of interfaces) are expected to follow by similar arguments but are not included in this paper. They also note that their current approach is perturbative; it relies on the model being close to a product measure (small $\beta$) to utilize the continuity argument. They acknowledge that extending these results to a non-perturbative regime or to models with fewer symmetries (like bond percolation on the square lattice) remains an open question beyond the scope of their current methods.

In summary, the paper establishes a robust framework for analyzing critical phenomena in dependent percolation models without positive association, successfully applying it to the antiferromagnetic Ising model and laying the groundwork for future investigations into the universality of critical exponents in this broader class of models.