Comparison of arm exponents in planar FK-percolation

TL;DR

The paper addresses comparing decay rates of arm events in critical planar FK-percolation and establishes a polynomial-factor improvement of the alternating two-arm inequality, namely $\phi_{p_c,q}[A_{01}(r,R)] \le (r/R)^c\,\phi_{p_c,q}[A_1(r,R)]\phi_{p_c,q}[A_0(r,R)]$, which, if the one-arm and alternating two-arm exponents exist, implies $\alpha_{01} > \alpha_0+\alpha_1$. The authors develop an explicit increasing coupling between conditioned and unconditioned configurations using well-separated flower domains and a density of good scales, showing that conditioning on a dual arm imposes a polynomial extra cost on the primal arm. This leads to the key result that the alternating two-arm exponent strictly exceeds the sum of the one-arm exponents, $\alpha_{01} > \alpha_0+\alpha_1$, under the usual existence assumptions. Importantly, the method provides a general technique to extract polynomial factors in FKG-type arm inequalities without relying on conjectured scaling limits, with potential implications for exceptional times and related models.

Abstract

By the FKG inequality for FK-percolation, the probability of the alternating two-arm event is smaller than the product of the probabilities of having a primal arm and a dual arm, respectively. In this paper, we improve this inequality by a polynomial factor for critical planar FK-percolation in the continuous phase transition regime ($1 \leq q \leq 4$). In particular, we prove that if the alternating two-arm exponent $α_{01}$ and the one-arm exponents $α_0$ and $α_1$ exist, then they satisfy the strict inequality $α_{01} > α_0 + α_1$. The question was formulated by Garban and Steif in the context of exceptional times and was brought to our attention by Radhakrishnan and Tassion, who obtained the same result for planar Bernoulli percolation through different methods.

Figures (4)

• Figure 1: An inner flower domain, an outer flower domain and a double four-petal flower domain. The plain and dashed lines represent primal and dual paths respectively.
• Figure 2: A depiction of a configuration in which the scale $k$ is good.
• Figure 3: In grey the explored region $E$. If $H \cap H' \cap A_{01}(2^m,2^{k}) \cap A_{01}(2^{k+1},2^{n})$ occurs in $E$, then the additional red connections outside of $E$ are sufficient to ensure that $G_k \cap A_{01}(2^m,2^{n})$ occurs.
• Figure 4: A configuration with some good scales in series. The central square is $\Lambda_{2^m}$ and each annulus represent a different scale. The grey region is the complementary of the explored region $E$. It is the union of all the flower domains lying within a good scale.

Theorems & Definitions (10)

• Theorem 1.1
• Remark 1.2
• Definition 2.1
• Lemma 2.2: DM2022
• Definition 3.1
• Lemma 3.2: Positive density of good scales
• proof : Proof of Lemma \ref{['lem:density_good_scales']}
• Lemma 4.1
• proof
• proof : Proof of Theorem \ref{['thm:comparison_exponents']}