Comparison of arm exponents in planar FK-percolation

TL;DR

This paper improves the understanding of arm exponents in planar FK-percolation by establishing strict inequalities among them, advancing the theoretical framework of critical phase transitions in statistical physics.

Contribution

It proves a polynomial improvement of the FKG inequality for arm events and establishes strict inequalities among arm exponents in critical planar FK-percolation.

Findings

Established polynomial bounds for arm event probabilities.

Proved strict inequalities among arm exponents.

Extended results to the entire critical regime for q between 1 and 4.

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 $\alpha_{01}$ and the one-arm exponents $\alpha_0$ and $\alpha_1$ exist, then they satisfy the strict inequality $\alpha_{01} > \alpha_0 + \alpha_1$. 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.