Comparison of arm exponents in planar FK-percolation

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 Christophe Garban and Jeffrey E. Steif (2014) in the context of exceptional times and was brought to our attention by Ritvik Ramanan Radhakrishnan and Vincent Tassion, who obtained the same result for planar Bernoulli percolation through different methods.