On the four-arm exponent for 2D percolation at criticality

Abstract

For two-dimensional percolation at criticality, we discuss the inequality ${\alpha }_4>1$ for the polychromatic four-arm exponent (and stronger versions, the strongest so far being ${\alpha }_4\ge 1+\frac{{\alpha }_2}{2}$, where ${\alpha }_2$ denotes the two-arm exponent). We first briefly discuss five proofs (some of them implicit and not self-contained) from the literature. Then we observe that, by combining two of them, one gets a completely self-contained (and yet quite short) proof.

23 pages, 3 figures; in memory of Vladas Sidoravicius