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

TL;DR

This paper examines the four-arm exponent in 2D critical percolation, establishing inequalities and providing a concise, self-contained proof by synthesizing existing methods.

Contribution

It offers a new, simplified proof of the inequality for the four-arm exponent, improving understanding of critical percolation properties.

Findings

Proves $oldsymbol{oldsymbol{ ext{α}_4 > 1}}$ for 2D critical percolation

Establishes $oldsymbol{ ext{α}_4 extgreater 1 + rac{ ext{α}_2}{2}}$ as a lower bound

Provides a self-contained proof by combining existing approaches

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 $\alpha_4 \geq 1 + \frac{\alpha_2}{2}$, 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.