A lower bound on the two-arms exponent for critical percolation on the lattice

TL;DR

This paper improves the lower bound on the two-arms exponent for critical percolation on lattices and provides a quantitative estimate of long-range order in finite boxes, advancing understanding of percolation phase transitions.

Contribution

It refines the lower bound on the two-arms exponent for all dimensions and derives a new quantitative estimate of long-range order without slab technology.

Findings

Lower bound on two-arms exponent is slightly improved for all d ≥ 2.

Establishes long-range order in finite boxes under the condition θ(p)>0.

Provides a new approach avoiding slab technology for percolation analysis.

Abstract

We consider the standard site percolation model on the $d$-dimensional lattice. A direct consequence of the proof of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman [Comm. Math. Phys. 111 (1987) 505-531] is that the two-arms exponent is larger than or equal to $1/2$. We improve slightly this lower bound in any dimension $d\geq2$. Next, starting only with the hypothesis that $\theta(p)>0$, without using the slab technology, we derive a quantitative estimate establishing long-range order in a finite box.