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

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≥2. Next, starting only with the hypothesis that θ(p)>0, without using the slab technology, we derive a quantitative estimate establishing long-range order in a finite box.