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.
"A lower bound on the two-arms exponent for critical percolation on the lattice." Ann. Probab. 43 (5) 2458 - 2480, September 2015. https://doi.org/10.1214/14-AOP940