 Square-lattice one-arm lower bound from disjoint occurrence

At density $1/2$, planar duality gives crossing probability $1/2$ for the $2n$-by-$(2n-1)$ lattice rectangle. A self-avoiding left-right crossing passes through one of its $2n$ middle-column vertices. At that vertex its two halves are disjoint edge witnesses for two copies of the radius-$n$ connection event. The BK inequality and the union bound give $1/2\le 2n{\mathbb{P}}_{1/2}(0\leftrightarrow \partial [-n,n{]}^2{)}^2$, proving the estimate.