= Square-lattice one-arm lower bound from disjoint occurrence
{title2=$\mathbb P_{1/2}(0\leftrightarrow\partial[-n,n]^2)\geq(2\sqrt n)^{-1}$}
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 $1/2\leq2n\mathbb P_{1/2}(0\leftrightarrow\partial[-n,n]^2)^2$, proving the estimate.
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