= One-arm extension estimate
In <independent> <bond percolation> on the <square lattice> at $p=1/2$, write $\pi(r)$ for the <one-arm probability> to maximum-norm distance $r$. There is a constant $c>0$, <independent> of $r$, such that
$$
\pi(2r)\geq c\pi(r).
$$
To see the geometry, require an <open path> to distance $r$, an open <graph cycle> surrounding the origin in an annulus between distances about $r/2$ and $3r/4$, and a long crossing from inside that cycle to distance $2r$. The initial arm and the long crossing both intersect the cycle, producing a connected route to the larger distance.
The <Russo-Seymour-Welsh theorem> bounds the cycle and long-crossing <probabilities> below by positive constants because their aspect ratios remain fixed. All the required events are increasing, so the <Harris-FKG inequality> bounds their joint <probability> below by a constant times $\pi(r)$. Integer rounding affects only the constants. The argument extends an existing arm with uniformly positive conditional cost; it does not bound the arm event itself below by a positive constant.<Percolation probabilities on the square lattice by P. D. Seymour and D. J. A. Welsh>[Section 5, rectangle extensions and gluing; annular arm extension derived here]{ref}
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