= RSW gluing lemma for two one-arm events
{c}
For critical <independent> <bond percolation> on the <square lattice>, two one-arm events can be joined at a scale-independent additional cost. With $y=(r,0)$, there is $c>0$ such that
$$
\mathbb P_{1/2}(0\leftrightarrow y)\geq c\pi(r)^2.
$$
The <one-arm probability> factors remain present: this is a lower bound proportional to the two arm <probabilities>, not a uniform positive bound for the unconditional connection <probability>.
For a concrete construction at large integer $r$, put $s=\lfloor r/12\rfloor$. Require arms from $0$ and $y$ to distance $4s$, open <graph cycles> around each root between distances $2s$ and $4s$, and a left-to-right crossing of $[-s,r+s]\times[-s,s]$. Each end of that crossing lies inside the corresponding cycle, so it crosses both cycles. Each arm also meets its cycle, and the combined configuration connects the roots.
The <Russo-Seymour-Welsh theorem> gives fixed positive lower bounds for the cycles and corridor crossing. The <Harris-FKG inequality> gives a joint lower bound $c\pi(4s)^2\geq c\pi(r)^2$. Finitely many small $r$ can be absorbed into the constant. This makes the geometric and <probability> costs of the gluing explicit.<Percolation probabilities on the square lattice by P. D. Seymour and D. J. A. Welsh>[Sections 4–5, fixed-shape crossing and gluing inequalities; two-arm construction derived here]{ref}
Back to article page
