= Russo-Seymour-Welsh theorem
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= RSW lemma
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{synonym}
For <independent> <bond percolation> on the <square lattice> at $p=1/2$, the <Russo-Seymour-Welsh theorem> gives nontrivial crossings at every scale. For each fixed aspect ratio $\rho\geq1$, there is $c_\rho>0$ such that
$$
c_\rho\leq h_{1/2}(\lceil\rho n\rceil,n)\leq1-c_\rho\qquad(n\geq1),
$$
where $h_p(m,n)$ is the left-to-right <rectangle crossing in bond percolation> <probability>. The constant can depend on $\rho$, but not on the rectangle's size.
Reflection extensions and transverse crossings in overlapping rectangles permit longer crossings to be assembled from square crossings. The <Harris-FKG inequality> controls intersections of these <increasing events>; <dual bond percolation> supplies the complementary crossing bound. These are the geometric ingredients of the original square-lattice crossing analysis.<Percolation probabilities on the square lattice by P. D. Seymour and D. J. A. Welsh>[Sections 4–5, sponge crossings and Lemmas 5.2–5.4; modern fixed-aspect formulation]{ref}
For example, a rectangle twice as wide as it is high retains a positive crossing <probability> as both dimensions grow. The bounds imply positive <probabilities> for open or closed circuits in fixed-shape annuli and support the <one-arm extension estimate>. Their critical-parameter assumption is essential: a corresponding uniform lower bound would contradict <exponential decay of subcritical percolation>.
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