RSW reflection extension lemma
ID: rsw-reflection-extension-lemma
For independent bond percolation on the square lattice at , square crossing probabilities bounded below by self-duality extend to a uniform positive bound for rectangles of aspect ratio . Expose an extremal vertical square crossing, reflect its geometry, and attach it to an exterior side through an unexposed independent region. Reflection symmetry bounds the attachment probability below. Two attachments and a horizontal square crossing join by the Harris-FKG inequality. Overlapping rectangles and transverse overlap-square crossings then give all fixed aspect ratios. Reflecting a geometrical graph path does not require its reflected edges to be open.
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