= Reflection comparison of random-cluster connections
Choose a <graph vertex> $x=(n,x_2,\ldots,x_d)$ on the boundary of a <lattice> box. Translation and reflection invariance identify the <probabilities> of $0\leftrightarrow x$ and $x\leftrightarrow2ne_1$. Their intersection implies $0\leftrightarrow2ne_1$, so <positive association of random variables> gives $C_{2n}\ge\phi(0\leftrightarrow x)^2$. A <union bound> and <lattice> symmetries choose such an $x$ with <probability> at least the box-boundary connection <probability> divided by the number of boundary <graph vertices>. This polynomial factor disappears in the exponential rate, identifying boundary and axial inverse correlation lengths.
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