 Random-cluster inverse correlation length

Positive association of random variables and translation invariance give $C_{m+n}\ge C_m{C}_n$ for axis connection probabilities. The Fekete lemma applied to $-\log C_n$ proves the displayed limit of a sequence and the bound $C_n\le e^{-n\alpha }$. The lower single-edge conditional bound gives $C_n\ge [p/(p+q(1-p)){]}^n$, so the decay rate is finite and nonnegative. It may be zero.