= Spin-connectivity identity for the Potts model
{title2=$\pi(\sigma_x=\sigma_y)=q^{-1}+(1-q^{-1})\phi(x\leftrightarrow y)$}
Given a <random-cluster> configuration, its components receive independent uniform colors under the <Edwards-Sokal coupling>. Two vertices have equal <Potts> spins with probability $1$ if connected and $1/q$ otherwise. Averaging proves the identity. With a wired component fixed to color $1$, the same argument gives normalized <spontaneous magnetization> $(q\pi^1(\sigma_0=1)-1)/(q-1)=\phi^{\rm w}(0\leftrightarrow\infty)$ in infinite volume. Thus its onset is the wired <random-cluster critical probability>, transformed to $\beta_c=-\log(1-p_c)$ in this convention.
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