= Spin-connectivity identity for the Potts model
{title2=$\pi(\sigma_x=\sigma_y)=q^{-1}+(1-q^{-1})\phi(x\leftrightarrow y)$}
For integer $q\geq2$, given a <random-cluster> configuration, its components receive <independent> uniform colors under the <Edwards-Sokal coupling>. Two <graph vertices>[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>, with dimensionless <Potts model> coupling $K_c=-\log(1-p_c)$ in the convention of the <Edwards-Sokal coupling>.
For $q=1$, the equal-spin <probability> is identically one and the first identity reduces to $1=1$. The normalized magnetization expression divides by $q-1$ and therefore does not apply to that degenerate one-color case. The conversion to a physical <Ising model> <inverse temperature> must also account for the interaction convention.
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