Spin-connectivity identity for the Potts model ($\pi ({\sigma }_x={\sigma }_y)=q^{-1}+(1-q^{-1})\varphi (x\leftrightarrow y)$)

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 $(q{\pi }^1({\sigma }_0=1)-1)/(q-1)={\varphi }^{\mathrm{w}}(0\leftrightarrow \infty )$ in infinite volume. Thus its onset is the wired random-cluster critical probability, transformed to ${\beta }_c=-\log (1-p_c)$ in this convention.