 Spin-connectivity identity for the Potts model

For integer $q\ge 2$, 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, 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.