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Use consistently wired boundary conditions in increasing finite boxes and let ${\varphi }_{p,q}^{\mathrm{w}}$ be the usual wired infinite-volume random-cluster measure. The finite-volume comparison just proved applies with boundary graph vertices identified, since it is still a comparison on a finite graph. Passing to the infinite-volume limit preserves the inequality for every increasing event determined by finitely many edges.

For fixed $m$, let $A_m$ be the event that the origin connects by an open path in $[-m,m{]}^2$ to its boundary. It is such a finite increasing event. Therefore, for $q_1\le q_2$,The events decrease to the event that the origin has an infinite percolation cluster, by local finiteness. Continuity from above of a measure givesDefine the random-cluster critical probability by $p_c(q)=\inf \{p:{\theta }^{\mathrm{w}}(p,q)>0\}$. The set in which percolation occurs at $q_2$ is a subset of that at $q_1$, whenceThe same proof works if free boundary conditions are used consistently throughout. It compares critical probabilities; it does not presume whether an infinite percolation cluster exists at the critical parameter itself.

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.