Bernoulli bounds for the random-cluster model (${\varphi }_{p^{\prime },1}{\le }_{\mathrm{st}}{\varphi }_{p,q}{\le }_{\mathrm{st}}{\varphi }_{p,1},p^{\prime }=p/(p+q(1-p))$)

For $q\ge 1$, the random-cluster measure lies between the two Bernoulli product measures displayed above. The upper Holley condition ratio is $q^{k(\omega \wedge \eta )-k(\eta )}\ge 1$. The lower ratio is $q^{k(\omega \vee \eta )-k(\omega )+|\eta \setminus \omega |}\ge 1$, because adding an edge reduces the component count by at most one. The comparisons persist under wired limits and give $p_c(1)\le p_c(q)\le q{p}_c(1)/(1+(q-1)p_c(1))$.