= Bernoulli bounds for the random-cluster model
{c}
{title2=$\phi_{p',1}\leq_{\rm st}\phi_{p,q}\leq_{\rm st}\phi_{p,1},\quad p'=p/(p+q(1-p))$}
For $q\geq1$, the <random-cluster measure> lies between the two <Bernoulli> product measures displayed above. The upper <Holley condition> ratio is $q^{k(\omega\wedge\eta)-k(\eta)}\geq1$. The lower ratio is $q^{k(\omega\vee\eta)-k(\omega)+|\eta\setminus\omega|}\geq1$, because adding an edge reduces the component count by at most one. The comparisons persist under wired limits and give $p_c(1)\leq p_c(q)\leq qp_c(1)/(1+(q-1)p_c(1))$.
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