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Identify a configuration with its set $S$ of open edges. Let $k(S)$ count the connected components of a graph on the full graph vertex set, including isolated graph vertices. The random-cluster measure isFor $p\in (0,1)$ and finite $q\ge 1$ all its weights are positive. We compare $1\le q_1\le q_2$, taking ${\mu }_1={\mu }_{p,q_1}$ and ${\mu }_2={\mu }_{p,q_2}$.

The key fact is supermodularity of graph component count:To prove it, orient the graph edges arbitrarily and let $U_S$ be the real span of their incidence vectors for edges in $S$. Its dimension of a vector space is $r(S)=|V|-k(S)$: on each connected component of a graph these vectors span the vectors whose coordinates sum to zero. Since $U_{S\cup T}=U_S+U_T$ and $U_{S\cap T}\subset U_S\cap U_T$, the dimension formula for a sum of subspaces giveswhich is exactly the component-count inequality.

Put $a=k(S)-k(S\cup T)\ge 0$ and $b=k(S\cap T)-k(T)\ge 0$. The preceding inequality gives $b\ge a$. In the ratio of the two sides of Holley's condition, the normalization constants and Bernoulli edge factors cancel, because $|S\cup T|+|S\cap T|=|S|+|T|$. The ratio is thereforeBy the Holley inequality, ${\mu }_{p,q_1}$ stochastically dominates ${\mu }_{p,q_2}$. Thus increasing cluster weight suppresses every increasing event:The random-cluster single-edge conditional probability confirms the mechanism: it is $p$ when the endpoints are already connected, and $p/(p+q(1-p))$ otherwise. Larger $q$ favours configurations with more separate components. The condition $q\ge 1$ enters explicitly in the last factor $q_2^{b-a}$.

For fixed $p\in (0,1)$ and $1\le q_1\le q_2$, the random-cluster measure at $q_1$ stochastically dominates that at $q_2$. In Holley's condition, Bernoulli factors cancel. Put $a=k(S)-k(S\cup T)$ and $b=k(S\cap T)-k(T)$; supermodularity of graph component count gives $b\ge a\ge 0$, and the weight ratio is $(q_2/q_1{)}^a{q}_2^{b-a}\ge 1$. The comparison also holds with the same boundary wiring in both measures.