= Boundary monotonicity of the random-cluster measure
{title2=$\xi\preceq\zeta\ \Longrightarrow\ \phi^\xi_{\Lambda,p,q}\preceq\phi^\zeta_{\Lambda,p,q}\quad(q\geq1)$}
Here $\xi\preceq\zeta$ means each block of $\xi$ lies in a block of $\zeta$, and the measure order is <stochastic domination of probability measures>. The <random-cluster single-edge conditional probability> is increasing in both exterior open edges and wiring for $q\geq1$. Couple two <heat-bath Markov chains> by identical update edges and uniform random variables, starting from ordered states. Order persists, and convergence of the finite chains to their stationary laws proves the displayed inequality. At $q=1$, the law does not depend on the boundary partition.
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