Boundary monotonicity of the random-cluster measure ($\xi ⪯\zeta ⟹{\varphi }_{\Lambda ,p,q}^{\xi }⪯{\varphi }_{\Lambda ,p,q}^{\zeta }(q\ge 1)$)

Here $\xi ⪯\zeta$ 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 $q\ge 1$. 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.