Random-cluster boundary condition ($\xi \text{ partitions }\partial \Lambda$)

A boundary condition for a finite-box random-cluster measure is a partition of the boundary vertices. All vertices in one block are identified before counting open components; the identifications carry no Bernoulli edge factors. The weight is $p^{o(\omega )}(1-p{)}^{|E|-o(\omega )}{q}^{k_{\xi }(\omega )}$. A coarser partition makes more identifications. General partitions need not have a planar realization.

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.

All boundary vertices are identified into one vertex for component counting. This is the most wired random-cluster boundary condition on a fixed boundary, and maximizes increasing-event probabilities when $q\ge 1$ by boundary monotonicity of the random-cluster measure.

The boundary partition has singleton blocks, so no different boundary vertices are identified. Open component counting is the ordinary counting in the finite graph, including isolated vertices. This is the least wired random-cluster boundary condition.