Finite FK random-cluster weights for real q and boundary partitions

Let $G=(V,E,\mathrm{\partial })$ be a finite undirected multigraph: $V$ and the edge-label set $E$ are finite, and each $e\in E$ has a singleton or two-element endpoint set $\mathrm{\partial }e\subseteq V$. Distinct labels may have the same endpoint set. Let $B\subseteq V$ be a designated boundary set and let $\xi$ be an equivalence relation on $B$; vertices in each $\xi$-class are wired together, while vertices outside $B$ remain separate. For $A\subseteq E$, let $k_{\xi }(A)$ be the number of connected components, including isolated vertices, after wiring the $\xi$-classes and opening exactly the edges in $A$. A loop does not join distinct vertices.

For $q\in [1,\mathrm{\infty })$ and $p\in (0,1)$, the finite FK weight of $A$ and its partition function are

$w_{G,p,q}^{\xi }(A):=p^{\mathrm{\mid }A\mathrm{\mid }}(1-p{)}^{\mathrm{\mid }E\setminus A\mathrm{\mid }}{q}^{k_{\xi }(A)},Z_{G,p,q}^{\xi }:={\sum }_{A\subseteq E}{w}_{G,p,q}^{\xi }(A)\mathrm{.}$

The finite FK random-cluster probability is ${\varphi }_{G,p,q}^{\xi }(A):=w_{G,p,q}^{\xi }(A)\mathrm{/}{Z}_{G,p,q}^{\xi }$. Every weight is positive and finite, and the finite sum $Z_{G,p,q}^{\xi }$ is positive and finite. The free boundary condition is the identity partition on $B$; the wired boundary condition identifies all vertices of $B$ in one class.

When $G$ is a finite simple graph with the Potts/FK convention of the earlier finite Potts definition, $\xi$ is free, $q=Q$ is an integer at least two, and every activity equals $v=p\mathrm{/}(1-p)$, this definition gives the same normalized FK law, since $w_{FK}(A)=(1-p{)}^{-\mathrm{\mid }E\mathrm{\mid }}{w}_{G,p,Q}^{\xi }(A)$.

Finite Potts and Fortuin–Kasteleyn weights

Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph