Finite FK boundary wiring

Let $G=(V,E)$ be a finite simple graph, let $Q\ge 2$ be an integer, and give each edge a finite activity $v_e\ge 0$. Choose a set $B\subseteq V$ of boundary vertices and an equivalence relation $\xi$ on $B$. The relation $\xi$ specifies which boundary vertices are identified before any edges are occupied. For $A\subseteq E$, let ${\sim }_{\xi ,A}$ be the least equivalence relation on $V$ containing every pair in $\xi$ and both endpoints of every edge in $A$. Write $k_{\xi }(A):=\mathrm{\mid }V\mathrm{/}{\sim }_{\xi ,A}\mathrm{\mid }$ and $v_A:={\prod }_{e\in A}{v}_e$, with empty product $1$. Define $w_{FK}^{\xi }(A):=Q^{k_{\xi }(A)}{v}_A,Z_{FK}^{\xi }(Q,\mathbf{v}):={\sum }_{A\subseteq E}{w}_{FK}^{\xi }(A)\mathrm{.}$ The free boundary choice is the discrete relation on $B$ (each boundary vertex is a singleton); the wired classes are the equivalence classes of $\xi$, with a fully wired boundary given by one class when $B\mathrm{\ne }\mathrm{\varnothing }$. Thus $k_{\xi }(A)$ counts components after the prescribed identifications and occupied edges, including isolated vertices and boundary classes. The sum is finite and positive because its empty-edge term is $Q^{k_{\xi }(\mathrm{\varnothing })}>0$. For discrete $\xi$, $k_{\xi }(A)=k_G(A)$ and $Z_{FK}^{\xi }=Z_{FK}$. This is a mathematical partition polynomial; no wired Potts spin marginal is asserted.

Finite Potts and Fortuin–Kasteleyn weights