Finite Potts and Fortuin–Kasteleyn weights

Let $G=(V,E)$ be a finite simple graph, so $V$ is finite and $E$ is a set of two-element subsets of $V$. Fix an integer $Q\ge 2$, write $[Q]=\{1,\dots ,Q\}$, and assign to each edge $e\in E$ a finite activity $v_e\in [0,\mathrm{\infty })$. A spin configuration is a function $\sigma :V\to [Q]$. Its unnormalized Potts weight and partition function are $w_P(\sigma ):={\prod }_{e=\{u_e,w_e\}\in E}(1+v_e{\mathbf{1}}_{\{\sigma (u_e)=\sigma (w_e)\}}),Z_P(Q,\mathbf{v}):={\sum }_{\sigma :V\to [Q]}{w}_P(\sigma )\mathrm{.}$ For $A\subseteq E$, let $G_A=(V,A)$, and let $k_G(A)$ be its number of connected components, counting isolated vertices; equivalently, these are the equivalence classes generated by the endpoint pairs of edges in $A$. Put $v_A:={\prod }_{e\in A}{v}_e$, with empty product $1$. The unnormalized Fortuin–Kasteleyn (FK) weight and partition function are $w_{FK}(A):=Q^{k_G(A)}{v}_A,Z_{FK}(Q,\mathbf{v}):={\sum }_{A\subseteq E}{w}_{FK}(A)\mathrm{.}$ Both normalizers are finite and positive: the spin sum is finite with every term at least $1$, while the empty-edge FK term is $Q^{\mathrm{\mid }V\mathrm{\mid }}>0$. These are finite mathematical weights; no physical units or empirical status are assigned.

A finite simple graph is a finite vertex set together with a set of two-element vertex subsets