Finite Potts Swendsen–Wang update

Input a finite simple graph $G=(V,E)$, an integer $Q\ge 2$, finite activities $v_e\ge 0$ for $e\in E$, and a spin configuration $\sigma :V\to [Q]$, using the Potts and FK conventions of the finite Potts/FK weight definition. For each edge $e=\{u,v\}$, if $\sigma (u)\mathrm{\ne }\sigma (v)$ set $e$ absent. If $\sigma (u)=\sigma (v)$, draw an independent uniform $U_e\in [0,1]$ and place $e$ in $A$ exactly when $U_e<v_e\mathrm{/}(1+v_e)$. For each connected component of $(V,A)$, draw an independent uniform $U_C\in [0,1]$ and assign its vertices the common color $1+\lfloor Q{U}_C\rfloor$, with the value $Q$ used when $U_C=1$. Return the resulting spin configuration ${\sigma }^{\mathrm{\prime }}$. The finite graph has finitely many edges and components, so the procedure uses finitely many draws and operations. At $v_e=0$ the edge is never activated; finite $v_e$ ensures the activation probability is less than one.

Finite Potts and Fortuin–Kasteleyn weights

The algorithm defines one full update of a finite-state chain. Its definition asserts neither rapid mixing nor any bound uniform in graph size, temperature, or critical parameters.