Finite Swendsen–Wang update preserves the Potts law

For each fixed spin configuration $\sigma$, summing $W(\sigma ,A)$ over $A\subseteq E$ factors over edges. An unequal-spin edge must be absent and contributes $1$; an equal-spin edge contributes either $1$ or $v_e$. Thus ${\sum }_{A}W(\sigma ,A)={\prod }_{e\in E}(1+v_e{\mathbf{1}}_{\{\sigma (u_e)=\sigma (w_e)\}})=w_P(\sigma )$, which is positive by [F1]. Dividing by this conditional normalizer shows that under $\nu$, given $\sigma$, unequal-spin edges are forced absent and equal-spin edges are independent Bernoulli variables with activation probabilities $v_e\mathrm{/}(1+v_e)$, including probability zero when $v_e=0$. This is exactly the first stage of [F3].

Every edge set produced in step 1.1 has $W(\sigma ,A)>0$: all active edges have $v_e>0$ and equal endpoint spins, while inactive edges contribute $1$. Therefore $\nu (A)>0$. By [F2], conditional on this edge set, its components receive independent uniform colors. This is exactly the second stage of [F3].

Start with $\sigma$ distributed as the spin marginal ${\pi }_P$ of $\nu$. By step 1.1, after the first stage the pair $(\sigma ,A)$ has joint law $\nu$. By step 2.1, the new configuration ${\sigma }^{\mathrm{\prime }}$ is then drawn from the spin conditional law $\nu (\cdot \mid A)$. Summing over the finite edge-set space shows that its marginal is the spin marginal of $\nu$, namely ${\pi }_P$. Hence the update preserves ${\pi }_P$.

The graph has finitely many edges, so the first stage performs one draw per edge; the second stage performs one draw per connected component of a finite graph. Both stages therefore terminate after finitely many operations.