Potts model (${\pi }_{\beta ,q}(\sigma )\propto \exp (\beta {\sum }_{\{x,y\}\in E}{1}_{{\sigma }_x={\sigma }_y})$)

The ferromagnetic Potts model on a finite graph assigns $q$ possible states to each graph vertex and rewards neighbouring equal states by the displayed weight, for $\beta \ge 0$. The interaction sum counts each unoriented edge once. The Edwards-Sokal coupling identifies its spin clusters with the random-cluster model at $p=1-e^{-\beta }$ and integer $q\ge 2$.

The ferromagnetic Potts model on a labelled ring has $Q$ possible site states, $J>0$ and ${\sigma }_{L+1}={\sigma }_1$. Its spin-chain transfer matrix is $T=(e^{\beta J}-1)I+P$, with $P$ the $Q$-dimensional all-ones matrix. The canonical partition function is $(e^{\beta J}+Q-1{)}^L+(Q-1)(e^{\beta J}-1{)}^L$. The convention counts two bonds when $L=2$.

At fixed finite $L$, the Helmholtz free energy of a periodic Potts chain receives its first thermal correction from a pair of domain walls. The two-domain-wall degeneracy of a Potts ring divided by the ground state degeneracy factor supplies the coefficient. Expanding the exact canonical partition function proves the formula. The expansion is not uniform in $L$: taking the thermodynamic limit first gives $F/L=-J-{\beta }^{-1}\log [1+(Q-1)e^{-\beta J}]$, whose leading correction is of order $e^{-\beta J}$.

For $Q,L\ge 2$, a periodic Potts chain has $Q$ uniform ground states with energy $-JL$. A single domain wall cannot close around the ring. Its first excited states have two domain walls and energy $-JL+2J$. Choose the wall bonds in $\left(\genfrac{}{}{0ex}{}{L}{2}\right)$ ways and assign distinct states to the two resulting arcs in $Q(Q-1)$ ways. This proves the displayed degeneracy factor without identifying globally relabelled configurations.