Square-lattice Potts critical inverse temperature

Put $v=e^{\beta J}-1$ and $p=v\mathrm{/}(1+v)=1-e^{-\beta J}$. On the finite spin-edge configurations with $\sigma {\mathrm{\mid }}_{B_n}=1$, define the joint weight $W_n^{+}(\sigma ,A)=v^{\mathrm{\mid }A\mathrm{\mid }}{\prod }_{\{x,y\}\in A}{\mathbf{1}}_{\{{\sigma }_x={\sigma }_y\}}\mathrm{.}$ For fixed $\sigma$, summing over $A\subseteq E_n$ gives ${\prod }_{e\in E_n}(1+v{\mathbf{1}}_{\{{\sigma }_x={\sigma }_y\}})=e^{-\beta H_n(\sigma )}$. For fixed $A$, every component meeting $B_n$ has color $1$ and each remaining component has $Q$ choices. After wiring all of $B_n$ into one class, the number of unpinned components is $k_{{\xi }_n}(A)-1$. Thus the edge marginal has weight $Q^{k_{{\xi }_n}(A)-1}{v}^{\mathrm{\mid }A\mathrm{\mid }}$, proportional by the configuration-independent factor $Q^{-1}$ to the wired FK weight in [F3]. Normalization therefore gives exactly the finite wired FK law. This supplies the pinned version of [F2] by an explicit finite sum.

Conditional on $A$, the origin has color $1$ with probability one if it connects to $B_n$, and with probability $1\mathrm{/}Q$ otherwise. Indeed every unpinned component has its independent uniform color by the finite count of step 1.1. Consequently ${\mathbb{P}}_{\beta ,n}^{+}({\sigma }_0=1)=Q^{-1}+(1-Q^{-1})a_n(p,Q)\mathrm{.}$ By [F4] the right side has a numerical limit. Hence $m_Q^{+}(\beta )=(1-Q^{-1})L(1-e^{-\beta J},Q)$.

The map $\beta \mapsto 1-e^{-\beta J}$ is strictly increasing from $0$ to $1$. By [F4] and the positive prefactor in step 2.1, $m_Q^{+}(\beta )$ is zero below the unique parameter solving $1-e^{-\beta J}=\sqrt{Q}\mathrm{/}(1+\sqrt{Q})$ and positive above it. Its defining positive-set infimum is therefore that parameter, regardless of its value at the critical point. Solving gives $e^{-{\beta }_{c}J}=1\mathrm{/}(1+\sqrt{Q})$ and ${\beta }_{c}J=\log (1+\sqrt{Q})$.