Another proof of $p_c = \frac{\sqrt{q}}{1+\sqrt{q}}$ on $\mathbb{Z}^2$ with $q \in [1,4]$ for random-cluster model

In this paper we give another proof of $p_c = \frac{\sqrt{q}}{1+\sqrt{q}}$ on $\mathbb{Z}^2$ with $q \in [1,4]$, based on the method of parafermionic observables.
Comment: 29 pages, 6 figures