The random-cluster model (or Fortuin-Kasteleyn percolation) plays a key role in studies of models on lattices, as it is connected to many of them, and the results obtained for RCM can be though applied for other models.

In this talk I will present another proof of the well-known fact that for the square lattice the critical probability of the random-cluster model p_{cr} is equal to \frac{\sqrt{q}}{1+\sqrt{q}} for q in [1,4]. Unlike other proofs, this one involves the method of parafermionic observables applied to exploration paths in boxes and strips of growing size.

This result was presented in a joint work with E. Mukoseeva during my PhD under the supervision of H. Duminil-Copin.