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A new computation of the critical point for the planar random-cluster model with $$q\geq1$$. (English. French summary) Zbl 1395.82043
Authors’ abstract: We present a new computation of the critical value of the random-cluster model with cluster weight $$q\geq1$$ on $$\mathbb{Z}^{2}$$. This provides an alternative approach to the result in [V. Beffara and H. Duminil-Copin, Probab. Theory Relat. Fields 153, No. 3–4, 511–542 (2012; Zbl 1257.82014)]. We believe that this approach has several advantages. First, most of the proof can easily be extended to other planar graphs with sufficient symmetries. Furthermore, it invokes RSW-type arguments which are not based on self-duality. And finally, it contains a new way of applying sharp threshold results which avoid the use of symmetric events and periodic boundary conditions. Some of the new methods presented in this paper have a larger scope than the planar random-cluster model, and may be useful to investigate sharp threshold phenomena for more general dependent percolation processes in arbitrary dimensions.

##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B26 Phase transitions (general) in equilibrium statistical mechanics 82B43 Percolation 82B27 Critical phenomena in equilibrium statistical mechanics
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