an:06880060
Zbl 1395.82043
Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent
A new computation of the critical point for the planar random-cluster model with \(q\geq1\)
EN
Ann. Inst. Henri Poincar??, Probab. Stat. 54, No. 1, 422-436 (2018).
00386300
2018
j
82B20 60K35 82B26 82B43 82B27
phase transition; random-cluster model; Potts model; critical point; sharp phase transition
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 [\textit{V. Beffara} and \textit{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.
E. Ahmed (Mansoura)
Zbl 1257.82014