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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.

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
Full Text: DOI
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