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The self-dual point of the two-dimensional random-cluster model is critical for \(q \geqslant 1\). (English) Zbl 1257.82014
The random-cluster model on a finite connected graph is a model on the edges of the graph, each one being either closed or open with the edge-weight \(p\in [0,1]\) and the cluster-weight \(q\in (0,\infty)\). The authors prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with a parameter \(q\geq 1\) on \(\mathbb{Z}^2\) is equal to the self-dual point \(p_{sd}(q)=\sqrt{q}/(1+\sqrt{q})\).

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