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Crossing probabilities in topological rectangles for the critical planar FK-Ising model. (English) Zbl 1341.60124
Summary: We consider the FK-Ising model in two dimensions at criticality. We obtain bounds on crossing probabilities of arbitrary topological rectangles, uniform with respect to the boundary conditions, generalizing results of H. Duminil-Copin et al. [Commun. Pure Appl. Math. 64, No. 9, 1165–1198 (2011; Zbl 1227.82015)] and D. Chelkak and S. Smirnov [Invent. Math. 189, No. 3, 515–580 (2012; Zbl 1257.82020)]. Our result relies on new discrete complex analysis techniques, introduced by D. Chelkak [Ann. Probab. 44, No. 1, 628–683 (2016; Zbl 1347.60050)]. We detail some applications, in particular the computation of so-called universal exponents, the proof of quasi-multiplicativity properties of arm probabilities, and bounds on crossing probabilities for the classical Ising model.

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