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Conformal invariance of crossing probabilities for the Ising model with free boundary conditions. (English. French summary) Zbl 1355.60119

Summary: We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by R. Langlands et al. [Bull. Am. Math. Soc., New Ser. 30, No. 1, 1–61 (1994; Zbl 0794.60109)]. We do so by establishing the convergence of certain exploration processes towards \(\operatorname{SLE}(3,\frac{-3}{2},\frac{-3}{2})\). We also construct an exploration tree for free boundary conditions, analogous to the one introduced by S. Sheffield [Duke Math. J. 147, No. 1, 79–129 (2009; Zbl 1170.60008)].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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