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

Benoist, Stéphane; Duminil-Copin, Hugo; Hongler, Clément

Ann. Inst. Henri Poincaré, Probab. Stat. 52, No. 4, 1784-1798 (2016).

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

Keywords:

Ising model; interfaces; Schramm-Loewner evolution; phase transition; crossing probabilities; exploration trees

Citations:

Zbl 0794.60109; Zbl 1170.60008