I am wondering which models are conjectured (eg. numerically) to converge to SLE(6) (Schramm-Loewner evolution with $\kappa=6$) or CLE(6) (conformal loop ensemble). I am searching for a research topic in this very interesting area.
The following are from Smirnov's 2007 review and elsewhere:
1) Critical percolation for lattices other than triangular eg. the square lattice. Any detailed review on this?
2) Fortuin-Kasteleyn random cluster model for $n=1$ and $x>x_{c}$, where $x_{c}$ the critical edge weight.
3) For $O(1)$ but for $x\neq 1$ and $p\neq 1/2$, which correspond to the critical percolation case.
4) In the article "conformal invariance of hydrodynamic turbulence", they showed numerically that zero-vorticity isolines correspond to SLE(6).
5) Random-field Ising model by Stevenson (2011).
