What discrete processes/models have been proven to have scaling limits to $\text{SLE}(\kappa)$, for various $\kappa$? I know about loop-erased random walk and uniform spanning trees. What about conjectures in this direction? (Such as the double-dimer-cover cycles, which I read are conjectured to be $\text{SLE}(4)$)
I'm very new to this, so if you please could, together with the answer refer to a paper/article/survey accompanying the result, that would be greatly appreciated!
There are several other models proved to converge to SLE: critical percolation on the triangular lattice, Gaussian Free Field, Harmonic Explorer, and recently also the critical Ising model. You can check the paper Kevin linked or Schramm's slides from ICM2006 for some highlights. Just keep in mind that this is a fast changing field at the moment so there has been some progress since 2006.
From Cardy's article http://arxiv.org/abs/cond-mat/0503313
Some important special cases are therefore:
$\kappa = 2$: loop-erased random walks (proven in [24]);
$\kappa = 8/3$: self-avoiding walks, as already suggested by the restriction property, Sec. 3.5.2; unproven, but see [22] for many consequences;
$\kappa = 3$: cluster boundaries in the Ising model, however as yet unproven;
$\kappa = 4$: BCSOS model of roughening transition (equivalent to the 4-state Potts model and the double dimer model), as yet unproven; also certain level lines of a gaussian random field and the ‘harmonic explorer’ (proven in [23]); also believed to be dual to the Kosterlitz-Thouless transition in the XY model;
$\kappa = 6$: cluster boundaries in percolation (proven in [7]);
$\kappa = 8$: dense phase of self-avoiding walks; boundaries of uniform spanning trees (proven in [24]).
It should be noted that no lattice candidates for κ > 8, or for the dual values κ < 2, have been proposed.
There are several examples in this readable survey article by Schramm.
