In "Percolation in the hyperbolic plane" the authors study the properties of percolation in the hyperbolic plane. Smirnov and others proved convergence of isotropic percolation to SLE(6).
Do these results follow for the hyperbolic case too?
1)L. Arosio, F. Bracci, "Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds,"
So there is a natural candidate.