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.



Smirnov's theorem assert the convergence to SLE in the scaling limit: one discretizes the domain with the triangular lattice of mesh size $\delta$, and lets $\delta$ go to zero. It is only proven for the triangular lattice; it's a major open problem to prove universality of this result (in fact, even to extend it to the square lattice).

Now, in the hyperbolic plane, you cannot discretize a domain by scaled copies of the same lattice, since there is no natiral scaling. So, it seems that even to formulate the problem sensibly, one has to deal with a large class of graphs. I don't see why proving it should be any easier than establishing universality in the Euclidean case.


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