Percolation on mesoscopic lattices

In a celebrated paper, Smirnov proved that critical site-percolation on the regular triangular lattice has a non-trivial, conformally invariant scaling limit and that this can be used to derive for instance the value of critical exponents. The argument is unfortunately very specific to this particular lattice, and so far has not been generalized to any other natural case — in particular, percolation on $\mathbb Z^2$ is much beyond reach of current methods. I will present one direction in which the proof can be extended into a non-trivial class of models that somehow interpolate between the triangular lattice and general planar cases.