The BTW mechanism on a self-similar image of a square: A path to unexpected exponents

We define the BTW mechanism on a two-dimensional heterogeneous self-similar lattice. Our model exhibits the power distribution of avalanches with the exponent τ = 2 − 2 / ν , where ν is the similarity exponent of the lattice. The inequality τ < 1 , for the first time detected in this paper inside a broad class of sand-piles, is ensured by random loading uniformly distributed over the lattice.