Sandpile on uncorrelated site-diluted percolation lattice; from three to two dimensions

Abstract

The BTW sandpile model is considered on the three dimensional percolation lattice which is tuned by the occupation parameter p. Along with the three-dimensional avalanches, we study the avalanches in two-dimensional cross-sections. We use the moment analysis (along with some other methods) to extract the exponents for two separate cases: the lattice at critical percolation () and the supercritical one (). Our numerical data is consistent with the conjecture that the three-dimensional avalanches at p = pc have nearly the same exponents as the regular 2D BTW model. The moment analysis shows that finite size scaling theory is fulfilled, and some hyper-scaling relations hold. The main finding of the paper is the logarithmic dependence of the exponents on p − pc, for which the cut-off exponents ν change discontinuously from p = pc to the values for the supercritical case. Moreover we show that there is a singular point ( and being three- and two-dimensional percolation thresholds) for 2D cross-sections, which separate the behaviors to two distinct intervals: which, due to the lack of 2D percolation cluster, has no thermodynamic limit, and which involves the percolated clusters.