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

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 (P = P-c (math) P-c(3D)) and the supercritical one (p(c) < p <= 1). Our numerical data is consistent with the conjecture that the three-dimensional avalanches at p = p c 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 - p(c), for which the cut-off exponents v change discontinuously from p = p(c) to the values for the supercritical case. Moreover we show that there is a singular point p(0) approximate to p(c)(2D) (p(c)(3D) and p(c)(2D) being three- and two-dimensional percolation thresholds) for 2D cross-sections, which separate the behaviors to two distinct intervals: p(c)3D <= P p(c)(2D) which, due to the lack of 2D percolation cluster, has no thermodynamic limit, and p >= p(c)(2D) which involves the percolated clusters.