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

The BTW sandpile model is considered on three dimensional percolation lattice which is tunned with the occupation parameter $p$. Along with the three-dimensional avalanches, we study the energy propagation in two-dimensional cross-sections. We use the moment analysis to extract the exponents for two separate cases: the critical ($p=p_c\equiv p_c^{3D}$) and the off-critical ($p_c<p\leq 1$) cases. The three-dimensional avalanches at $p=p_c$ has exponents like the regular 2D BTW model, whereas the exponents for the 2D cross-sections have serious similarities with the 2D critical Ising model. The moment analysis show that finite size scaling theory is the fulfilled, and some hyper-scaling relations are confirmed. For the off-critical lattice, the exponents change logarithmically with $p-p_c$, for which the cut-off exponents $\nu$ drop discontinuously from $p=p_c$ to the other values. The analysis for the 2D cross-sections show a singular behavior at some $p_0\approx p_c^{2D}$ ($p_c^{3D}$ and $p_c^{2D}$ being three- and two-dimensional percolation thresholds). We argue that there are two separate phases in the cross-sections, namely $p_c^{3D}\leq p<p_c^{2D}$ which, due to lack of 2D percolation cluster, has no thermodynamic limit, and $p\geq p_c^{2D}$ having the chance to involve percolated clusters.