Sandpile on uncorrelated site-diluted percolation lattice; from three to two dimensions. (23rd February 2018) - Najafi, M N; Dashti-Naserabadi, H
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 ( p = p c ≡ p c 3 D ) 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 ν change discontinuously from p   =   p c to the values for the supercritical case. Moreover we show that there is a singular point p 0 ≈ p c 2 D ( p c 3 D and p c 2 D being three- and two-dimensional percolation thresholds) for 2D cross-sections, which separate the behaviors to two distinct intervals: p c 3 D ⩽ p < p c 2 D which, due to the lack of 2D percolation cluster, has no thermodynamic limit, and p ⩾ p c 2 D which involves the percolated clusters.