Critical properties of deterministic and stochastic sandpile models on two-dimensional percolation backbone

Abstract

Deterministic Bak Tang Wiesenfeld (BTW) model and stochastic sandpile model (SSM), a modified Manna model, are studied on the percolation backbone, a random fractal with well known fractal features, generated on a square lattice in 2-dimensions. In spite of the underlying random structure of the backbone, the BTW model preserves its positive time auto-correlation and multifractal behaviour due to its complete toppling balance. In contrast, the critical properties of the SSM still exhibits finite size scaling (FSS) as it manifests on the regular lattices. Various scaling relations are developed analysing the topography of the avalanches. The extended set of critical exponents obtained for the SSM is found to obey various scaling relations in terms of the fractal dimension dfB of the backbone, on the other hand, the BTW model does not exhibit any such scaling relations. As the critical exponents of the SSM defined on the backbone depend on dfB, they are found to be entirely different from those known for the SSM defined on the regular lattice as well as on deterministic fractals. The SSM on the percolation backbone is found to obey FSS and belongs to a new universality class.