Comparison of two-dimensional isotropic conservative self-organized critical sand-pile models

We study BTW and Manna models – 2D self-organized critical sandpile models with conservative isotropic rules. In spite of the similarity of the rules and the identity of their symmetries, these models have different set of critical indices. The equality of indices of the dependence between the area and the perimeter of avalanche is the only nontrivial coincidence. We determine this index from the condition of scale invariance of stochastic differential equation determining avalanche growth for both models. We find that for this process outside and inside directions are equivalent for Manna model but are not for BTW one. Thereby we reveal the symmetry difference separating properties of these models.