Sandpile models are usually characterized by studying the "macroscopic" critical properties of its non-equilibrium steady state. However, most of the information of a sandpile avalanche remains stored in a "microscopic" parameter, the toppling number si of each sand column at a lattice site i. In terms of si, a number of physical quantities such as distribution of si, fluctuation and correlation in si, density distribution of si etc.; are defined and analyzed for a variety of sandpile models at their respective steady states. A set of new critical exponents is estimated and scaling relations among them are established. The values of the critical exponents are found to satisfy the scaling relations and able to characterize the system properties distinctly for different sandpile models. The analysis not only reveals the complex geometrical properties associated with the toppling surface but also provides deeper insight into the avalanche properties.

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In the rotational sandpile model, either the clockwise or the anti-clockwise toppling rule is assigned to all the lattice sites. It has all the features of a stochastic sandpile model but belongs to a different universality class than the Manna class. A crossover from rotational to Manna universality class is studied by constructing a random rotational sandpile model and assigning randomly clockwise and anti-clockwise rotational toppling rules to the lattice sites. The steady state and the respective critical behaviour of the present model are found to have a strong and continuous dependence on the fraction of the lattice sites having the anti-clockwise (or clockwise) rotational toppling rule. As the anti-clockwise and clockwise toppling rules exist in equal proportions, it is found that the model reproduces critical behaviour of the Manna model. It is then further evidence of the existence of the Manna class, in contradiction with some recent observations of the non-existence of the Manna class.

A continuous phase transition occurs in the topography of toppling surfaces of stochastic and rotational sandpile models when they are flooded with liquid, say water. The toppling surfaces are extracted from the sandpile avalanches that appear due to sudden burst of toppling activity in the steady state of these sandpile models. Though a wide distribution of critical flooding heights exists, a critical point is defined by merging the flooding thresholds of all the toppling surfaces. The criticality of the transition is characterized by power-law distribution of island area in the critical regime. A finite size scaling theory is developed and verified by calculating several new critical exponents. The flooding transition is found to be an interesting phase transition and does not belong to the percolation universality class. The universality class of this transition is found to depend on the degree of self-affinity of the toppling surfaces characterized by the Hurst exponent H and the fractal dimension D(f) of critical spanning islands. The toppling surfaces of different stochastic sandpile models are found to have a single Hurst exponent, whereas those of different rotational sandpile models have another Hurst exponent. As a consequence, the universality class of different sandpile models remains preserved within the same symmetry of the models.

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