Crossover Phenomenon And Universality: From Random Walk To Deterministic Sand-Piles Through Random Sand-Piles

The sand-pile models exhibit a great diversity of behavioral patterns. The average height as a function on time points out the variety of the sand-pile's criticality. We define the classes of universality for the sand-piles in terms of the spectrum of the average height. The definition, in particular, distinguishes two classes with Manna's and Baket al.sand-piles as the typical representatives. The one-parametric family of the models investigated in this paper develops a crossover between the classes and introduces the infinite quantity of other sand-pile's classes. The parameter is the number of grains that any unstable cell transfers to its neighbors. The sand-piles admit a certain normalization generating unified laws. The spectral approach increases the precision of the universalities from our size distribution analysis. The latter describes the sand-piles' typical distribution, sets the correspondence between the sand-piles with small and large parameter values, but fails to determine their diversity.