Lattice Animals and Self-Organized Criticality

Abstract

The paper considers one model of SOC close to BTW and slider blocks models. In addition, it introduces an additional time parameter and imposes special restrictions on the avalanche geometrical structure. The generalization and modification of the avalanche's concept allows us to apply H. Weyl's theorem in the dynamical system theory so as to obtain the strong and exact results in this area. We introduce some combinatorial characteristic of clusters and use it as a tool for estimating the frequency of the avalanches. The results obtained give the asymptotically exact expressions for the asymptotical frequency as well as two special types of such extended avalanches. In some special cases, they reduce the determination of the frequency of the avalanches to combinatorial enumerative problem for lattice animals on the d-dimensional torus. The other two results are related to the one-dimensional model and establish the connection between the SOC and the theory of number partitions.