Directed self-organized critical patterns emerging from fractional Brownian paths

We discuss a family of clusters C corresponding to the region whose boundary is formed by a fractional Brownian path y(i) and by the moving average function yn(i)≡1n∑k=0n−1y(i−k). Our model generates fractal directed patterns showing spatio-temporal complexity, and we demonstrate that the cluster area, length and duration exhibit the characteristic scaling behavior of SOC clusters. The function Cn(i) acts as a magnifying lens, zooming in (or out) the ‘avalanches’ formed by the cluster construction rule, where the magnifying power of the zoom is set by the value of the amplitude window n. On the basis of the construction rule of the clusters Cn(i)≡y(i)−yn(i) and of the relationship among the exponents, we hypothesize that our model might be considered to be a generalized stochastic directed model, including the Dhar–Ramaswamy (DR) model and the stochastic models as particular cases. As in the DR model, the growth and annihilation of our clusters are obtained from the set of intersections of two random walk paths, and we argue that our model is a variant of the directed self-organized criticality scheme of the DR model.