Avalanches and the directed percolation depinning model: Experiments, simulations, and theory

L. A.N. Amaral, A. L. Barabási, S. V. Buldyrev, S. T. Harrington, S. Havlin, R. Sadr-Lahijany, H. E. Stanley

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We study the recently introduced directed percolation depinning (DPD) model for interface roughening with quenched disorder for which the interface becomes pinned by a directed percolation (DP) cluster for d=1 or a directed surface for d>1. The mapping to DP enables us to predict some of the critical exponents of the growth process. For the case of 1+1 dimensions, the theory predicts that the roughness exponent α is given by α=ν/ν, where ν and ν are the exponents governing the divergence of perpendicular and parallel correlation lengths of the DP incipient infinite cluster. The theory also predicts that the dynamical exponent z equals the exponent dmin characterizing the scaling of the shortest path on an isotropic percolation cluster. For the case of 1+1 dimensions, our simulations give ν=1.73±0.02, α=0.63±0.01, and z=1.01±0.02, in good agreement with the theory. For the case of 2+1 dimensions, we find ν=1.16±0.05, α=0.48±0.03, and z=1.15±0.05, also in accord with the theory. For higher dimensions, α decreases monotonically but does not seem to approach zero for any dimension calculated (d≤6), suggesting that the DPD model has no upper critical dimension for the static exponents. On the other hand, z appears to approach 2 as d→6, as expected by the result z=dmin, suggesting that dc=6 for the dynamics. We also perform a set of inhibition experiments, in both 1+1 and 2+1 dimensions, that can be used to test the DPD model. We find good agreement between experimental, theoretical, and numerical approaches. Further, we study the properties of avalanches in the context of the DPD model. In 1+1 dimensions, our simulations for the critical exponent characterizing the duration of the avalanches give τsurv=1.46±0.02, and for the exponent characterizing the number of growth cells in the interface δ=0.60±0.03. In 2+1 dimensions, we find τsurv=2.18±0.03 and δ=1.14±0.06. We relate the scaling properties of the avalanches in the DPD model to the scaling properties for the self-organized depinning model, a variant of the DPD model. We calculate the exponent characterizing the avalanches distribution τaval for d=1-6 and compare our results with recent theoretical predictions. Finally, we discuss a variant of the DPD model, the ''gradient DPD model,'' in which the concentration of pinning cells increases with height. We perform a set of experiments in 1+1 dimensions that are well described by the gradient DPD model.

Original languageEnglish
Pages (from-to)4655-4673
Number of pages19
JournalPhysical Review E
Issue number5
StatePublished - 1995


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