Mapping of the Bak, Tang, and Wiesenfeld sandpile model on a two-dimensional Ising-correlated percolation lattice to the two-dimensional self-avoiding random walk

Abstract

The self-organized criticality on the random fractal networks has many motivations, like the movement pattern of fluid in the porous media. In addition to the randomness, introducing correlation between the neighboring portions of the porous media has some nontrivial effects. In this paper, we consider the Ising-like interactions between the active sites as the simplest method to bring correlations in the porous media, and we investigate the statistics of the BTW model in it. These correlations are controlled by the artificial "temperature" T and the sign of the Ising coupling. Based on our numerical results, we propose that at the Ising critical temperature T_{c} the model is compatible with the universality class of two-dimensional (2D) self-avoiding walk (SAW). Especially the fractal dimension of the loops, which are defined as the external frontier of the avalanches, is very close to D_{f}^{SAW}=4/3. Also, the corresponding open curves has conformal invariance with the root-mean-square distance R_{rms}∼t^{3/4} (t being the parametrization of the curve) in accordance with the 2D SAW. In the finite-size study, we observe that at T=T_{c} the model has some aspects compatible with the 2D BTW model (e.g., the 1/log(L)-dependence of the exponents of the distribution functions) and some in accordance with the Ising model (e.g., the 1/L-dependence of the fractal dimensions). The finite-size scaling theory is tested and shown to be fulfilled for all statistical observables in T=T_{c}. In the off-critical temperatures in the close vicinity of T_{c} the exponents show some additional power-law behaviors in terms of T-T_{c} with some exponents that are reported in the text. The spanning cluster probability at the critical temperature also scales with L^{1/2}, which is different from the regular 2D BTW model.

Dynamical Crossover in Invasion Percolation

“…1. It is consistent with the fractal dimension of the self-avoiding walks D SAW f = 4 3 [23,24], which is related to the fractal dimension of the interfaces of percolation…”

Scaling relations and finite-size scaling in gravitationally correlated lattice percolation models

“…The gravitationally correlated lattice percolation models (GCLPMs) introduced in this paper are new models of long-range correlated percolation [22][23][24][25], and they are in different universality class from the existing correlation percolation model, e.g. the scaling law mentioned in [22] is violated and the PR is merged into the bond-occupation schemes.…”

“…By contrast, it is necessary to have a geometric controllability in network percolation, i.e., to facilitate or inhibit network percolation in link-adding processes based on the geometric distance, which motivates us to free ourselves from the constraint of purely topological connection between nodes in previous models. As the consequence, intervening strategies for this kind of correlated percolation [22][23][24][25] lead to new scaling relations and finite-size scaling.…”

Multifractal analysis of Earthquakes in Central Alborz, Iran; A phenomenological self-organized critical Model

“…Also we have r ij = 4.52km in the x 1 direction, r ij = 5.55km in the x 2 direction, and r ij varies from 2km to 4km in the x 3 direction. Burst dynamics is a popular property of sandpiles, manifested by prominent avalanches which occur with a low frequency, called sometimes rare events [40][41][42][43][44][45]. Avalanches in our model are defined as a chain of local relaxations (topplings) occurring as a consequence of an external stimulus.…”