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

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 $D_f^{\text{SAW}}=\frac{4}{3}$. Also, the corresponding open curves has conformal invariance with the root-mean-square distance $R_{\text{rms}}\sim t^{3/4}$ ($t$ 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 $1/log\left(L\right)$-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 $L^{\frac{1}{2}}$, which is different from the regular 2D BTW model.

Figure 1

A avalanche sample in an Ising sample media in a $512\times 512$ lattice at $T=T_c$. The red sites represent the forbidden (inactive) sites, and the white sites are representative of the active ones. The toppled sites in an avalanche have been shown by the gray ones.

Figure 2

The fitted plot of the distribution function of the (a) loop length (b) gyration radius with their $\beta$, $\nu$, and $\tau$ exponents for various rates of lattice sizes. (c) The fractal dimension $D_f\equiv {\gamma }_{lr}$ at $T=T_c$ as the slope of the $〈log\left(l\right)〉-〈log\left(r\right)〉$ plot. Inset: the exponent ${\gamma }_{mr}$. (d) root-mean-square distance $R_{\text{rms}}$ of the cut curves in terms of $t$ ($t\equiv$ the time parametrization of the curve) with the exponent $\frac{3}{4}$ which is compatible with ${\nu }_{\text{SAW}}=0.75$.

Figure 3

(a) The plot of the distribution function of the gyration radius for various rates of the temperature $T$. Inset: the collapse graphs of the main graph. The exponents of the distribution functions of (b) the gyration radius $r$ (c) the number of topplings in an avalanche $n$, (d) the mass of the avalanches $m$, and (e) the loop length $l$ length of the frontier of avalanche in terms of $T$ for various system sizes. Insets: The linear fits of the exponents at $T=T_c$ in terms of $1/log\left(L\right)$, $L$ being the lattice size, from which the thermodynamic values can be extrapolated, and the power-law behaviors of exponents near the critical temperatures with the exponents represented by ${\gamma }_x$ ($x\equiv r,n,m$, and $l$).

Figure 4

(a) The ${\gamma }_{lr}$ and (b) the ${\gamma }_{mr}$ in terms of $T$ for various system sizes. The lower insets show the finite-dependence of the exponents in terms of $1/L$ at $T=T_c$, $L$ being the system size, from which the thermodynamic values can be extrapolated. The upper insets show the power-law behavior of the exponents in the vicinity of the critical temperature with the exponents shown by $\kappa$.

Figure 5

The spanning cluster probability $\mathrm{SCP}(T,L)$ in terms of $T$ for various rates of lattices sizes. The inset shows the power-law behavior of this function in terms of $L$ at the critical temperature with the exponent $\frac{1}{2}$.