Bak-Tang-Wiesenfeld model in the upper critical dimension: Induced criticality in lower-dimensional subsystems

We present extensive numerical simulations of Bak-Tang-Wiesenfeld (BTW) sandpile model on the hypercubic lattice in the upper critical dimension $D_u=4$. After re-extracting the critical exponents of avalanches, we concentrate on the three- and two-dimensional (2D) cross sections seeking for the induced criticality which are reflected in the geometrical and local exponents. Various features of finite-size scaling (FSS) theory have been tested and confirmed for all dimensions. The hyperscaling relations between the exponents of the distribution functions and the fractal dimensions are shown to be valid for all dimensions. We found that the exponent of the distribution function of avalanche mass is the same for the $d$-dimensional cross sections and the $d$-dimensional BTW model for $d=2$ and 3. The geometrical quantities, however, have completely different behaviors with respect to the same-dimensional BTW model. By analyzing the FSS theory for the geometrical exponents of the two-dimensional cross sections, we propose that the 2D induced models have degrees of similarity with the Gaussian free field (GFF). Although some local exponents are slightly different, this similarity is excellent for the fractal dimensions. The most important one showing this feature is the fractal dimension of loops $d_f$, which is found to be $1.50\pm 0.02\approx \frac{3}{2}=d_f^{\text{GFF}}$.

Figure 1

(a) The avalanche region of 3D cross section of a 4D avalanche on a hypercubic lattice with size $L=16$, and also a cut from its 2D cross section. (b) The boundary loop of an avalanche of 2D cross section of a 4D avalanche on a hypercubic lattice with size $L=144$.

Figure 2

The distribution function of (a) the number of relaxation events $s_4$ (which is equivalent to size of avalanche $S_4$), with the exponent ${\tau }_{s_4}=1.50\left(3\right)$ (derived using ${\chi }^2$ method), and (b) the gyration radius $R_4$ with the exponent ${\tau }_{R_4}=3.00\left(4\right)$. (c) The $q\mathrm{th}$ moment of $s_4$ versus different sizes $L$. (d) The exponent ${\sigma }_{s_4}$ versus $q$, which is derived from Eq. (3). Inset: The exponent ${\sigma }_{R_4}$ versus $q$. (e) The function $S_4$ versus $R_4$ with slope (fractal dimension) ${\gamma }_{S_4{R}_4}^{L=120}=3.90\left(3\right)$. Inset: the finite-size effect for ${\gamma }_{S_4{R}_4}^{\infty }=4.00\left(5\right)$.

Figure 3

The distribution function of (a) the number of relaxation events $s_3$ with the exponent ${\tau }_{s_3}=1.33\left(3\right)$, (b) the gyration radius $R_3$ with the exponent ${\tau }_{R_3}=3.30\left(4\right)$, and (c) the size of the clusters $S_3$ with the exponent ${\tau }_{S_3}=1.85\left(4\right)$. (d) The function $S_3$ versus $R_3$ with slope (fractal dimension) ${\gamma }_{S_3{R}_3}^{L=120}=2.75\left(3\right)$. Inset: the finite-size effect of ${\gamma }_{S_3{R}_3}$ with ${\gamma }_{S_3{R}_3}^{\infty }=2.80\left(5\right)$. (e) The $q\mathrm{th}$ moment of $s_3$ versus different sizes $L$. (f) The exponent ${\sigma }_{s_3}$ versus $q$, which is derived from Eq. (3). Inset: The exponent ${\sigma }_{R_3}$ versus $q$.

Figure 4

The distribution function of (a) the number of relaxation events $s_2$ with the exponent ${\tau }_{s_2}=1.20\left(10\right)$, (b) the gyration radius $R_2$ with the exponent ${\tau }_{R_2}=3.60\left(9\right)$, and (c) the cluster size $S_2$ with the exponent ${\tau }_{S_2}=2.45\left(10\right)$. (d) The function $S_2$ versus $R_2$ with slope (fractal dimension) ${\gamma }_{S_2{R}_2}^{L=120}=1.85\left(5\right)$. Inset: the finite-size effect of ${\gamma }_{S_2{R}_2}$ with ${\gamma }_{S_2{R}_2}^{\infty }=1.86\left(5\right)$. (e) The $q\mathrm{th}$ moment of $s_2$ versus different sizes $L$. (f) The exponent ${\sigma }_{s_2}$ versus $q$, which is derived from Eq. (3). Inset: The exponent ${\sigma }_{R_2}$ versus $q$.

Figure 5

(a) The Green function in terms of $r$ for various rates of lattice sizes with the exponent $x_l=2.5\left(2\right)$. The distribution function of (b) the length of loops $l$ with the exponent ${\tau }_l=3.15\left(13\right)$, and (c) the gyration radius $r$ with exponent ${\tau }_r=4.20\left(11\right)$. (d) The function $l$ versus $r$ with slope (fractal dimension) ${\gamma }_{rl}=1.50\left(2\right)$. Inset: The function $l$ versus $a$ with slope ${\gamma }_{al}=0.75\left(2\right)$. (e) The $q\mathrm{th}$ moment of $l$ versus different sizes $L$. (f) The exponent ${\sigma }_l$ versus $q$, which is derived from Eq. (3).