Physics and Accelerators Research School, NSTRI, AEOI 11365-3486, Tehran, Iran.
Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran.
Phys Rev E. 2017 Oct;96(4-1):042115. doi: 10.1103/PhysRevE.96.042115. Epub 2017 Oct 9.
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±0.02≈3/2=d_{f}^{GFF}.
我们对超立方晶格上的 Bak-Tang-Wiesenfeld（BTW）沙堆模型进行了广泛的数值模拟，处于上临界维度 Du=4。在重新提取了雪崩的临界指数后，我们专注于三维和二维（2D）横截面对诱导临界性的研究，这在几何和局部指数中有所体现。已经对所有维度进行了各种有限尺寸标度（FSS）理论的测试和验证。分布函数指数和分形维数之间的超尺度关系对于所有维度都是有效的。我们发现，对于 d=2 和 3，雪崩质量分布函数的指数在 d 维横截面上与 d 维 BTW 模型相同。然而，几何量相对于相同维度的 BTW 模型具有完全不同的行为。通过分析二维横截面上几何指数的 FSS 理论，我们提出二维诱导模型与高斯自由场（GFF）具有一定程度的相似性。尽管一些局部指数略有不同，但这种相似性对于分形维数来说非常好。显示这种特征的最重要一个是环的分形维数 df，发现它为 1.50±0.02≈3/2=dfGFF。
