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Phys Rev E. 2017 Oct;96(4-1):042115. doi: 10.1103/PhysRevE.96.042115. Epub 2017 Oct 9.

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

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Physics and Accelerators Research School, NSTRI, AEOI 11365-3486, Tehran, Iran.
Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran.


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}.


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