Renormalization-group approach to an Abelian sandpile model on planar lattices.

One important step in the renormalization-group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in the RG approach to the Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. 59, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. 72, 1690 (1994)], and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. 76, 3368 (1996)]. Using this algorithm, we are able to carry out RG transformations more quickly with large cell size, e.g., 3x3 cell for the square (SQ) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E 51, 1711 (1995)]. For SQ and plane triangular (PT) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent tau and the dynamical exponent z. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.