1 research outputs found
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 RG approach to Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett.
{\bf 59}, 381 (1987)] and consider both the reduced-high RG equations proposed
by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. {\bf 72}, 1690
(1994)] and the real-height RG equations proposed by Ivashkevich [Phys. Rev.
Lett. {\bf 76}, 3368 (1996)]. Using this algorithm we are able to carry out RG
transformations more quickly with large cell size, e.g. 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 {\bf 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
and the dynamical exponent . 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.Comment: 29 pages, 6 figure