Renormalization group approach to an Abelian sandpile model on planar lattices

TL;DR

This paper introduces a computer algorithm for exact enumeration of toppling processes in a lattice sandpile model, enabling larger cell size RG transformations and analysis of fixed points and critical exponents.

Contribution

The authors develop a novel algorithm for efficient exact enumeration in RG analysis of sandpile models, allowing larger cell sizes and correction of previous errors.

Findings

Identified the only attractive fixed point for square and triangular lattices.

Calculated avalanche exponent $ au$ and dynamical exponent $z$ for these lattices.

Increasing cell size does not necessarily improve RG result accuracy.

Abstract

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. $3 \times 3$ 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 $\tau$ 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.