2003/03/24 by Yehiel Shilo, Ofer Biham · 15 citations
Mathematics · Physics and Astronomy · #Abelian sandpile model #Combinatorics #Complex Network Analysis Techniques #Connection (principal bundle) #Geometry #Lattice (music) #Mathematics #Physics #Quantum mechanics #Random walk #Statistical physics #Statistics #Stochastic process #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Universality (dynamical systems) #cond-mat
paper · pdf · doi:10.1103/physreve.67.066102
published in Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 67(6), 066102 (American Physical Society) · 18 pages, 6 figures
arxiv created 2003/03/24 · openalex publication_date 2003/06/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abelian sandpile models, both deterministic, such as the Bak, Tang, Wiesenfeld (BTW) model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] and stochastic, such as the Manna model [S.S. Manna, J. Phys. A 24, L363 (1991)] are studied on finite square lattices with open boundaries. The avalanche size distribution P(L)(n) is calculated for a range of system sizes, L. The first few moments of this distribution are evaluated numerically and their dependence on the system size is examined. The sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries. It is shown that the conservation law provides an interesting connection between the sandpile models and random-walk models. Using this connection, it is shown that the average avalanche sizes <n>(L) for the BTW and Manna models are equal to each other, and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size L. This is in spite of the fact that the sandpile models with deterministic (BTW) and stochastic (Manna) toppling rules exhibit different critical exponents, indicating that they belong to different universality classes.