TY - JOUR

T1 - Sandpile models and random walkers on finite lattices

AU - Shilo, Yehiel

AU - Biham, Ofer

PY - 2003

Y1 - 2003

N2 - 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 [Formula presented] 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 [Formula presented] 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.

AB - 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 [Formula presented] 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 [Formula presented] 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.

UR - http://www.scopus.com/inward/record.url?scp=84924612146&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.67.066102

DO - 10.1103/PhysRevE.67.066102

M3 - Article

AN - SCOPUS:84924612146

SN - 1063-651X

VL - 67

SP - 8

JO - Physical Review E

JF - Physical Review E

IS - 6

ER -