1994/08/05 by D. Dhar, P. Ruelle, Dhar, D. +5 · 1 citation
Mathematics · Physics and Astronomy · #Condensed Matter (cond-mat) #FOS: Physical sciences #Geometric and Algebraic Topology #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat
paper · pdf · doi:10.48550/arxiv.cond-mat/9408020
39 pages, TIFR/TH/94-30
arxiv created 1994/08/05 · openalex publication_date 1994/08/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The abelian sandpile models feature a finite abelian group G generated by the operators corresponding to particle addition at various sites. We study the canonical decomposition of G as a product of cyclic groups G = Zd1 X Zd2 X Zd3...X Zdg, where g is the least number of generators of G, and di is a multiple of di+1. The structure of G is determined in terms of toppling matrix. We construct scalar functions, linear in height variables of the pile, that are invariant toppling at any site. These invariants provide convenient coordinates to label the recurrent configurations of the sandpile. For an L X L square lattice, we show that g = L. In this case, we observe that the system has nontrivial symmetries coming from the action of the cyclotomic Galois group of the (2L+2)th roots of unity which operates on the set of eigenvalues of the toppling matrix. These eigenvalues are algebraic integers, whose product is the order |G|. With the help of this Galois group, we obtain an explicit factorizaration of |G|. We also use it to define other simpler, though under-complete, sets of toppling invariants.