Dhar, D. ; Ruelle, P. ; Sen, S. ; Verma, D. N. (1995) Algebraic aspects of Abelian sandpile models Journal of Physics A: Mathematical and General, 28 (4). pp. 805831. ISSN 03054470

PDF
 Publisher Version
299kB 
Official URL: http://iopscience.iop.org/03054470/28/4/009
Related URL: http://dx.doi.org/10.1088/03054470/28/4/009
Abstract
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=Z(d_{1})^{∗}Z(d_{2})^{∗}Z(d_{3})...^{∗}Z(d_{g}), where g is the least number of generators of G, and d_{i} is a multiple of d_{i+1}. The structure of G is determined in terms of the toppling matrix Delta . We construct scalar functions, linear in the height variables of the pile, that are invariant under toppling at any site. These invariants provide convenient coordinates to label the recurrent configurations of the sandpile. For an L^{∗}L square lattice, we show that g=L. In this case, we observe that the system has nontrivial symmetries, transcending the obvious symmetries of the square, namely those coming from the action of the cyclotomic Galois group GalL of the 2(L+1)th roots of unity (which operates on the set of eigenvalues of h). These eigenvalues are algebraic integers, the product of which is the order mod G mod . With the help of Gal_{L} we are able to group the eigenvalues into certain subsets the products of which are separately integers, and thus obtain an explicit factorization of mod G mod . We also use Gal_{L} to define other simpler sets of toppling invariants.
Item Type:  Article 

Source:  Copyright of this article belongs to Institute of Physics Publishing. 
ID Code:  9375 
Deposited On:  02 Nov 2010 12:20 
Last Modified:  16 May 2016 19:11 
Repository Staff Only: item control page