Algebraic aspects of Abelian sandpile models

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. 805-831. ISSN 0305-4470

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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(d1)Z(d2)Z(d3)...Z(dg), 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 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 LL square lattice, we show that g=L. In this case, we observe that the system has non-trivial 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 GalL 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 GalL 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
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