The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph @math . By viewing this chain as a (nonreversible) random walk on an abelian group, we give a formula for its eigenvalues and eigenvectors in terms of multiplicative harmonic functions' on the vertices of @math . We show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest non-integer vector in the dual Laplacian lattice, while the mixing time is at most a constant times the smoothing parameter of the Laplacian lattice. We find a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on @math : If the latter has a sufficiently large spectral gap, then the former has a small gap! In the case where @math is the complete graph on @math vertices, we show that the sandpile chain exhibits cutoff at time @math .
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 L*L 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.