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 pseudoinverse @math has appeared before in the context of sandpiles: It is used by Bj " o rner, Lov ' a sz and Shor @cite_20 to bound the number of topplings until a configuration stabilizes; see @cite_14 for a recent improvement. In addition, the pseudoinverse is a crucial ingredient in the energy pairing' of Baker and Shokrieh @cite_32 .
@cite_14 @cite_32 @cite_20
2158827560 2116480361 2165425090 2058525073
We prove a precise relationship between the threshold state of the fixed-energy sandpile and the stationary state of Dhar’s abelian sandpile: in the limit as the initial condition s 0 tends to ( - ) , the former is obtained by size-biasing the latter according to burst size, an avalanche statistic. The question of whether and how these two states are related has been a subject of some controversy since 2000. A polynomial ideal encoding topplings in the abelian sandpile model on a graph is introduced. A Grbner basis of this ideal is interpreted combinatorially in terms of well-connected subgraphs. This gives rise to algorithms to determine the identity and the operation in the group of recurrent configurations. We define stabilizability of an infinite volume height configuration and of a probability measure on height configurations. We show that for high enough densities, a probability measure cannot be stabilized. We also show that in some sense the thermodynamic limit of the uniform measures on the recurrent configurations of the abelian sandpile model (ASM) is a maximal element of the set of stabilizable measures. In that sense the self-organized critical behavior of the ASM can be understood in terms of an ordinary transition between stabilizable and non-stabilizable. Key-words: Self-organized criticality, abelian sandpile model, activated random walkers, stabilizability. AMS classification: 60K35 (primary), 60G60 (secondary) 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.