Mixing time and eigenvalues of the abelian sandpile Markov chain

Abstract

The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph G G . 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 G G . We show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest noninteger 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 G G : If the latter has a sufficiently large spectral gap, then the former has a small gap! In the case where G G is the complete graph on n n vertices, we show that the sandpile chain exhibits cutoff at time 1 4 π 2 n 3 log ⁡ n \frac {1}{4\pi ^{2}}n^{3}\log n .