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 $G$. 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 $G$. 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 $G$: If the latter has a sufficiently large spectral gap, then the former has a small gap! In the case where $G$ is the complete graph on $n$ vertices, we show that the sandpile chain exhibits cutoff at time $\frac{1}{4\pi^{2}}n^{3}\log n$.