Mixing time and eigenvalues of the abelian sandpile Markov chain

TL;DR

This paper analyzes the spectral properties and mixing times of the abelian sandpile Markov chain, revealing relationships with graph Laplacian lattices and demonstrating cutoff phenomena on complete graphs.

Contribution

It provides explicit formulas for eigenvalues and eigenvectors, relates spectral gap to Laplacian lattice parameters, and uncovers an inverse relationship between sandpile and random walk spectral gaps.

Findings

Spectral gap relates to shortest non-integer vectors in the dual Laplacian lattice.

Mixing time is bounded by the Laplacian lattice's smoothing parameter.

On complete graphs, the chain exhibits cutoff at a specific mixing time.

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 $\frac{1}{4\pi^{2}}n^{3}\log n$.