2015/11/02 by Jerison, Daniel C., Levine, Lionel, Pike, John
#05C50 (Secondary) #60J10 (Primary) 82C20 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.1511.00666
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 \frac14π2n3log n.