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 .
We were drawn to the topic of sandpile mixing times in part by its intrinsic mathematical interest and in part by questions arising in statistical physics. Many questions about the sandpile model, even if they appear unrelated to mixing, seem to lead inevitably to the study of its mixing time. For instance, the failure of the density conjecture' @cite_0 @cite_12 @cite_3 and the distinction between the stationary and threshold states @cite_22 are consequences of slow mixing. Our characterization of the eigenvalues of the sandpile chain may also help explain the recent finding of memory on two time scales' for the sandpile dynamics on the @math -dimensional square grid @cite_2 .
@cite_22 @cite_3 @cite_0 @cite_2 @cite_12
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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. We consider the Abelian sandpile model (ASM) on the square lattice with a single dissipative site (sink). Particles are added one by one per unit time at random sites and the resulting density of particles is calculated as a function of time. We observe different scenarios of evolution depending on the value of initial uniform density (height) h0. During the first stage of the evolution, the density of particles increases linearly. Reaching a critical density ρc(h0), the system changes its behavior and relaxes exponentially to the stationary state of the ASM with density ρs. Considering initial heights −1≤h0≤4, we observe a dramatic decrease of the difference ρc(h0)−ρs when h0 is zero or negative. In parallel with the ASM, we consider the conservative fixed energy sandpile (FES). The extensive Monte Carlo simulations show that the threshold density ρth(h0) of the FES converges rapidly to ρs for h0<1. A popular theory of self-organized criticality relates driven dissipative systems to systems with conservation. This theory predicts that the stationary density of the Abelian sandpile model equals the threshold density of the fixed-energy sandpile. We refute this prediction for a wide variety of underlying graphs, including the square grid. Driven dissipative sandpiles continue to evolve even after reaching criticality. This result casts doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the Abelian sandpile model at stationarity. We report results of a numerical analysis of the memory effects in two-dimensional Abelian sandpiles. It is found that a sandpile forgets its instantaneous configuration in two distinct stages: a fast stage and a slow stage, whose durations roughly scale as N and N2 respectively, where N is the linear size of the sandpile. We confirm the presence of the longer time-scale by an independent diagnostic based on analysing emission probabilities of a hidden Markov model applied to a time-averaged sequence of avalanche sizes. The application of hidden Markov modelling to the output of sandpiles is novel. It discriminates effectively between a sandpile time series and a shuffled control time series with the same time-averaged event statistics and hence deserves further development as a pattern-recognition tool for Abelian sandpiles.