Memory on multiple time-scales in an Abelian sandpile

Abstract

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.