Kadanoff sand pile model. Avalanche structure and wave shape

Sand pile models are dynamical systems describing the evolution from N stacked grains to a stable configuration. It uses local rules to depict grain moves and iterate it until reaching a fixed configuration from which no rule can be applied. Physicists L. Kadanoff et al inspire KSPM, extending the well known Sand Pile Model (SPM). In KSPM(D), we start from a pile of N stacked grains and apply the rule: D−1 grains can fall from column i onto columns i+1, i+2, . . . , i+D−1 if the difference of height between columns i and i+1 is greater or equal to D. Toward the study of fixed points (stable configurations on which no grain can move) obtained from N stacked grains, we propose an iterative study of KSPM evolution consisting in the repeated addition of one grain on a heap of sand, triggering an avalanche at each iteration. We develop a formal background for the study of avalanches, resumed in a finite state word transducer, and explain how this transducer may be used to predict the form of fixed points. Further precise developments provide a plain formula for fixed points of KSPM(3), showing the emergence of a wavy shape.