Abstract

In this paper we define an extension of the sand pile model (SPM) and, more generally, of the ice pile model (IPM) by adding a further dimension to the system. By drawing a parallel between these unidimensional and bidimensional models we will find some common features and some differences. We will show that, like for \(\text{SPM}(n)\), not all plane partitions are accessible in \(\text{BSPM}(n)\) starting from the initial state. However, it appears to be much more difficult to characterize the partitions that are accessible in \(\text{BSPM}(n)\): we are able to give some necessary but not sufficient conditions for a partition to be accessible. On the other hand, we show how several properties of the ice pile model in one dimension can be generalized when one adds a second dimension.