Measures for transient configurations of the sandpile-model

Abstract

The Abelian Sandpile-Model (ASM) is a well-studied model for Self-Organized Criticality (SOC), for which many interesting algebraic properties have been proved. This paper deals with the process of starting with the empty configuration and adding grains of sand, until a recurrent configuration is reached. The notion studied in this paper is that the configurations at the beginning of the process are in a sense very far from being recurrent, while the configurations near the end of the process are quite close to being recurrent; this leads to the idea of ordering the transient configurations, such that configurations closer to being recurrent generally are greater than configurations far from recurrent. Then measures are defined which increase monotonically with respect to these orderings and can be interpreted as "degrees of recurrence". Diagrams for these measures are shown and briefly discussed. © Springer-Verlag Berlin Heidelberg 2006.