Dissipation in the Abelian Sandpile Model

Abstract

The Abelian Sandpile model was originally introduced by Bak, Tang and Wiesenfeld in 1987 as a paradigm for self-organized criticality. In this thesis, we study a variant of this model, both from the point of view of mathematics, as well as from the point of view of physics. The effect of dissipation and creation of mass is investigated. By linking the avalanche dynamics of the infinite-volume sandpile model to random walks, we derive some criteria on the amount of dissipation and creation of mass in order for the model to be critical or non-critical. As an example we prove that a finite amount of conservative sites on a totally dissipative lattice is not critical, and more generally, if the distance to a dissipative site is uniformly bounded from above, then the model is not critical. We apply also applied a renormalisation method to the model in order to deduce its critical exponents and to determine whether a constant bulk dissipation destroys critical behaviour. Numerical simulations and a statistical analysis are performed to estimate critical exponents. Finally, we give a short discussion on self-organized criticality.