The Abelian Sandpile Model: The connections between probabilistic and group theoretical approaches.

Abstract

The Abelian sandpile model (ASM) is a toy model from statistical physics used to study self-organized criticality. It has been studied both in terms of a Markov chain and in terms of divisors. In this thesis we will introduce both this probabilistic and graph geometric approach to the ASM and discuss some connections between the two approaches. We will discuss a new Torelli theorem for graphs studied by Griffith in 2023 and a newfound relation between the non-special divisors from this theorem and the minimal configurations of the Markov chain. We will give a new version of this Torelli theorem, which will for certain graphs state that two graphs are isomorphic if and only if their groups of recurrent configurations admit an isomorphism that induces a bijection on minimal configurations.