The abelian sandpile and related models

TL;DR

This paper reviews the Abelian sandpile model, highlighting its exact solvability, critical exponents, and connections to other models, including stochastic and non-abelian variants, emphasizing its role in understanding self-organized criticality.

Contribution

It provides a comprehensive review of known results, including exact calculations of critical exponents and extensions to stochastic and non-abelian variants, advancing the theoretical understanding of the model.

Findings

Exact calculation of critical exponents for the directed model in all dimensions.

Relation of the undirected model to the q=0 Potts model enabling some exact exponent calculations.

Introduction of stochastic and non-abelian variants with different universality classes.

Abstract

The Abelian sandpile model is the simplest analytically tractable model of self-organized criticality. This paper presents a brief review of known results about the model. The abelian group structure allows an exact calculation of many of its properties. In particular, one can calculate all the critical exponents for the directed model in all dimensions. For the undirected case, the model is related to q= 0 Potts model. This enables exact calculation of some exponents in two dimensions, and there are some conjectures about others. We also discuss a generalization of the model to a network of communicating reactive processors. This includes sandpile models with stochastic toppling rules as a special case. We also consider a non-abelian stochastic variant, which lies in a different universality class, related to directed percolation.