1998/08/31 by Deepak Dhar · 8 citations
Mathematics · Physics and Astronomy · #Abelian group #Abelian sandpile model #Algebraic structure #Critical exponent #Exponent #Generalization #Markov Chains and Monte Carlo Methods #Potts model #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Universality (dynamical systems) #cond-mat.stat-mech
paper · pdf · doi:10.1016/s0378-4371(98)00493-2
published as Physica A 263(1999) 4. · Typos and minor errors fixed and some references added
arxiv created 1998/10/22 · openalex publication_date 1999/02/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.