Mathematical aspects of the abelian sandpile model

In 1988, Bak, Tang and Wiesenfeld (BTW) introduced a lattice model of what they called “self-organized criticality”. Since its appearance, this model has been studied intensively, both in the physics and in the mathematics literature. It shows how a simple dynamics can lead to the emergence of very complex structures and drive the system towards a stationary state which shares several properties of equilibrium systems at the critical point, e.g. power-law decay of cluster sizes and of correlations of the height-variables. Some years later, Deepak Dhar generalized the model, discovered the “abelian group structure of addition operators” in it and called it “the abelian sandpile model”( abbreviated from now on ASM). He studied the self-organized critical nature of the stationary measure and gave an algorithmic characterization of recurrent configurations, the so-called “burning algorithm”. This algorithm gives a one-to one correspondence between the recurrent configurations of the ASM and rooted spanning trees. The correspondence with spanning trees allowed Priezzhev to compute the height probabilities in dimension 2 in the infinite-volume limit. Probabilities of certain special events -so-called “weakly allowed clusters”- can be computed exactly in the infinite-volume limit using the “Bombay-trick”. In the physics literature people studied critical exponents with scaling arguments, renormalization group method and conformal field theory (in d = 2), and it is argued that the upper critical dimension of the model is d = 4 [35]. Dhar and Majumdar studied the model on the Bethe lattice where they computed various correlation functions and avalanche cluster-size distributions exactly in the thermodynamic limit, using a transfer matrix approach. Since the discovery of the abelian group structure of the set of of recurrent configurations, in the mathematics literature (especially combinatorics and algebraic combinatorics) one (re)introduces the model under the names “chip-firing game, Dirichlet

[1]  C. Maes,et al.  The Infinite Volume Limit of Dissipative Abelian Sandpiles , 2002 .

[2]  Herbert Heyer,et al.  Probability Measures on Locally Compact Groups , 1977 .

[3]  C. Maes,et al.  On the thermodynamic limit for a one-dimensional sandpile process , 1998, math/9810093.

[4]  P. Ruelle,et al.  Logarithmic scaling for height variables in the Abelian sandpile model , 2004, cond-mat/0410253.

[5]  R. Pemantle Choosing a Spanning Tree for the Integer Lattice Uniformly , 1991, math/0404043.

[6]  S Mahieu,et al.  Scaling fields in the two-dimensional Abelian sandpile model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Norman Biggs,et al.  Chip-Firing and the Critical Group of a Graph , 1999 .

[8]  V. Priezzhev,et al.  Introduction to the sandpile model , 1998, cond-mat/9801182.

[9]  V. Priezzhev Structure of two-dimensional sandpile. I. Height probabilities , 1994 .

[10]  The Upper Critical Dimension of the Abelian Sandpile Model , 1999, cond-mat/9904054.

[11]  Fan Chung Graham,et al.  A chip-firing game and Dirichlet eigenvalues , 2002, Discret. Math..

[12]  D. Turcotte,et al.  Self-organized criticality , 1999 .

[13]  Deepak Dhar,et al.  Studying Self-Organized Criticality with Exactly Solved Models , 1999, cond-mat/9909009.

[14]  Russell Lyons,et al.  Uniform spanning forests , 2001 .

[15]  David M. Raup,et al.  How Nature Works: The Science of Self-Organized Criticality , 1997 .

[16]  Yvan Le Borgne,et al.  On the identity of the sandpile group , 2002, Discret. Math..

[17]  M. A. Muñoz,et al.  Paths to self-organized criticality , 1999, cond-mat/9910454.

[18]  The random geometry of equilibrium phases , 1999, math/9905031.

[19]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[20]  Christian Maes,et al.  The Abelian sandpile model on an infinite tree , 2002 .

[21]  Fhj Frank Redig,et al.  Infinite volume limits of high-dimensional sandpile models , 2004 .

[22]  A. Fey,et al.  Organized versus self-organized criticality in the abelian sandpile model , 2005, math-ph/0510060.

[23]  A. Járai Thermodynamic limit of the Abelian sandpile model on Z^d , 2005 .

[24]  S. Athreya,et al.  Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models , 2004 .

[25]  R. Meester,et al.  Connections between 'self-organised' and 'classical' criticality , 2005 .

[26]  Eugene R. Speer,et al.  Asymmetric abeiian sandpile models , 1993 .

[27]  Dhar,et al.  Self-organized critical state of sandpile automaton models. , 1990, Physical review letters.

[28]  R. Meester,et al.  The Abelian sandpile : a mathematical introduction , 2003, cond-mat/0301481.

[29]  Satya N. Majumdar,et al.  Abelian sandpile model on the bethe lattice , 1990 .

[30]  Satya N. Majumdar,et al.  Equivalence between the Abelian sandpile model and the q→0 limit of the Potts model , 1992 .

[31]  N. Biggs Algebraic Potential Theory on Graphs , 1997 .