Abstract
We present a construction of an entropy-preserving equivariant surjective map from the d-dimensional critical sandpile model to a certain closed, shift-invariant subgroup of \({\mathbb{T}^{\mathbb{Z}^d}}\) (the ‘harmonic model’). A similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the Bernoulli property of the measure of maximal entropy for that model.
Article PDF
Similar content being viewed by others
References
Athreya S.R., Járai A.A.: Infinite volume limit for the stationary distribution of abelian sandpile models. Commun. Math. Phys. 249(1), 197–213 (2004)
Athreya S.R., Járai A.A.: Erratum: Infinite volume limit for the stationary distribution of abelian sandpile models. Commun. Math. Phys. 264(3), 843 (2006)
Bak P., Tang C., Wiesenfeld K.: Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987)
Bak P., Tang C., Wiesenfeld K.: Self-organized criticality. Phys. Rev. A 38, 364–374 (1988)
Burton R., Pemantle R.: Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21, 1329–1371 (1993)
de Boor C., Höllig K., Riemenschneider S.: Fundamental solutions for multivariate difference equations. Amer. J. Math. 111, 403–415 (1989)
Daerden F., Vanderzande C.: Dissipative abelian sandpiles and random walks. Phys. Rev. E 63, 30301–30304 (2001)
Dhar D.: Self organized critical state of Sandpile Automaton models. Phys. Rev. Lett. 64, 1613–1616 (1990)
Dhar D.: The abelian sandpiles and related models. Phys. A 263, 4–25 (1999)
Dhar D.: Theoretical studies of self-organized criticality. Phys. A 369, 29–70 (2006)
Dhar D., Ruelle P., Sen S., Verma D.-N.: Algebraic aspects of abelian sandpile models. J. Phys. A 28, 805–831 (1995)
Fukai Y., Uchiyama K.: Potential kernel for the two-dimensional random walk. Ann. Probab. 24, 1979–1992 (1996)
Járai A., Redig F.: Infinite volume limit of the Abelian sandpile model in dimensions d ≥ 3. Probab. Theor. Relat. Fields 141, 181–212 (2008)
Lind D., Schmidt K.: Homoclinic points of algebraic Z d-actions. J. Amer. Math. Soc. 12, 953–980 (1999)
Lind D., Schmidt K., Ward T.: Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101, 593–629 (1990)
Maes C., Redig F., Saada E.: The infinite volume limit of dissipative abelian sandpiles. Commun. Math. Phys. 244, 395–417 (2004)
Pemantle R.: Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19, 1559–1574 (1991)
van der Put M., Tsang F.L.: Discrete Systems and Abelian Sandpiles. J. Alg. 322(1), 153–161 (2009)
Redig, F.: Mathematical aspects of the abelian sandpile model. Mathematical Statistical Physics, Volume Session LXXXIII: Lecture Notes of the Les Houches Summer School 2005 (Les Houches), Bovier, A., Dunlop, F., den Hollander, F., van Enter, A., Dalibard, J. (eds.), Amsterdam: Elsevier, (2006), pp. 657–728
Rudolph D.J., Schmidt K.: Almost block independence and Bernoullicity of \({\mathbb Z^{d}}\) -actions by automorphisms of compact groups. Invent. Math. 120, 455–488 (1995)
Schmidt K.: Dynamical Systems of Algebraic Origin. Basel-Berlin-Boston, Birkhäuser Verlag (1995)
Sheffield S.: Uniqueness of maximal entropy measure on essential spanning forests. Ann. Probab. 34, 857–864 (2006)
Sinai Ya.G.: On a weak isomorphism of transformations with invariant measure. Mat. Sb. 63(105), 23–42 (1964)
Solomyak R.: On coincidence of entropies for two classes of dynamical systems. Ergod. Th. & Dynam. Sys. 18, 731–738 (1998)
Spitzer F.: Principles of random walks. van Nostrand Reinhold, New York (1964)
Tsuchiya T., Tomori M.: Proof of breaking of self-organized criticality in a nonconservative abelian sandpile model. Phys. Rev. E 61, 1183–1188 (2000)
Uchiyama K.: Green’s function for random walks on Z N. Proc. London Math. Soc. 77, 215–240 (1998)
Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics, Vol. 79, Berlin- Heidelberg-New York: Springer Verlag, 1982
Acknowledgments
E.V. would like to acknowledge the hospitality of the Erwin Schrödinger Institute (Vienna), where part of this work was done. E.V. is also grateful to Frank Redig, Marius van der Put and Thomas Tsang for illuminating discussions. K.S. would like to thank EURANDOM (Eindhoven) and MSRI (Berkeley), for hospitality and support during part of this work.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Gallavotti
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Schmidt, K., Verbitskiy, E. Abelian Sandpiles and the Harmonic Model. Commun. Math. Phys. 292, 721–759 (2009). https://doi.org/10.1007/s00220-009-0884-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-009-0884-3