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Infinite volume limit for the stationary distribution of Abelian sandpile models. (English) Zbl 1085.82005

The abelian sandpile model (ASM) was introduced by P. Bak, C. Tang and K. Wiesenfield [Self-organized criticality. Phys. Rev. A 38, 364–374 (1988)]. It is a dynamical model in which dynamics drives a system towards a stationary state characterized by power law correlations. Thus far to prove this fact rigorously was possible only in 1D. Present work was inspired by earlier work by S. N. Majumdar and D. Dhar [Equivalence between the Abelian sandpile model and the \(q\to 0\) limit of the Potts model. Physica A 185, 129-145 (1992)] in which these authors demonstrated the equivalence between the ASM and \(q\to 0\) limit of the Potts model which, as is well known, produces the spaning tree configurations. Let \(\nu_\Lambda\) be a uniform measure on a set of recurrent states then, because of the mentioned equivalence, it can be mapped onto the uniform spanning tree measure on \(\Lambda\) for as long as the dimensionality d lies between 2 and 4. The arguments need serious adjustment for d greater than 4. Both cases were studied in this work

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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References:

[1] Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. A 59, 381–384 (1987)
[2] Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364–374 (1988) · Zbl 1230.37103 · doi:10.1103/PhysRevA.38.364
[3] Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Uniform spanning forests. Ann. Probab. 29, 1–65 (2001) · Zbl 1016.60009
[4] Dhar, D., Ramaswamy, R.: Exactly solved model of self-organized critical phenomena. Phys. Rev. Lett. 63, 1659–1662 (1989) · doi:10.1103/PhysRevLett.63.1659
[5] Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64, 1613–1616 (1990) · Zbl 0943.82553 · doi:10.1103/PhysRevLett.64.1613
[6] Dhar, D.: The Abelian sandpile and related models. Phys. A 263, 4–25 (1999) · doi:10.1016/S0378-4371(98)00493-2
[7] Dhar, D.: Studying Self-organized criticality with exactly solved models. Preprint (1999) http://arXiv.org/abs/cond-mat/9909009
[8] Dhar, D., Majumdar, S.N.: Abelian sandpile models on the Bethe lattice. J. Phys. A 23, 4333–4350 (1990) · doi:10.1088/0305-4470/23/19/018
[9] Ivashkevich, E.V., Priezzhev, V.B.: Introduction to the sandpile model. Phys. A 254, 97–116 (1998) · doi:10.1016/S0378-4371(98)00012-0
[10] Járai, A.A., Redig, F.: Infinite volume limits of high-dimensional sandpile models. In preparation
[11] Jensen, H.J.: Self-organized criticality. Emergent complex behavior in physical and biological systems. Cambridge Lecture Notes in Physics, 10, Cambridge: Cambridge University Press, 2000
[12] Lawler, G.F.: Intersections of random walks. Basel-Boston: Birkhäuser, softcover edition (1996) · Zbl 0925.60078
[13] Loève, M.: Probability theory I–II. Graduate Texts in Mathematics, 45–46, Berlin-Heidelberg-New York: Springer-Verlag, 4th edition 1977
[14] Maes, C., Redig, F., Saada, E.: The Abelian sandpile model on an infinite tree. Ann. Probab. 30, 2081–2107 (2002) · Zbl 1013.60075 · doi:10.1214/aop/1039548382
[15] Maes, C., Redig, F., Saada, E.: The infinite volume limit of dissipative abelian sandpiles. Commun. Math. Phys. 244, 395–417 (2004) · Zbl 1075.82013 · doi:10.1007/s00220-003-1000-8
[16] Maes, C., Redig, F., Saada, E., Van Moffaert, A.: On the thermodynamic limit for a one-dimensional sandpile process. Markov Process. Related Fields 6, 1–22 (2000) · Zbl 1005.82021
[17] Mahieu, S., Ruelle, P.: Scaling fields in the two-dimensional Abelian sandpile model. Phys. Rev. E 64, 066130 (2001) · doi:10.1103/PhysRevE.64.066130
[18] Majumdar, S.N., Dhar, D.: Height correlations in the Abelian sandpile model. J. Phys. A 24, L357–L362 (1991)
[19] Majumdar, S.N., Dhar, D.: Equivalence between the Abelian sandpile model and the q 0 limit of the Potts model. Physica A 185, 129–145 (1992) · doi:10.1016/0378-4371(92)90447-X
[20] Meester, R., Redig, F. and Znamenski, D.: The Abelian sandpile; a mathematical introduction. Markov Proccess. Related Fields 7, 509–523 (2002) · Zbl 0995.60094
[21] Pemantle, R.: Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19, 1559–1574 (1991) · Zbl 0758.60010 · doi:10.1214/aop/1176990223
[22] Priezzhev, V.B.: Structure of two-dimensional sandpile. I. Height Probabilities. J. Stat. Phys. 74, 955–979 (1994)
[23] Priezzhev, V.B.: The upper critical dimension of the Abelian sandpile model. J. Stat. Phys. 98, 667–684 (2000) · Zbl 1056.82008 · doi:10.1023/A:1018619323983
[24] Tebaldi, C., De Menech, M., Stella, A.L.: Multifractal scaling in the Bak-Tang-Wiesenfeld sandpile and edge events. Phys. Rev. Letters 83, 3952–3955 (1999) · doi:10.1103/PhysRevLett.83.3952
[25] Vespignani, A., Zapperi S.: How Self-organised criticality works: A unified mean-field picture. Phys. Rev. A 57, 6345–6361 (1988)
[26] Wilson, D.B.: Generating random spanning trees more quickly than the cover time. In: Proceedings of the Twenty-Eighth ACM Symposium on the Theory of Computing, New York: ACM, pp. 296–303 (1996) · Zbl 0946.60070
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