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Stabilizability and percolation in the infinite volume sandpile model. (English) Zbl 1165.60033

Summary: We study the sandpile model in infinite volume on \(\mathbb Z^d\). In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure \(\mu \), are \(\mu \)-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In \(d=1\) and \(\mu \) a product measure with density \(\rho =1\) (the known critical value for stabilizability in \(d=1)\) with a positive density of empty sites, we prove that \(\mu \) is not stabilizable.
Furthermore, we study, for values of \(\rho \) such that \(\mu \) is stabilizable, percolation of toppled sites. We find that for \(\rho >0\) small enough, there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail, as is the case in the subcritical regime for ordinary percolation.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
60G99 Stochastic processes
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References:

[1] Bak, P., Tang, C. and Wiesenfeld, K. (1988). Self-organized criticality. Phys. Rev. A (3) 38 364-374. · Zbl 1230.37103
[2] Diaconis, P. and Fulton, W. (1991). A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Rend. Sem. Mat. Univ. Politec. Torino 49 95-119. Commutative algebra and algebraic geometry, II (Italian) (Turin, 1990). · Zbl 0776.60128
[3] Dickman, R., Muñoz, M., Vespagnani, A. and Zapperi, S. (2000). Paths to self-organized criticality. Brazilian Journal of Physics 30 27-41.
[4] Fey-den Boer, A. and Redig, F. (2005). Organized versus self-organized criticality in the abelian sandpile model. Markov Process. Related Fields 11 425-442. · Zbl 1093.60069
[5] Grimmett, G. (1999). Percolation , 2nd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 321 . Springer, Berlin. · Zbl 0926.60004
[6] Ivashkevich, E. V. and Priezzhev, V. B. (1998). Introduction to the sandpile model. Physica A 254 97-116.
[7] Priezzhev, V. B. and Ktitarev, D. V. (1997). Minimal sandpiles on hexagonal lattice. J. Statist. Phys. 88 781-793. · Zbl 0945.82541
[8] Meester, R. and Quant, C. (2005). Connections between ‘self-organised’ and ‘classical’ criticality. Markov Process. Related Fields 11 355-370. · Zbl 1084.82008
[9] Redig, F. (2005). Mathematical aspects of the Abelian Sandpile Model. Les Houches lecture notes. · Zbl 1411.60150
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