×

Universality and sharpness in activated random walks. (English) Zbl 1478.60269

Summary: We consider the activated random walk model in any dimension with any sleep rate and jump distribution and ergodic initial state. We show that the stabilization properties depend only on the average density of particles, regardless of how they are initially located on the lattice.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Amir, G., Gurel-Gurevich, O.: On fixation of activated random walks. Electron. Commun. Probab. 15, 119-123 (2010). https://doi.org/10.1214/ECP.v15-1536 · Zbl 1231.60110 · doi:10.1214/ECP.v15-1536
[2] Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364-374 (1988). https://doi.org/10.1103/PhysRevA.38.364 · Zbl 1230.37103 · doi:10.1103/PhysRevA.38.364
[3] Basu, R., Ganguly, S., Hoffman, C.: Non-fixation for conservative stochastic dynamics on the line. Commun. Math. Phys. 358, 1151-1185 (2018). https://doi.org/10.1007/s00220-017-3059-7 · Zbl 1393.60117 · doi:10.1007/s00220-017-3059-7
[4] Basu, R., Ganguly, S., Hoffman, C., Richey, J.: Activated random walk on a cycle. Ann Inst H Poincaré Probab. Statist. (to appear). arXiv:1709.09163 · Zbl 1442.60097
[5] Cabezas, M., Rolla, L.T., Sidoravicius, V.: Non-equilibrium phase transitions: activated random walks at criticality. J. Stat. Phys. 155, 1112-1125 (2014). https://doi.org/10.1007/s10955-013-0909-3 · Zbl 1297.82025 · doi:10.1007/s10955-013-0909-3
[6] Cabezas, M., Rolla, L.T., Sidoravicius, V.: Recurrence and density decay for diffusion-limited annihilating systems. Probab. Theory Relat. Fields 170, 587-615 (2018). https://doi.org/10.1007/s00440-017-0763-3 · Zbl 1429.60071 · doi:10.1007/s00440-017-0763-3
[7] Dickman, R., Muñoz, M.A., Vespignani, A., Zapperi, S.: Paths to self-organized criticality. Braz. J. Phys. 30, 27 (2000). https://doi.org/10.1590/S0103-97332000000100004 · doi:10.1590/S0103-97332000000100004
[8] Dickman, R., Rolla, L.T., Sidoravicius, V.: Activated random walkers: facts, conjectures and challenges. J. Stat. Phys. 138, 126-142 (2010). https://doi.org/10.1007/s10955-009-9918-7 · Zbl 1187.82104 · doi:10.1007/s10955-009-9918-7
[9] Fey, A., Levine, L., Wilson, D.B.: Driving sandpiles to criticality and beyond. Phys. Rev. Lett. 104, 145703 (2010). https://doi.org/10.1103/PhysRevLett.104.145703 · doi:10.1103/PhysRevLett.104.145703
[10] Fey, A., Meester, R.: Critical densities in sandpile models with quenched or annealed disorder. Markov Process. Relat. Fields 21, 57-83 (2015). arXiv:1211.4760 · Zbl 1326.60136
[11] Fey-den Boer, A., Redig, F.: Organized versus self-organized criticality in the Abelian sandpile model. Markov Process. Relat. Fields 11:425-442, (2005). arXiv:math-ph/0510060 · Zbl 1093.60069
[12] Jo, H.-H., Jeong, H.-C.: Comment on “driving sandpiles to criticality and beyond”. Phys. Rev. Lett. 105, 019601 (2010). https://doi.org/10.1103/PhysRevLett.105.019601 · doi:10.1103/PhysRevLett.105.019601
[13] Kerr, D., Li, H.: Ergodic Theory: Independence and Dichotomies. Springer Monographs in Mathematics. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-49847-8 · Zbl 1396.37001 · doi:10.1007/978-3-319-49847-8
[14] Levine, L.: Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle. Commun. Math. Phys. 335, 1003-1017 (2015). https://doi.org/10.1007/s00220-014-2216-5 · Zbl 1320.82039 · doi:10.1007/s00220-014-2216-5
[15] Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42. Cambridge University Press, New York (2016). https://doi.org/10.1017/9781316672815 · Zbl 1376.05002 · doi:10.1017/9781316672815
[16] Poghosyan, S.S., Poghosyan, V.S., Priezzhev, V.B., Ruelle, P.: Numerical study of the correspondence between the dissipative and fixed-energy abelian sandpile models. Phys. Rev. E 84, 066119 (2011). https://doi.org/10.1103/PhysRevE.84.066119 · doi:10.1103/PhysRevE.84.066119
[17] Rolla, L.T.: Activated random walks, 2015. Preprint. arXiv:1507.04341 · Zbl 1454.60144
[18] Rolla, L.T., Sidoravicius, V.: Absorbing-state phase transition for driven-dissipative stochastic dynamics on \[Z\] Z. Invent. Math. 188, 127-150 (2012). https://doi.org/10.1007/s00222-011-0344-5 · Zbl 1242.60104 · doi:10.1007/s00222-011-0344-5
[19] Rolla, L.T., Tournier, L.: Non-fixation for biased activated random walks. Ann. Inst. H. Poincaré Probab. Statist. 54, 938-951 (2018). https://doi.org/10.1214/17-AIHP827 · Zbl 1391.60242 · doi:10.1214/17-AIHP827
[20] Shellef, E.: Nonfixation for activated random walks. ALEA Lat. Am. J. Probab. Math. Stat. 7:137-149, 2010. http://alea.impa.br/articles/v7/07-07.pdf · Zbl 1276.60118
[21] Sidoravicius, V., Teixeira, A.: Absorbing-state transition for stochastic sandpiles and activated random walks. Electron. J. Probab. 22, 33 (2017). https://doi.org/10.1214/17-EJP50 · Zbl 1362.60089 · doi:10.1214/17-EJP50
[22] Stauffer, A., Taggi, L.: Critical density of activated random walks on transitive graphs. Ann. Probab. 46, 2190-2220 (2018). https://doi.org/10.1214/17-AOP1224 · Zbl 1397.82038 · doi:10.1214/17-AOP1224
[23] Taggi, L.: Absorbing-state phase transition in biased activated random walk. Electron. J. Probab. 21, 13 (2016). https://doi.org/10.1214/16-EJP4275 · Zbl 1336.60195 · doi:10.1214/16-EJP4275
[24] Taggi, L.: Active phase for activated random walks on \[{Z}^d\] Zd, \[d \ge 3\] d≥3, with density less than one and arbitrary sleeping rate. Ann. Inst. H. Poincaré Probab. Stat. (to appear). arXiv:1712.05292 · Zbl 1427.82025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.