Activated random walk on a cycle. (English. French summary) Zbl 1442.60097

Summary: We consider activated random walk (ARW), a particle system with mass conservation, on the cycle \(\mathbb{Z}/n\mathbb{Z} \). One starts with a mass density \(\mu >0\) of initially active particles, each of which performs a simple symmetric random walk at rate one and falls asleep at rate \(\lambda >0\). Sleepy particles become active on coming in contact with other active particles. There have been several recent results concerning fixation/non-fixation of the ARW dynamics on infinite systems depending on the parameters \(\mu\) and \(\lambda \). On the finite graph \(\mathbb{Z}/n\mathbb{Z} \), unless there are more than \(n\) particles, the process fixates (reaches an absorbing state) almost surely in finite time. In a first rigorous result for a finite system, establishing well known beliefs in the statistical physics literature, we show that the number of steps the process takes to fixate is linear in \(n\) (up to poly-logarithmic terms), when the density is sufficiently low compared to the sleep rate, and exponential in \(n\) when the sleep rate is sufficiently small compared to the density, reflecting the fixation/non-fixation phase transition in the corresponding infinite system as established in [L. T. Rolla and V. Sidoravicius, Invent. Math. 188, No. 1, 127–150 (2012; Zbl 1242.60104)].


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics


Zbl 1242.60104
Full Text: DOI arXiv Euclid


[1] G. Amir and O. Gurel-Gurevich. On fixation of activated random walks. Electron. Commun. Probab.12 (2010) 119-123. · Zbl 1231.60110 · doi:10.1214/ECP.v15-1536
[2] R. Basu, S. Ganguly and C. Hoffman. Non-fixation of symmetric activated random walk on the line for small sleep rate. Comm. Math. Phys. To appear, 2018. Preprint. Available at arXiv:1508.05677.
[3] P. Diaconis and W. Fulton. A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Rend. Semin. Mat. Univ. Politec. Torino49 (1) (1991) 95-119. · Zbl 0776.60128
[4] R. Dickman. Nonequilibrium phase transitions in epidemics and sandpiles. Phys. A306 (2002) 90-97. · Zbl 0994.82058 · doi:10.1016/S0378-4371(02)00488-0
[5] R. Dickman, M. Alava, M. A. Muñoz, J. Peltola, A. Vespignani and S. Zapperi. Critical behaviour of a one-dimensional fixed-energy stochastic sandpile. Phys. Rev. E64 (2001), 56104.
[6] R. Dickman, M. A. Muñoz, A. Vespignani and S. Zapperi. Paths to self-organized criticality. Braz. J. Phys.30 (2000) 27-41.
[7] R. Dickman, A. Vespignani and S. Zapperi. Self-organized criticality as an absorbing-state phase transition. Phys. Rev. E57 (1998) 5095-5105.
[8] R. Dickman, L. T. Rolla and V. Sidoravicius. Activated random walkers: Facts, conjectures and challenges. J. Stat. Phys.138 (1-3) (2010) 126-142. · Zbl 1187.82104 · doi:10.1007/s10955-009-9918-7
[9] K. Eriksson. Chip-firing games on mutating graphs. SIAM J. Discrete Math.9 (1) (1996) 118-128. · Zbl 0844.90140 · doi:10.1137/S0895480192240287
[10] A. Fey, L. Levine and D. B. Wilson. Driving sandpiles to criticality and beyond. Phys. Rev. Lett.104 (14) (2010) Article ID 145703.
[11] B. Hough, D. Jerison and L. Levine. Sandpiles on the square lattice. Preprint, 2017. Available at arXiv:1703.00827. · Zbl 1419.82014 · doi:10.1007/s00220-019-03408-5
[12] S. Janson. Tail bounds for sums of geometric and exponential random variables. Statist. Probab. Lett.135 (2018) 1-6. · Zbl 1392.60042 · doi:10.1016/j.spl.2017.11.017
[13] H. Kesten and V. Sidoravicius. Branching random walk with catalysts. Electron. J. Probab.8 (2003) 1-51. · Zbl 1064.60196 · doi:10.1214/EJP.v8-127
[14] H. Kesten and V. Sidoravicius. The spread of a rumor or infection in a moving population. Ann. Probab.33 (2005) 2402-2462. · Zbl 1111.60074 · doi:10.1214/009117905000000413
[15] H. Kesten and V. Sidoravicius. A phase transition in a model for the spread of an infection. Illinois J. Math.50 (2006) 547-634. · Zbl 1101.92040 · doi:10.1215/ijm/1258059486
[16] H. Kesten and V. Sidoravicius. A shape theorem for the spread of an infection. Ann. of Math. (2)167 (2008) 701-766. · Zbl 1202.92077 · doi:10.4007/annals.2008.167.701
[17] L. Levine. Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle. Comm. Math. Phys.335 (2) (2015) 1003-1017. · Zbl 1320.82039 · doi:10.1007/s00220-014-2216-5
[18] L. Levine and Y. Peres. Internal erosion and the exponent \(3/4\). Available at http://www.math.cornell.edu/ levine/erosion.pdf.
[19] S. Lübeck. Universal scaling behavior of non-equilibrium phase transitions. Internat. J. Modern Phys. B18 (31-32) (2004) 3977-4118. · Zbl 1111.82040
[20] S. S. Manna. Large-scale simulation of avalanche cluster distribution in sand pile model. J. Stat. Phys.59 (1990) 509-521.
[21] S. S. Manna. Two-state model of self-organized criticality. J. Phys. A: Math. Gen.24 (1991) L363-L369.
[22] L. T. Rolla and V. Sidoravicius. Absorbing-state phase transition for driven-dissipative stochastic dynamics on \(\mathbb{Z} \). Invent. Math.188 (1) (2012) 127-150. · Zbl 1242.60104 · doi:10.1007/s00222-011-0344-5
[23] E. Shellef. Nonfixation for activated random walks. ALEA Lat. Am. J. Probab. Math. Stat.7 (2010) 137-149. · Zbl 1276.60118
[24] V. Sidoravicius and A. Teixeira. Absorbing-state transition for stochastic sandpiles and activated random walks. Electron. J. Probab.22 (33) (2017) Article ID 22. · Zbl 1362.60089 · doi:10.1214/17-EJP50
[25] A. Stauffer and L. Taggi. Critical density of activated random walks on transitive graphs. Ann. Probab.46 (2018) 2190-2220. · Zbl 1397.82038 · doi:10.1214/17-AOP1224
[26] L. T. Rolla and L. Tournier. Non-fixation for biased activated random walks. Ann. Inst. Henri Poincaré Probab. Stat.54 (2) (2018) 938-951. · Zbl 1391.60242 · doi:10.1214/17-AIHP827
[27] L. Taggi. Absorbing-state phase transition in biased activated random walk. Electron. J. Probab.21 (2016) Article ID 13. · Zbl 1336.60195 · doi:10.1214/16-EJP4275
[28] A. Vespignani, R. Dickman, M. A. Muñoz and S. Zapperi. Driving, conservation, and absorbing states in sandpiles. Phys. Rev. Lett.81 (1998) 5676-5679.
[29] A. Vespignani, R. Dickman, M. A. Muñoz and S. Zapperi. Absorbing-state phase transitions in fixed-energy sandpiles. Phys. Rev. E62 (2000) 4564-4582.
[30] R. Vidigal and R. Dickman. Asymptotic behavior of the order parameter in a stochastic sandpile. J. Stat. Phys.118 (1) (2005) 1-25. · Zbl 1130.82024 · doi:10.1007/s10955-004-8775-7
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.