Critical density of activated random walks on transitive graphs. (English) Zbl 1397.82038

Summary: We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density \(\mu_{c}\) for sustained activity is strictly between 0 and 1. It was known that \(\mu_{c}>0\) on \(\mathbb{Z}^{d}\), \(d\geq1\), and that \(\mu_{c}<1\) on \(\mathbb{Z}\) for small enough sleeping rate. We show that \(\mu_{c}\rightarrow0\) as \(\lambda\rightarrow0\) in all vertex-transitive transient graphs, implying that \(\mu_{c}<1\) for small enough sleeping rate. We also show that \(\mu_{c}<1\) for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that \(\mu_{c}>0\) in any vertex-transitive amenable graph, and that \(\mu_{c}\in(0,1)\) for any sleeping rate on regular trees.


82C22 Interacting particle systems in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C27 Dynamic critical phenomena in statistical mechanics
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[1] Amir, G. and Gurel-Gurevich, O. (2010). On fixation of activated random walks. Electron. Commun. Probab.15 119-123. · Zbl 1231.60110
[2] Basu, R., Ganguly, S. and Hoffman, C. (2015). Non-fixation of symmetric activated random walk on the line for small sleep rate. Available at arXiv:1508.05677.
[3] Dickman, R., Rolla, L. T. and Sidoravicius, V. (2010). Activated random walkers: Facts, conjectures and challenges. J. Stat. Phys.138 126-142. · Zbl 1187.82104
[4] Gurel-Gurevich, O. and Nachmias, A. (2013). Nonconcentration of return times. Ann. Probab.41 848-870. · Zbl 1268.05183
[5] Rolla, L. T. and Sidoravicius, V. (2012). Absorbing-state phase transition for driven-dissipative stochastic dynamics on \({\mathbb{Z}}\). Invent. Math.188 127-150. · Zbl 1242.60104
[6] Rolla, L. T. and Tournier, L. (2015). Sustained activity for biased activated random walks at arbitrarily low density. Available at arXiv:1507.04732.
[7] Shellef, E. (2010). Nonfixation for activated random walks. ALEA Lat. Am. J. Probab. Math. Stat.7 137-149. · Zbl 1276.60118
[8] Sidoravicius, V. and Teixeira, A. (2014). Absorbing-state transitions for stochastic sandpiles and activated random walk. Available at arXiv:1412.7098. · Zbl 1362.60089
[9] Taggi, L. (2016). Absorbing-state phase transition in biased activated random walk. Electron. J. Probab.21 Paper No. 13. · Zbl 1336.60195
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