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Refined universality for critical KCM: lower bounds

Part of: Special processes Markov processes Combinatorial probability Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  03 March 2022

Ivailo Hartarsky Open the ORCID record for Ivailo Hartarsky [Opens in a new window]  and
Laure Marêché
Ivailo Hartarsky
Affiliation:
CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, PSL University, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France
Laure Marêché*
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000 Strasbourg, France
*
*Corresponding author. Email: [email protected]
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Abstract

We study a general class of interacting particle systems called kinetically constrained models (KCM) in two dimensions tightly linked to the monotone cellular automata called bootstrap percolation. There are three classes of such models, the most studied being the critical one. In a recent series of works by Martinelli, Morris, Toninelli and the authors, it was shown that the KCM counterparts of critical bootstrap percolation models with the same properties split into two classes with different behaviour. Together with the companion paper by the first author, our work determines the logarithm of the infection time up to a constant factor for all critical KCM, which were previously known only up to logarithmic corrections. This improves all previous results except for the Duarte-KCM, for which we give a new proof of the best result known. We establish that on this level of precision critical KCM have to be classified into seven categories instead of the two in bootstrap percolation. In the present work, we establish lower bounds for critical KCM in a unified way, also recovering the universality result of Toninelli and the authors and the Duarte model result of Martinelli, Toninelli and the second author.

Keywords

Kinetically constrained modelsbootstrap percolationuniversalityclassificationGlauber dynamicsspectral gap

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82C22: Interacting particle systems 60J27: Continuous-time Markov processes on discrete state spaces 60C05: Combinatorial probability
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 5 , September 2022 , pp. 879 - 906
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Aizenman, M. and Lebowitz, J. L. (1988) Metastability effects in bootstrap percolation. J. Phys. A 21(19) 3801–3813.CrossRefGoogle Scholar
Balister, P., Bollobás, B., Przykucki, M. and Smith, P. (2016) Subcritical $\,\mathcal {U}$ -bootstrap percolation models have non-trivial phase transitions. Trans. Amer. Math. Soc. 368(10) 7385–7411.CrossRefGoogle Scholar
Bollobás, B., Duminil-Copin, H., Morris, R. and Smith, P. (2017) The sharp threshold for the Duarte model. Ann. Probab. 45(6B) 4222–4272.CrossRefGoogle Scholar
Bollobás, B., Duminil-Copin, H., Morris, R. and Smith, P. (to appear) Universality of two-dimensional critical cellular automata. Proc. Lond. Math. Soc. Google Scholar
Bollobás, B., Smith, P. and Uzzell, A. (2015) Monotone cellular automata in a random environment. Combin. Probab. Comput. 24(4), 687–722.CrossRefGoogle Scholar
Cancrini, N., Martinelli, F., Roberto, C. and Toninelli, C. (2008) Kinetically constrained spin models. Probab. Theory Related Fields 140(3–4) 459–504.CrossRefGoogle Scholar
Cerf, R. and Cirillo, E. N. M. (1999) Finite size scaling in three-dimensional bootstrap percolation. Ann. Probab. 27(4) 1837–1850.CrossRefGoogle Scholar
Chung, F., Diaconis, P. and Graham, R. (2001) Combinatorics for the East model. Adv. Appl. Math. 27(1) 192–206.CrossRefGoogle Scholar
de Gregorio, P., Lawlor, A. and Dawson, K. A. (2009) Bootstrap percolation. In Encyclopedia of Complexity and Systems Science (R. A. Meyers, ed.), Springer, New York, NY, pp. 608–626.CrossRefGoogle Scholar
Duminil-Copin, H. and van Enter, A. C. D. (2013) Sharp metastability threshold for an anisotropic bootstrap percolation model. Ann. Probab. 41(3A) 1218–1242.CrossRefGoogle Scholar
Faggionato, A., Martinelli, F., Roberto, C. and Toninelli, C. (2013) The East model: recent results and new progresses. Markov Process. Related Fields 19(3), 407–452.Google Scholar
Fredrickson, G. H. and Andersen, H. C. (1984) Kinetic Ising model of the glass transition. Phys. Rev. Lett. 53(13) 1244–1247.CrossRefGoogle Scholar
Fredrickson, G. H. and Andersen, H. C. (1985) Facilitated kinetic Ising models and the glass transition. J. Chem. Phys. 83(11) 5822–5831.CrossRefGoogle Scholar
Garrahan, P., Sollich, P. and Toninelli, C. (2011) Kinetically constrained models. In Dynamical Heterogeneities in Glasses, Colloids and Granular Media, Vol. 150 of International Series of Monographs on Physics (L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti and W. van Saarloos, eds), Oxford University Press, Oxford, pp. 341–369.CrossRefGoogle Scholar
Gravner, J. and Holroyd, A. E. (2008) Slow convergence in bootstrap percolation. Ann. Appl. Probab. 18(3) 909–928.CrossRefGoogle Scholar
Hartarsky, I. (2021) $\mathcal U$ -bootstrap percolation: critical probability, exponential decay and applications. Ann. Inst. Henri Poincaré Probab. Stat. 57(3) 1255–1280.CrossRefGoogle Scholar
Hartarsky, I. (2021) Refined universality for critical KCM: upper bounds. arXiv e-prints, available at arXiv:2104.02329.Google Scholar
Hartarsky, I. and Marêché, L. (2020) Refined universality for critical KCM: lower bounds. arXiv e-prints, available at arXiv:2011.06952.Google Scholar
Hartarsky, I., Marêché, L. and Toninelli, C. (2020) Universality for critical KCM: infinite number of stable directions. Probab. Theory Related Fields 178(1) 289–326.CrossRefGoogle Scholar
Hartarsky, I., Martinelli, F. and Toninelli, C. (2020) Sharp threshold for the FA-2f kinetically constrained model. arXiv e-prints, available at arXiv:2012.02557.Google Scholar
Hartarsky, I., Martinelli, F. and Toninelli, C. (2021) Universality for critical KCM: finite number of stable directions. Ann. Probab. 49(5) 2141–2174.CrossRefGoogle Scholar
Hartarsky, I. and Morris, R. (2019) The second term for two-neighbour bootstrap percolation in two dimensions. Trans. Amer. Math. Soc. 372(9) 6465–6505.CrossRefGoogle Scholar
Holroyd, A. E. (2003) Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields 125(2) 195–224.CrossRefGoogle Scholar
Liggett, T. M. (2005) Interacting Particle Systems, Classics in Mathematics, Springer, Berlin, Heidelberg. Originally published by Springer, New York (1985).Google Scholar
Marêché, L. (2020) Combinatorics for general kinetically constrained spin models. SIAM J. Discrete Math. 34(1) 370–384.CrossRefGoogle Scholar
Marêché, L., Martinelli, F. and Toninelli, C. (2020) Exact asymptotics for Duarte and supercritical rooted kinetically constrained models. Ann. Probab. 48(1) 317–342.CrossRefGoogle Scholar
Martinelli, F., Morris, R. and Toninelli, C. (2019) Universality results for kinetically constrained spin models in two dimensions. Comm. Math. Phys. 369(2) 761–809.CrossRefGoogle Scholar
Martinelli, F. and Toninelli, C. (2019) Towards a universality picture for the relaxation to equilibrium of kinetically constrained models. Ann. Probab. 47(1) 324–361.CrossRefGoogle Scholar
Morris, R. (2017) Bootstrap percolation, and other automata. Eur. J. Combin. 66 250–263.CrossRefGoogle Scholar
Morris, R. (2017) Monotone cellular automata. In Surveys in Combinatorics 2017, Vol. 440 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, pp. 312–371.CrossRefGoogle Scholar
Ritort, F. and Sollich, P. (2003) Glassy dynamics of kinetically constrained models. Adv. Phys. 52(4) 219–342.CrossRefGoogle Scholar
Sollich, P. and Evans, M. R. (1999) Glassy time-scale divergence and anomalous coarsening in a kinetically constrained spin chain. Phys. Rev. Lett. 83(16) 3238–3241.CrossRefGoogle Scholar
van den Berg, J. and Kesten, H. (1985) Inequalities with applications to percolation and reliability. J. Appl. Probab. 22(3) 556–569.CrossRefGoogle Scholar