Abstract
Kinetically constrained models (KCM) are reversible interacting particle systems on \({\mathbb{Z}^{d}}\) with continuous timeMarkov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as \({\mathcal{U}}\)-bootstrap percolation. KCM also display some of the peculiar features of the so-called “glassy dynamics”, and as such they are extensively used in the physics literature to model the liquid-glass transition, a major and longstanding open problem in condensed matter physics. We consider two-dimensional KCM with update rule \({\mathcal{U}}\), and focus on proving universality results for the mean infection time of the origin, in the same spirit as those recently established in the setting of \({\mathcal{U}}\)-bootstrap percolation. We first identify what we believe are the correct universality classes, which turn out to be different from those of \({\mathcal{U}}\)-bootstrap percolation. We then prove universal upper bounds on the mean infection time within each class, which we conjecture to be sharp up to logarithmic corrections. In certain cases, including all supercritical models, and the well-known Duarte model, our conjecture has recently been confirmed in Marêché et al. (Exact asymptotics for Duarte and supercritical rooted kinetically constrained models). In fact, in these cases our upper bound is sharp up to a constant factor in the exponent. For certain classes of update rules, it turns out that the infection time of the KCM diverges much faster than for the corresponding \({\mathcal{U}}\)-bootstrap process when the equilibrium density of infected sites goes to zero. This is due to the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the monotone bootstrap dynamics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aldous D., Diaconis P.: The asymmetric one-dimensional constrained Ising model: rigorous results. J. Stat. Phys. 107(5-6), 945–975 (2002)
Andersen H.C., Fredrickson G.H.: Kinetic Ising model of the glass transition. Phys. Rev. Lett. 53(13), 1244–1247 (1984)
Asselah A., Dai Pra P.: Quasi-stationary measures for conservative dynamics in the infinite lattice. Ann. Prob. 29(4), 1733–1754 (2001)
Balister P., Bollobás B., Przykucki M.J., Smith P.: Subcritical \({\mathcal{U}}\)-bootstrap percolation models have non-trivial phase transitions. Trans. Am. Math. Soc. 368, 7385–7411 (2016)
Bollobás, B., Duminil-Copin, H., Morris, R., Smith, P.: Universality of two-dimensional critical cellular automata. In: Proceedings of the London Mathematical Society (2016, to appear). arXiv:1406.6680
Bollobás B., Duminil-Copin H., Morris R., Smith P.: The sharp threshold for the Duarte model. Ann. Prob. 45, 4222–4272 (2017)
Blondel O., Cancrini N., Martinelli F., Roberto C., Toninelli C.: Fredrickson-Andersen one spin facilitated model out of equilibrium. Markov Proc. Rel. Fields 19, 383–406 (2013)
Bollobás B., Smith P., Uzzell A.: Monotone cellular automata in a random environment. Comb. Probab. Comput. 24(4), 687–722 (2015)
Cancrini, N., Martinelli, F., Roberto, C., Toninelli, C.: Facilitated spin models: recent and new results, methods of contemporary mathematical statistical physics. Lecture Notes in Math., Vol. 1970, pp. 307–340. Springer, Berlin (2009)
Cancrini N., Martinelli F., Roberto C., Toninelli C.: Kinetically constrained spin models. Prob. Theory Rel. Fields 140(3-4), 459–504 (2008)
Cancrini N., Martinelli F., Schonmann R., Toninelli C.: Facilitated oriented spin models: some non equilibrium results. J. Stat. Phys. 138(6), 1109–1123 (2010)
Chleboun P., Faggionato A., Martinelli F.: Mixing time and local exponential ergodicity of the East-like process in \({\mathbb{Z}^{d}}\). Ann. Fac. Sci. Toulouse Math. Sér 6 24(4), 717–743 (2015)
Chleboun P., Faggionato A., Martinelli F.: Time scale separation and dynamic heterogeneity in the low temperature East model. Commun. Math. Phys. 328, 955–993 (2014)
Chleboun P., Faggionato A., Martinelli F.: Relaxation to equilibrium of generalized East processes on \({\mathbb{Z}^{d}}\):Renormalisation group analysis and energy-entropy competition.Ann. Prob. 44(3), 1817–1863 (2016)
Chung F., Diaconis P., Graham R.: Combinatorics for the East model. Adv. Appl. Math. 27(1), 192–206 (2001)
Duarte J.A.M.S.: Simulation of a cellular automaton with an oriented bootstrap rule. Phys. A. 157(3), 1075–1079 (1989)
Duminil-Copin H., van Enter A.C.D.: Sharp metastability threshold for an anisotropic bootstrap percolation model. Ann. Prob. 41, 1218–1242 (2013)
Duminil-Copin, H., van Enter, A.C.D., Hulshof, T.: Higher order corrections for anisotropic bootstrap percolation (2016). arXiv:1611.03294
van Enter A.C.D.: Proof of Straley’s argument for bootstrap percolation. J. Stat. Phys. 48, 943–945 (1987)
Faggionato A., Martinelli F., Roberto C., Toninelli C.: The East model: recent results and new progresses. Markov Proc. Rel. Fields 19, 407–458 (2013)
Faggionato A., Martinelli F., Roberto C., Toninelli C.: A ging through hierarchical coalescence in the East model. Commun. Math. Phys. 309, 459–495 (2012)
Garrahan, J.P., Sollich, P., Toninelli, C.: Kinetically constrained models. In: Berthier, L., Biroli, G., Bouchaud, J.-P., Cipelletti, L., van Saarloos, W. (eds.) Dynamical Heterogeneities in Glasses, Colloids, and Granular Media. Oxford University Press, Oxford (2011)
Jäckle J., Eisinger S.: A hierarchically constrained kinetic Ising model. Z. Phys. B: Condens. Matter 84(1), 115–124 (1991)
Levin D.A., Peres Y., Wilmer E.L.: Markov Chains and Mixing Times. AmericanMathematical Society, Providence (2008)
Liggett T.M.: Interacting Particle Systems. Springer, New York (1985)
Martinelli, F., Toninelli, C.: Towards a universality picture for the relaxation to equilibrium of kinetically constrained models. Ann. Prob. (2018, to appear). arXiv:1701.00107
Marêché, L.,Martinelli, F.,Toninelli, C.: Exact asymptotics for Duarte and supercritical rooted kinetically constrained models. arXiv:1807.07519
Mountford T.S.: Critical length for semi-oriented bootstrap percolation. Stoch. Proc. Appl. 56(2), 185–205 (1995)
Morris R.: Bootstrap percolation, and other automata. Eur. J. Combin. 66, 250–263 (2017)
Pillai N.S., Smith A.: Mixing times for a constrained Ising process on the torus at low density. Ann. Prob. 45(2), 1003–1070 (2017)
Ritort F., Sollich P.: Glassy dynamics of kinetically constrained models. Adv. Phys. 52(4), 219–342 (2003)
Saloff-Coste, L., Bernard, P.: Lectures on finite Markov chains. In: Bernard, P. (ed.) Lecture Notes in Mathematics, vol.1665. Springer, Berlin (1997)
Schonmann R.: On the behaviour of some cellular automata related to bootstrap percolation. Ann. Prob. 20, 174–193 (1992)
Schonmann R.: Critical points of two-dimensional bootstrap percolation-like cellular automata. J. Stat. Phys. 58, 1239–1244 (1990)
Sollich P., Evans M.R.: Glassy time-scale divergence and anomalous coarsening in a kinetically constrained spin chain. Phys. Rev. Lett 83, 3238–3241 (1999)
Acknowledgements
This work has been supported by the ERC Starting Grant 680275 “MALIG”, ANR-15-CE40-0020-01 and by the PRIN 20155PAWZB “Large Scale Random Structures”. RM is also partially supported by CNPq (Proc. 303275/2013-8) and by FAPERJ (Proc. 201.598/2014).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Duminil-Copin
Rights and permissions
About this article
Cite this article
Martinelli, F., Morris, R. & Toninelli, C. Universality Results for Kinetically Constrained Spin Models in Two Dimensions. Commun. Math. Phys. 369, 761–809 (2019). https://doi.org/10.1007/s00220-018-3280-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-018-3280-z