## Annals of Probability

### Exact asymptotics for Duarte and supercritical rooted kinetically constrained models

#### Abstract

Kinetically constrained models (KCM) are a class of interacting particle systems which represent a natural stochastic (and nonmonotone) counterpart of the family of cellular automata known as $\mathcal{U}$-bootstrap percolation. A key issue for KCM is to identify the divergence of the characteristic time scales when the equilibrium density of empty sites, $q$, goes to zero. In (Ann. Probab. 47 (2019) 324–361; Comm. Math. Phys. 369 (2019) 761–809), a general scheme was devised to determine a sharp upper bound for these time scales. Our paper is devoted to developing a (very different) technique which allows to prove matching lower bounds. We analyse the class of two-dimensional supercritical rooted KCM and the Duarte KCM. We prove that the relaxation time and the mean infection time diverge for supercritical rooted KCM as $e^{\Theta ((\log q)^{2})}$ and for Duarte KCM as $e^{\Theta ((\log q)^{4}/q^{2})}$ when $q\downarrow 0$. These results prove the conjectures put forward in (European J. Combin. 66 (2017) 250–263; Comm. Math. Phys. 369 (2019) 761–809) for these models, and establish that the time scales for these KCM diverge much faster than for the corresponding $\mathcal{U}$-bootstrap processes, the main reason being the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the bootstrap dynamics.

#### Article information

Source
Ann. Probab., Volume 48, Number 1 (2020), 317-342.

Dates
First available in Project Euclid: 25 March 2020

https://projecteuclid.org/euclid.aop/1585123330

Digital Object Identifier
doi:10.1214/19-AOP1362

Mathematical Reviews number (MathSciNet)
MR4079438

Zentralblatt MATH identifier
07206760

#### Citation

Marêché, Laure; Martinelli, Fabio; Toninelli, Cristina. Exact asymptotics for Duarte and supercritical rooted kinetically constrained models. Ann. Probab. 48 (2020), no. 1, 317--342. doi:10.1214/19-AOP1362. https://projecteuclid.org/euclid.aop/1585123330

#### References

• [1] Aldous, D. and Diaconis, P. (2002). The asymmetric one-dimensional constrained Ising model: Rigorous results. J. Stat. Phys. 107 945–975.
• [2] Asselah, A. and Dai Pra, P. (2001). Quasi-stationary measures for conservative dynamics in the infinite lattice. Ann. Probab. 29 1733–1754.
• [3] 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 7385–7411.
• [4] Berthier, L. and Biroli, G. (2011). Theoretical perspective on the glass transition and amorphous materials. Rev. Modern Phys. 83 587–645.
• [5] Bollobás, B., Duminil-Copin, H., Morris, R. and Smith, P. (2016). Universality of two-dimensional critical cellular automata. Proc. London Math. Soc. To appear. arXiv.org:1406.6680.
• [6] Bollobás, B., Duminil-Copin, H., Morris, R. and Smith, P. (2017). The sharp threshold for the Duarte model. Ann. Probab. 45 4222–4272.
• [7] Bollobás, B., Smith, P. and Uzzell, A. (2015). Monotone cellular automata in a random environment. Combin. Probab. Comput. 24 687–722.
• [8] Cancrini, N., Martinelli, F., Roberto, C. and Toninelli, C. (2008). Kinetically constrained spin models. Probab. Theory Related Fields 140 459–504.
• [9] Chleboun, P., Faggionato, A. and Martinelli, F. (2014). Time scale separation and dynamic heterogeneity in the low temperature East model. Comm. Math. Phys. 328 955–993.
• [10] Chleboun, P., Faggionato, A. and Martinelli, F. (2016). Relaxation to equilibrium of generalized East processes on $\mathbb{Z}^{d}$: Renormalization group analysis and energy-entropy competition. Ann. Probab. 44 1817–1863.
• [11] Chung, F., Diaconis, P. and Graham, R. (2001). Combinatorics for the East model. Adv. in Appl. Math. 27 192–206.
• [12] Duarte, J. A. M. S. (1989). Simulation of a cellular automat with an oriented bootstrap rule. Phys. A 157 1075–1079.
• [13] Faggionato, A., Martinelli, F., Roberto, C. and Toninelli, C. (2013). The East model: Recent results and new progresses. Markov Process. Related Fields 19 407–452.
• [14] Garrahan, J. P., Sollich, P. and Toninelli, C. (2011). Kinetically constrained models. In Dynamical Heterogeneities in Glasses, Colloids, and Granular Media (L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti and W. van Saarloos, eds.) Oxford Univ. Press, Oxford.
• [15] Jäckle, J. and Eisinger, S. (1991). A hierarchically constrained kinetic Ising model. Z. Phys. B, Condens. Matter 84 115–124.
• [16] Keys, A. S., Garrahan, J. P. and Chandler, D. (2013). Calorimetric glass transition explained by hierarchical dynamic facilitation. Proc. Natl. Acad. Sci. USA 110 4482–4487.
• [17] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
• [18] Marêché, L. (2020). Combinatorics for general kinetically constrained spin models. SIAM J. Discrete Math. 34 370–384.
• [19] Martinelli, F., Morris, R. and Toninelli, C. (2019). Universality results for kinetically constrained spin models in two dimensions. Comm. Math. Phys. 369 761–809.
• [20] Martinelli, F. and Toninelli, C. (2019). Towards a universality picture for the relaxation to equilibrium of kinetically constrained models. Ann. Probab. 47 324–361.
• [21] Morris, R. (2017). Bootstrap percolation, and other automata. European J. Combin. 66 250–263.
• [22] Mountford, T. S. (1995). Critical length for semi-oriented bootstrap percolation. Stochastic Process. Appl. 56 185–205.
• [23] Reed, M. and Simon, B. (1973). Methods of Modern Mathematical Physics: Functional Analysis. Academic Press, New York.
• [24] Ritort, F. and Sollich, P. (2003). Glassy dynamics of kinetically constrained models. Adv. Phys. 52 219–342.
• [25] Sollich, P. and Evans, M. R. (2003). Glassy dynamics in the asymmetrically constrained kinetic Ising chain. Phys. Rev. E 031504.