Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Mixing times for a constrained Ising process on the two-dimensional torus at low density

Natesh S. Pillai and Aaron Smith

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We study a kinetically constrained Ising process (KCIP) associated with a graph $G$ and density parameter $p$; this process is an interacting particle system with state space $\{0,1\}^{G}$, the locations of the particles. The ‘constraint’ in the name of the process refers to the rule that a vertex cannot change its state unless it has at least one neighbour in state ‘1’. The KCIP has been proposed by statistical physicists as a model for the glass transition. In this note, we study the mixing time of a KCIP on the 2-dimensional torus $G=\mathbb{Z}_{L}^{2}$ in the low-density regime $p=\frac{c}{L^{2}}$ for arbitrary $0<c<\infty$, extending our previous results for the analogous process on the torus $\mathbb{Z}_{L}^{d}$ in dimension $d\geq3$. Our general approach is similar, but the extension requires more delicate bounds on the behaviour of the process at intermediate densities.


Nous étudions un processus d’Ising avec contraintes cinétiques (PICC) associé à un graphe $G$ et un paramètre de densité $p$. Ce processus est un système de particules en interaction avec espace d’états $\Omega=\{0,1\}^{G}$, décrivant les positions des particules. Les « contraintes » apparaissant dans le nom de ce processus réfèrent à la règle suivante: un sommet ne peut pas changer son état à moins qu’il ait un voisin dans l’état « 1 ». Le PICC a été proposé par des physiciens comme un modèle pour la transition vitreuse. Dans ce travail, nous analysons le temps de mélange d’un PICC sur le tore de dimension 2 $G=\mathbb{Z}_{L}^{2}$ dans le régime de faible densité $p=\frac{c}{L^{2}}$, où $0<c<\infty$. Ceci prolonge nos résultats au processus analogue sur le tore $G=\mathbb{Z}_{L}^{d}$, $d\geq3$. Notre approche générale est similaire, mais cette extension requiert des bornes plus délicates sur le comportement du processus aux densités intermédiaires.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1649-1678.

Received: 29 August 2017
Revised: 24 May 2018
Accepted: 4 September 2018
First available in Project Euclid: 25 September 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

Kinetically constrained process Interacting particle systems Markov chain Mixing time Glass transition


Pillai, Natesh S.; Smith, Aaron. Mixing times for a constrained Ising process on the two-dimensional torus at low density. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1649--1678. doi:10.1214/18-AIHP930.

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  • [1] D. Aldous. Open problems, 2017. Available at; last accessed 7-June-2017.
  • [2] M. Ben and Y. Peres. Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2) (2005) 245–266.
  • [3] O. Blondel, N. Cancrini, F. Martinelli, C. Roberto and C. Toninelli. Fredrickson–Andersen one spin facilitated model out of equilibrium. Markov Process. Related Fields 19 (2013) 383–406.
  • [4] N. Cancrini, F. Martinelli, C. Roberto and C. Toninelli. Kinetically constrained models. In New Trends in Mathematical Physics 741–752, 2009.
  • [5] P. Chleboun, A. Faggionato and F. Martinelli. Time scale separation and dynamic heterogeneity in the low temperature East model. Comm. Math. Phys. 328 (3) (2014) 955–993.
  • [6] P. Chleboun, A. Faggionato and F. Martinelli. Mixing time and local exponential ergodicity of the East-like process in $\mathbb{Z}^{d}$. Ann. Fac. Sci. Toulouse Math. (6) 24 (4) (2015) 717–743.
  • [7] P. Chleboun, A. Faggionato and F. Martinelli. Relaxation to equilibrium of generalized East processes on $\mathbb{Z}^{d}$: Renormalization group analysis and energy-entropy competition. Ann. Probab. 44 (3) (2016) 1817–1863.
  • [8] P. Chleboun and F. Martinelli. Mixing time bounds for oriented kinetically constrained spin models. Electron. J. Probab. 18 (2013) 1–9.
  • [9] P. Chleboun and A. Smith. Mixing of the square plaquette model on a critical length scale. Preprint, 2018. Available at arXiv:1807.00634.
  • [10] P. Clifford and A. Sudbury. A model for spatial conflict. Biometrika 60 (3) (1973) 581–588.
  • [11] M. Dyer, L. Goldberg, M. Jerrum and R. Martin. Markov chain comparison. Probab. Surv. 3 (2006) 89–111.
  • [12] P. Flajolet, D. Gardy and L. Thimonier. Birthday paradox, coupon collectors, caching algorithms and self-organizing search. Discrete Appl. Math. 39 (3) (1992) 207–229.
  • [13] G. Fredrickson and H. Andersen. Kinetic Ising model of the glass transition. Phys. Rev. Lett. 53 (1984) 1244–1247.
  • [14] G. Fredrickson and H. Andersen. Facilitated kinetic Ising models and the glass transition. J. Chem. Phys. 83 (1985) 5822–5831.
  • [15] J. Garrahan, P. Sollich and C. Toninelli. Dynamical Heterogeneities in Glasses, Colloids and Granular Media, Chapter 10. Oxford University Press, London, 2011.
  • [16] N. Hamamuki. A discrete isoperimetric inequality on lattices. Discrete Comput. Geom. 52 (2) (2014) 221–239.
  • [17] R. Holley and T. Liggett. Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3 (4) (1975) 643–663.
  • [18] N. Jain and S. Orey. On the range of random walk. Israel J. Math. 6 (4) (1968) 373–380.
  • [19] M. Jerrum, J.-B. Son, P. Tetali and E. Vigoda. Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains. Ann. Appl. Probab. 14 (4) (2004) 1741–1765.
  • [20] G. Kordzakhia and S. Lalley. Ergodicity and mixing properties of the Northeast model. J. Appl. Probab. 43 (3) (2006) 782–792.
  • [21] D. Levin, Y. Peres and E. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, Providence, RI, 2009.
  • [22] N. Madras and D. Randall. Markov chain decomposition for convergence rate analysis. Ann. Appl. Probab. 12 (2002) 581–606.
  • [23] F. Martinelli and C. Toninelli. Kinetically constrained spin models on trees. Ann. Appl. Probab. 23 (5) (2013) 1967–1987.
  • [24] F. Martinelli and C. Toninelli. Towards a universality picture for the relaxation to equilibrium of kinetically constrained models. Ann. Probab. 47 (1) (2019) 324–361.
  • [25] R. Oliveira. Mixing and hitting times for finite Markov chains. Electron. J. Probab. 17 (70) (2012) 1–12.
  • [26] R. Oliveira. On the coalescence time of reversible random walks. Trans. Amer. Math. Soc. 364 (4) (2012) 2109–2128.
  • [27] J. Palacios and P. Tetali. A note on expected hitting times for birth and death chains. Statist. Probab. Lett. 30 (1996) 119–125.
  • [28] Y. Peres and P. Sousi. Mixing times are hitting times of large sets. J. Theoret. Probab. 9 (2013) 459–510.
  • [29] D. Persi and L. Saloff-Coste. Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 (3) (1993) 696–730.
  • [30] D. Persi and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695–750.
  • [31] N. Pillai and A. Smith. Elementary bounds on mixing times for decomposable Markov chains. Stochastic Process. Appl. 127 (9) (2017) 3068–3109.
  • [32] N. S. Pillai and A. Smith. Mixing times for a constrained Ising process on the torus at low density. Ann. Probab. 45 (2) (2017) 1003–1070.
  • [33] A. Smith. Comparison theory for Markov chains on different state spaces and application to random walk on derangements. J. Theoret. Probab. 28 (4) (2015) 1406–1430.
  • [34] J. Theodore Cox. Coalescing random walks and voter model consensus times on the torus in $\mathbb{Z}^{d}$. Ann. Probab. 17 (4) (1989) 1333–1366.