Electronic Journal of Probability

Front evolution of the Fredrickson-Andersen one spin facilitated model

Oriane Blondel, Aurelia Deshayes, and Cristina Toninelli

Full-text: Open access


The Fredrickson-Andersen one spin facilitated model (FA-1f) on $\mathbb Z$ belongs to the class of kinetically constrained spin models (KCM). Each site refreshes with rate one its occupation variable to empty (respectively occupied) with probability $q$ (respectively $p=1-q$), provided at least one nearest neighbor is empty. Here, we study the non equilibrium dynamics of FA-1f started from a configuration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for $q$ larger than a threshold $\bar q<1$, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 1, 32 pp.

Received: 22 March 2018
Accepted: 16 November 2018
First available in Project Euclid: 4 January 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces

kinetically constrained models invariant measure coupling contact process

Creative Commons Attribution 4.0 International License.


Blondel, Oriane; Deshayes, Aurelia; Toninelli, Cristina. Front evolution of the Fredrickson-Andersen one spin facilitated model. Electron. J. Probab. 24 (2019), paper no. 1, 32 pp. doi:10.1214/18-EJP246. https://projecteuclid.org/euclid.ejp/1546571126

Export citation


  • [AD02] D. Aldous and P. Diaconis. The asymmetric one-dimensional constrained ising model: rigorous results. J. Stat. Phys., 107(5-6):945?975, 2002.
  • [BCM+13] 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(3):383–406, 2013.
  • [BD88] M.A. Burschka and R. Dickman. Nonequilibrium critical poisoning in a single-species model. Physics Letters A, 127(3):132–137, 1988.
  • [BFM78] R.C. Brower, M.A. Furman, and M. Moshe. Critical exponents for the reggeon quantum spin model. Physics Letters, 76B:213–219, 1978.
  • [Blo13] Oriane Blondel. Front progression in the East model. Stochastic Process. Appl., 123(9):3430–3465, 2013.
  • [Bol82] E. Bolthausen. On the central limit theorem for stationary mixing random fields. Ann. Probab., 10(4):1047–1050, 1982.
  • [DG83] Richard Durrett and David Griffeath. Supercritical contact processes on $\bf Z$. Ann. Probab., 11(1):1–15, 1983.
  • [Dur80] Richard Durrett. On the growth of one-dimensional contact processes. Ann. Probab., 8(5):890–907, 1980.
  • [GK11] Olivier Garet and Aline Kurtzmann. De l’intégration aux probabilités. Références sciences. Ellipses Marketing, 2011.
  • [GLM15] S. Ganguly, E. Lubetzky, and F. Martinelli. Cutoff for the east process. Comm. Math. Phys., 335(3):1287–1322, 2015.
  • [Har74] T. E. Harris. Contact interactions on a lattice. Ann. Probability, 2:969–988, 1974.
  • [LPW09] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov chains and mixing times. American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson.
  • [MV18] T. Mountford and G. Valle. Exponential convergence for the fredrikson-andersen one spin facilitated model. ArXiv e-prints 1609.01364, 2018.
  • [RS03] F. Ritort and P. Sollich. Glassy dynamics of kinetically constrained models. Advances in Physics, 52(4):219?342, 2003.