Abstract
We perform the rigorous analysis of the relaxation to equilibrium for some facilitated or kinetically constrained spin models (KCSM) when the initial distribution ν is different from the reversible one, μ. This setting has been intensively studied in the physics literature to analyze the slow dynamics which follows a sudden quench from the liquid to the glass phase. We concentrate on two basic oriented KCSM: the East model on ℤ, for which the constraint requires that the East neighbor of the to-be-update vertex is vacant and the AD model on the binary tree introduced in Aldous and Diaconis (J. Stat. Phys. 107(5–6):945–975, 2002), for which the constraint requires the two children to be vacant. It is important to observe that, while the former model is ergodic at any p≠1, the latter displays an ergodicity breaking transition at p c =1/2. For the East we prove exponential convergence to equilibrium with rate depending on the spectral gap if ν is concentrated on any configuration which does not contain a forever blocked site or if ν is a Bernoulli(p′) product measure for any p′≠1. For the model on the binary tree we prove similar results in the regime p,p′<p c and under the (plausible) assumption that the spectral gap is positive for p<p c . By constructing a proper test function, we also prove that if p′>p c and p≤p c convergence to equilibrium cannot occur for all local functions. Finally, in a short appendix, we present a very simple argument, different from the one given in Aldous and Diaconis (J. Stat. Phys. 107(5–6):945–975, 2002), based on a combination of some combinatorial results together with “energy barrier” considerations, which yields the sharp upper bound for the spectral gap of East when p↑1.
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)
Cancrini, N., Martinelli, F., Roberto, C., Toninelli, C.: Facilitated spin models: recent and new results. In Kotecky R. (ed.) Mathematics of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics, pp. 307–339. Springer, Berlin
Cancrini, N., Martinelli, F., Roberto, C., Toninelli, C.: Kinetically constrained spin models. Probab. Theory Relat. Fields 140(3–4), 459–504 (2008)
Chung, F., Diaconis, P., Graham, R.: Combinatorics for the east model. Adv. Appl. Math. 27(1), 192–206 (2001)
Fredrickson, G., Andersen, H.: Kinetic Ising model of the glass transition. Phys. Rev. Lett. 53, 1244–1247 (1984)
Fredrickson, G., Andersen, H.: Facilitated kinetic Ising models and the glass transition. J. Chem. Phys. 83, 5822–5831 (1985)
Giné, E., Grimmett, G.R., Saloff-Coste, L.: Lectures on probability theory and statistics. In: Lecture Notes in Mathematics, vol. 1665 (1997), pp. x+424
Grimmett, G.: Percolation Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, 2nd edn. Springer, Berlin (1999)
Haggstrom, O.: Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25(3), 1423–1436 (1997)
Jackle, J., Eisinger, S.: A hierarchically constrained kinetic Ising-model. Z. Phys. B, Condens. Mater 84(1), 115–124 (1991)
Kordzakhia, G., Lalley, S.P.: Ergodicity and mixing properties of the northeast model. J. Appl. Probab. 43(3), 782–792 (2006)
Leonard, S., Mayer, P., Sollich, P., Berthier, L., Garrahan, J.P.: Non-equilibrium dynamics of spin facilitated glass models. J. Stat. Mech.-Theory E P07017 (2007)
Martinelli, F.: Lectures on Glauber Dynamics for Discrete Spin Models. Lectures on Probability Theory and Statistics, Saint-Flour, 1997, pp. 93–191. Springer, Berlin (1999)
Mayer, P., Sollich, P.: Ageing in one-dimensional coagulation-diffusion processes and the Fredrickson-Andersen model. J. Phys. A, Math. Theor. 40(22), 5823–5856 (2007)
Ritort, F., Sollich, P.: Glassy dynamics of kinetically constrained models. Adv. Phys. 52(4), 219–342 (2003)
Sollich, P., Evans, M.: Glassy time-scale divergence and anomalous coarsening in a kinetically constrained spin chain. Phys. Rev. Lett. 83, 3238–3241 (1999)
Toninelli, C., Biroli, G.: Jamming percolation and glassy dynamics. J. Stat. Phys. 126(4–5), 731–763 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cancrini, N., Martinelli, F., Schonmann, R. et al. Facilitated Oriented Spin Models: Some Non Equilibrium Results. J Stat Phys 138, 1109–1123 (2010). https://doi.org/10.1007/s10955-010-9923-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-9923-x