The Annals of Applied Probability

Kinetically constrained spin models on trees

F. Martinelli and C. Toninelli

Full-text: Open access

Abstract

We analyze kinetically constrained $0$–$1$ spin models (KCSM) on rooted and unrooted trees of finite connectivity. We focus in particular on the class of Friedrickson–Andersen models FA-jf and on an oriented version of them. These tree models are particularly relevant in physics literature since some of them undergo an ergodicity breaking transition with the mixed first-second order character of the glass transition. Here we first identify the ergodicity regime and prove that the critical density for FA-jf and OFA-jf models coincide with that of a suitable bootstrap percolation model. Next we prove for the first time positivity of the spectral gap in the whole ergodic regime via a novel argument based on martingales ideas. Finally, we discuss how this new technique can be generalized to analyze KCSM on the regular lattice ${\mathbb{Z}}^{d}$.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 5 (2013), 1967-1987.

Dates
First available in Project Euclid: 28 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1377696303

Digital Object Identifier
doi:10.1214/12-AAP891

Mathematical Reviews number (MathSciNet)
MR3134727

Zentralblatt MATH identifier
1284.60170

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Keywords
Kinetically constrained models dynamical phase transitions glass transition bootstrap percolation stochastic models on trees interacting particle systems

Citation

Martinelli, F.; Toninelli, C. Kinetically constrained spin models on trees. Ann. Appl. Probab. 23 (2013), no. 5, 1967--1987. doi:10.1214/12-AAP891. https://projecteuclid.org/euclid.aoap/1377696303


Export citation

References

  • [1] Aldous, D. and Diaconis, P. (2002). The asymmetric one-dimensional constrained Ising model: Rigorous results. J. Stat. Phys. 107 945–975.
  • [2] Balogh, J., Peres, Y. and Pete, G. (2006). Bootstrap percolation on infinite trees and non-amenable groups. Combin. Probab. Comput. 15 715–730.
  • [3] Cancrini, N., Martinelli, F., Roberto, C. and Toninelli, C. (2008). Kinetically constrained spin models. Probab. Theory Related Fields 140 459–504.
  • [4] Cancrini, N., Martinelli, F., Roberto, C. and Toninelli, C. (2009). Facilitated spin models: Recent and new results. In Methods of Contemporary Mathematical Statistical Physics (R. Kotecky, ed.). Lecture Notes in Math. 1970 307–340. Springer, Berlin.
  • [5] Cancrini, N., Martinelli, F., Roberto, C. and Toninelli, C. (2012). Mixing time of a kinetically constrained spin model on trees: Power law scaling at criticality. Preprint.
  • [6] Cancrini, N., Martinelli, F., Schonmann, R. and Toninelli, C. (2010). Facilitated oriented spin models: Some non equilibrium results. J. Stat. Phys. 138 1109–1123.
  • [7] Chalupa, J., Leath, P. L. and Reich, G. R. (1979). Bootstrap percolation on a Bethe lattice. J. Phys. C: Solid State Phys. 12 L31–L35.
  • [8] Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Probab. 12 999–1040.
  • [9] Fontes, L. R. and Schonmann, R. H. (2008). Threshold $\theta\geq2$ contact processes on homogeneous trees. Probab. Theory Related Fields 141 513–541.
  • [10] Fredrickson, G. and Andersen, H. (1984). Kinetic Ising model of the Glass transition. Phys. Rev. Lett. 53 1244–1247.
  • [11] Fredrickson, G. and Andersen, H. (1985). Facilitated kinetic Ising models and the glass transition. J. Chem. Phys. 83 5822–5831.
  • [12] Garrahan, J., Sollich, P. and Toninelli, C. (2011). Dynamical Heterogeneities in Glasses, Colloids, and Granular Media. Oxford Univ. Press, Oxford. Available at arXiv:1009.6113.
  • [13] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [14] Jackle, J., Mauch, F. and Reiter, J. (1992). Blocking transitions in lattice spin models with directed kinetic constraints. Phys. A 184 458–476.
  • [15] Kordzakhia, G. and Lalley, S. P. (2006). Ergodicity and mixing properties of the northeast model. J. Appl. Probab. 43 782–792.
  • [16] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • [17] Martinelli, F. (1999). Lectures on Glauber dynamics for discrete spin models. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 93–191. Springer, Berlin.
  • [18] Martinelli, F., Sinclair, A. and Weitz, D. (2004). Glauber dynamics on trees: Boundary conditions and mixing time. Comm. Math. Phys. 250 301–334.
  • [19] Martinelli, F. and Wouts, M. (2012). Glauber dynamics for the quantum Ising model in a transverse field on a regular tree. J. Stat. Phys. 146 1059–1088.
  • [20] Ritort, F. and Sollich, P. (2003). Glassy dynamics of kinetically constrained models. Adv. Phys. 52 219–342.
  • [21] Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Math. 1665 301–413. Springer, Berlin.
  • [22] Sausset, F., Toninelli, C., Biroli, G. and Tarjus, G. (2010). Bootstrap percolation and kinetically constrained models on hyperbolic lattices. J. Stat. Phys. 138 411–430.
  • [23] Schonmann, R. H. (1992). On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20 174–193.
  • [24] Schwartz, J. M., Liu, A. J. and Chayes, L. Q. (2006). The onset of jamming as the sudden emergence of an infinite $k$-core cluster. Europhysics Lett. 73 560–566.
  • [25] Sellitto, M., Biroli, G. and Toninelli, C. (2005). Facilitated spin models on Bethe lattice: Bootstrap percolation, mode coupling transition and glassy dynamics. Europhysics Lett. 69 496–512.