Abstract
We study zero-temperature Glauber dynamics on \({\mathbb{Z}^d}\) , which is a dynamic version of the Ising model of ferromagnetism. Spins are initially chosen according to a Bernoulli distribution with density p, and then the states are continuously (and randomly) updated according to the majority rule. This corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field, to one at zero temperature with no external field. Define \({p_c(\mathbb{Z}^d)}\) to be the infimum over p such that the system fixates at ‘ + ’ with probability 1. It is a folklore conjecture that \({p_c(\mathbb{Z}^d) = 1/2}\) for every \({2 \le d \in \mathbb{N}}\) . We prove that \({p_c(\mathbb{Z}^d) \to 1/2}\) as d → ∞.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aizenman M., Lebowitz J.L.: Metastability effects in bootstrap percolation. J. Phys. A. 21, 3801–3813 (1988)
Arratia R.: Site recurrence for annihilating random walks on \({\mathbb{Z}_d}\) . Ann. Probab. 11, 706–713 (1983)
Balogh J., Bollobás B.: Bootstrap percolation on the hypercube. Probab. Theory Relat. Fields 134, 624–648 (2006)
Balogh J., Bollobás B., Morris R.: Majority bootstrap percolation on the hypercube. Comb. Probab. Comput. 18, 17–51 (2009)
Balogh J., Bollobás B., Morris R.: Bootstrap percolation in three dimensions. Ann. Probab. 37, 1329–1380 (2009)
Balogh, J., Bollobás, B., Duminil-Copin, H., Morris, R.: The sharp threshold for r-neighbour bootstrap percolation (in preparation)
Balogh, J., Bollobás, B., Morris, R.: Bootstrap percolation in high dimensions (submitted)
Balogh J., Peres Y., Pete G.: Bootstrap percolation on infinite trees and non-amenable groups. Comb. Probab. Comput. 15, 715–730 (2006)
Balogh J., Pittel B.: Bootstrap percolation on random regular graphs. Random Struct. Algorithms 30, 257–286 (2007)
Berger N., Kenyon C., Mossel E., Peres Y.: Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Relat. Fields 131, 311–340 (2005)
Biskup M., Schonmann R.H.: Metastable behavior for bootstrap percolation on regular trees. J. Stat. Phys. 136, 667–676 (2009)
Camia F., De Santis E., Newman C.M.: Clusters and recurrence in the two-dimensional zero-temperature stochastic Ising model. Ann. Appl. Probab. 12, 565–580 (2002)
Caputo P., Martinelli F.: Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree. Probab. Theory Relat. Fields 136, 37–80 (2006)
Cerf R., Cirillo E.N.M.: Finite size scaling in three-dimensional bootstrap percolation. Ann. Probab. 27, 1837–1850 (1999)
Cerf R., Manzo F.: The threshold regime of finite volume bootstrap percolation. Stoch. Proc. Appl. 101, 69–82 (2002)
Chalupa J., Leath P.L., Reich G.R.: Bootstrap percolation on a Bethe latice. J. Phys. C. 12, L31–L35 (1979)
Erdős P., Ney P.: Some problems on random intervals and annihilating particles. Ann. Probab. 2, 828–839 (1974)
Fontes L.R.G., Schonmann R.H.: Bootstrap percolation on homogeneous trees has 2 phase transitions. J. Stat. Phys. 132, 839–861 (2008)
Fontes L.R., Schonmann R.H., Sidoravicius V.: Stretched exponential fixation in stochastic Ising models at zero temperature. Commun. Math. Phys. 228, 495–518 (2002)
Holroyd A.: Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Relat. Fields 125, 195–224 (2003)
Howard C.D.: Zero-temperature Ising spin dynamics on the homogeneous tree of degree three. J. Appl. Probab. 37, 736–747 (2000)
Howard C.D., Newman C.M.: The percolation transition for the zero-temperature stochastic Ising model on the hexagonal lattice. J. Stat. Phys. 111, 57–72 (2003)
Janson S.: On percolation in random graphs with given vertex degrees. Electron. J. Probab. 14, 86–118 (2009)
Lootgieter J.C.: Problèmes de recurrence concernant des mouvements aléatoires de particules sur \({\mathbb{Z}}\) avec destruction. Ann. Inst. H. Poincaré 13, 127–139 (1977)
Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717, pp. 93–191. Springer, Berlin (1998)
Martinelli F., Sinclair A., Weitz D.: Glauber dynamics on trees: boundary conditions and mixing time. Commun. Math. Phys. 250, 301–334 (2004)
Morris, R.: The phase transition for bootstrap percolation in two dimensions (in preparation)
Nanda, S., Newman, C.M., Stein, D.: Dynamics of Ising spin systems at zero temperature. In: Minlos, R., Shlosman, S., Suhov, Y. (eds.) On Dobrushin’s Way (From Probability Theory to Statistical Mechanics). Am. Math. Soc. Transl. 198(2), 183–194 (2000)
Newman C.M., Stein D.L.: Zero-temperature dynamics of Ising spin systems following a deep quench: results and open problems. Physica A 279, 156–168 (2000)
Schonmann R.H.: On the behaviour of some cellular automata related to bootstrap percolation. Ann. Probab. 20, 174–193 (1992)
Schwartz D.: On hitting probabilities for an annihilating particle model. Ann. Probab. 6, 398–403 (1978)
Sidoravicius V., Camia F., Newman C.M.: Approach to fixation for zero-temperature stochastic Ising models on the hexagonal lattice. Prog. Probab. 53, 163–183 (2002)
Wu C.C.: Zero-temperature dynamics of Ising models on the triangular lattice. J. Stat. Phys. 106, 369–373 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was supported during this research by MCT grant PCI EV-8C. This work partly done whilst at the Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil.
Rights and permissions
About this article
Cite this article
Morris, R. Zero-temperature Glauber dynamics on \({\mathbb{Z}^d}\) . Probab. Theory Relat. Fields 149, 417–434 (2011). https://doi.org/10.1007/s00440-009-0259-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-009-0259-x