Abstract
We analyze the relaxation time of a ferromagnetic d-dimensional growth model on the lattice. The model is characterized by d parameters which represent the activation energies of a site, depending on the number of occupied nearest neighbours. This model is a natural generalisation of the model studied by Dehghanpour and Schonmann (Probab Theory Relat Fields. 107(1):123–135, 1997), where the activation energy of a site with more than two occupied neighbours is zero.
References
Aizenman M., Lebowitz J.L.: Metastability effects in bootstrap percolation. J. Phys. A 21(19), 3801–3813 (1988)
Cerf R., Cirillo Emilio N.M.: Finite size scaling in three-dimensional bootstrap percolation. Ann. Probab. 27(4), 1837–1850 (1999)
Cerf R., Manzo F.: The threshold regime of finite volume bootstrap percolation. Stoch. Process. Appl. 101(1), 69–82 (2002)
Cerf, R., Manzo, F.: Nucleation and growth for the ising model in d dimensions at very low temperatures (preprint, 2011)
Dehghanpour P., Schonmann R.H.: Metropolis dynamics relaxation via nucleation and growth. Commun. Math. Phys. 188(1), 89–119 (1997)
Dehghanpour P., Schonmann R.H.: A nucleation and growth model. Probab. Theory Relat. Fields 107(1), 123–135 (1997)
Eden, M.: A two-dimensional growth process. In: Proceedings of the 4th Berkeley symposium on Mathematics Statistics and Probability, vol. IV. University of California Press, Berkeley, pp. 223–239 (1961)
Holroyd A.E.: Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Relat. Fields 125(2), 195–224 (2003)
Kesten H., Schonmann R.H.: On some growth models with a small parameter. Probab. Theory Relat. Fields 101(4), 435–468 (1995)
Liggett, T.M.: Interacting particle systems. Classics in Mathematics. Springer, Berlin (2005) (reprint of the 1985 original)
Richardson D.: Random growth in a tessellation. Proc. Cambridge Philos. Soc. 74, 515–528 (1973)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cerf, R., Manzo, F. A d-dimensional nucleation and growth model. Probab. Theory Relat. Fields 155, 427–449 (2013). https://doi.org/10.1007/s00440-011-0402-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-011-0402-3
Mathematics Subject Classification (2000)
- 60K35
- 82C05