Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-08T19:29:51.042Z Has data issue: false hasContentIssue false
  • English
  • Français

Percolation of Words on Z d with Long-Range Connections

Part of: Equilibrium statistical mechanics Special processes

Published online by Cambridge University Press:  14 July 2016

B. N. B. de Lima ,
R. Sanchis  and
R. W. C. Silva
B. N. B. de Lima*
Affiliation:
Universidade Federal de Minas Gerais
R. Sanchis*
Affiliation:
Universidade Federal de Minas Gerais
R. W. C. Silva*
Affiliation:
Universidade Federal de Minas Gerais and Universidade Federal de Ouro Preto
*
∗ Postal address: Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, CP 702, CEP 30123-970 Belo Horizonte, MG, Brazil.
∗ Postal address: Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, CP 702, CEP 30123-970 Belo Horizonte, MG, Brazil.
∗∗∗ Postal address: Departamento de Estatística, UFMG, Av. Antônio Carlos 6627, CP 702, CEP 30123-970 Belo Horizonte, MG, Brazil.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider an independent site percolation model on Z d , with parameter p ∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameter p, there exists an integer K(p) such that all binary sequences (words) ξ ∈ {0, 1} N can be seen simultaneously, almost surely, even if all connections with length larger than K(p) are suppressed. We also show some results concerning how K(p) should scale with p as p goes to 0. Related results are also obtained for the question of whether or not almost all words are seen.

Keywords

Percolation of wordstruncation question

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B41: Random walks, random surfaces, lattice animals, etc. 82B43: Percolation
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 1152 - 1162
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Aizenmann, M. and Lebowitz, J. L. (1988). Metastability effects in bootstrap percolation. J. Phys. A 21, 3801–3813.Google Scholar
[2] Benjamini, I. and Kesten, H. (1995). Percolation of arbitrary words in {0,1}ℕ . Ann. Prob. 23, 1024–1060.Google Scholar
[3] Coppersmith, D., Tetali, P. and Winkler, P. (1993). Collisions among random walks on a graph. SIAM J. Discrete Math. 6, 363–374.Google Scholar
[4] De Lima, B. N. B. (2008). A note about the truncation question in percolation of words. Bull. Brazilian Math. Soc. 39, 183–189.Google Scholar
[5] Grimmett, G. (2010). Three problems for the clairvoyant demon. In Probability and Mathematical Genetics, eds Bingham, N. H. and Goldie, C. M., Cambridge University Press, pp. 380–396.Google Scholar
[6] Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. R. Soc. London A 430, 439–457.Google Scholar
[7] Grimmett, G. R., Liggett, T. M. and Richthammer, T. (2010). Percolation of arbitrary words in one dimension. Random Structures Algorithms 37, 85–99.Google Scholar
[8] Kesten, H. (1990). Asymptotics in high dimensions for percolation. In Disorder in Physical Systems, eds Grimmett, G. R. and Welsh, D. J. A., Oxford University Press, New York, pp. 219–240.Google Scholar
[9] Kesten, H., Sidoravicius, V. and Zhang, Y. (1998). Almost all words are seen in critical site percolation on the triangular lattice. Electron. J. Prob. 3, 75pp.Google Scholar
[10] Kesten, H., Sidoravicius, V. and Zhang, Y. (2001). Percolation of arbitrary words on the close-packed graph of {Z}2 . Electron. J. Prob. 6, 27pp.Google Scholar
[11] Peled, R. (2010). On rough isometries of Poisson processes on the line. Ann. Appl. Prob. 20, 462–494.Google Scholar
[12] Winkler, P. (2000). Dependent percolation and colliding random walks. Random Structures Algorithms 16, 58–84.Google Scholar