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Percolation of Words on Z d with Long-Range Connections

Part of: Special processes Equilibrium statistical mechanics

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.
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Abstract

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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 

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