Embedding binary sequences into Bernoulli site percolation on Z3

M.R. Hilário, B.N.B. de Lima, P. Nolin and V. Sidoravicius

Stochastic Processes and their Applications, 2014, vol. 124, issue 12, 4171-4181

Abstract: We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on Zd with parameter p. In 1995, I. Benjamini and H. Kesten proved that, for d⩾10 and p=1/2, all sequences can be embedded, almost surely. They conjectured that the same should hold for d⩾3. We consider d⩾3 and p∈(pc(d),1−pc(d)), where pc(d)<1/2 is the critical threshold for site percolation on Zd. We show that there exists an integer M=M(p), such that, a.s., every binary sequence, for which every run of consecutive 0s or 1s contains at least M digits, can be embedded.

Keywords: Percolation of words; Block renormalization (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1016/j.spa.2014.07.022

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