Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1156479 | Stochastic Processes and their Applications | 2014 | 11 Pages |
Abstract
We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on ZdZd with parameter pp. In 1995, I. Benjamini and H. Kesten proved that, for d⩾10d⩾10 and p=1/2p=1/2, all sequences can be embedded, almost surely. They conjectured that the same should hold for d⩾3d⩾3. We consider d⩾3d⩾3 and p∈(pc(d),1−pc(d))p∈(pc(d),1−pc(d)), where pc(d)<1/2pc(d)<1/2 is the critical threshold for site percolation on ZdZd. We show that there exists an integer M=M(p)M=M(p), such that, a.s., every binary sequence, for which every run of consecutive 0s or 1s contains at least MM digits, can be embedded.
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Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
M.R. Hilário, B.N.B. de Lima, P. Nolin, V. Sidoravicius,