Embedding binary sequences into Bernoulli site percolation on ℤ³

2013/10/19 by Marcelo R. Hilário, Bernardo N. B. de Lima, Hilário, Marcelo R. +6
Mathematics · #60K35 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60K35

paper · pdf · doi:10.48550/arxiv.1310.5262

13 pages, 5 figures

arxiv created 2013/10/19 · openalex publication_date 2013/10/19 · arxiv updated 2013/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on ℤ^d with parameter p, known also as percolation of words. 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. In this paper we consider d ≥ 3 and p ∈ (p_c(d), 1-p_c(d)), where p_c(d)<1/2 is the critical threshold for site percolation on ℤ^d. 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.

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