Embedding binary sequences into Bernoulli site percolation on

We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on Z d 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 ∈ ( 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.