Embedding binary sequences into Bernoulli site percolation on Z3Z3

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.