Almost All Words Are Seen in Critical Site Percolation on the Triangular Lattice

We consider critical site percolation on the triangular lattice, that is, we choose X(v) = 0 or 1 with probability 1/2 each, independently for all vertices v of the triangular lattice. We say that a word (ξ1, ξ2, . . . ) ∈ {0, 1}N is seen in the percolation configuration if there exists a selfavoiding path (v1, v2, . . . ) on the triangular lattice with X(vi) = ξi, i ≥ 1. We prove that with probability 1 ‘almost all’ words, as well as all periodic words, except the two words (1, 1, 1, . . . ) and (0, 0, 0, . . . ), are seen. ‘Almost all’ words here means almost all with respect to the measure μβ under which the ξi are i.i.d. with μβ{ξi = 0} = 1 − μβ{ξi = 1} = β (for an arbitrary 0 < β < 1). 1