Density dichotomy in random words

Word W is said to encounter word V provided there is a homomorphism φ mapping letters to nonempty words so that φ(V) is a substring of W. For example, taking φ such that φ(h) = c and φ(u) = ien, we see that " science " encounters " huh " since cienc = φ(huh). The density of V in W , δ(V, W), is the proportion of substrings of W that are homomorphic images of V. So the density of " huh " in " science " is 2/ 8 2. A word is doubled if every letter that appears in the word appears at least twice. The dichotomy: Let V be a word over any alphabet, Σ a finite alphabet with at least 2 letters, and Wn ∈ Σ n chosen uniformly at random. Word V is doubled if and only if E(δ(V, Wn)) → 0 as n → ∞. We further explore convergence for nondoubled words and concentration of the limit distribution for doubled words around its mean.