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 $2/{8 \choose 2}$. 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 $W_n \in Σ^n$ chosen uniformly at random. Word $V$ is doubled if and only if $\mathbb{E}(δ(V,W_n)) \rightarrow 0$ as $n \rightarrow \infty$. We further explore convergence for nondoubled words and concentration of the limit distribution for doubled words around its mean.

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