  Word $W$ is said to encounter word $V$ provided there is a homomorphism
$\phi$ mapping letters to nonempty words so that $\phi(V)$ is a substring of
$W$. For example, taking $\phi$ such that $\phi(h)=c$ and $\phi(u)=ien$, we see
that "science" encounters "huh" since $cienc=\phi(huh)$. The density of $V$ in
$W$, $\delta(V,W)$, 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, $\Sigma$ a finite
alphabet with at least 2 letters, and $W_n \in \Sigma^n$ chosen uniformly at
random. Word $V$ is doubled if and only if $\mathbb{E}(\delta(V,W_n))
\rightarrow 0$ as $n \rightarrow \infty$.
  We further explore convergence for nondoubled words and concentration of the
limit distribution for doubled words around its mean.
