Density dichotomy in random words

TL;DR

This paper investigates the density of homomorphic images of words within random words, establishing a dichotomy based on whether the word is doubled, and explores convergence and concentration properties.

Contribution

It introduces a density dichotomy for words in random sequences, linking the property of being doubled to the asymptotic behavior of homomorphic image density.

Findings

Doubled words have density tending to zero in large random words.

Non-doubled words exhibit different convergence behaviors.

Concentration results describe the distribution of densities for doubled words.

Abstract

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.