Density dichotomy in random words

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

Toward the Combinatorial Limit Theory of Free Words

Free words are elements of a free monoid, generated over an alphabet via the binary operation of concatenation. Casually speaking, a free word is a finite string of letters. Henceforth, we simply refer to them as words. Motivated by recent advances in the combinatorial limit theory of graphs-notably those involving flag algebras, graph homomorphisms, and graphons-we investigate the extremal and asymptotic theory of pattern containment and avoidance in words. Word V is a factor of word W provided V occurs as consecutive letters within W. W is an instance of V provided there exists a nonerasing monoid homomorphsism {\phi} with {\phi}(V) = W. For example, using the homomorphism {\phi} defined by {\phi}(P) = Ror, {\phi}(h) = a, and {\phi}(D) = baugh, we see that Rorabaugh is an instance of PhD. W avoids V if no factor of W is an instance of V. V is unavoidable provided, over any finite alphabet, there are only finitely many words that avoid V. Unavoidable words were classified by Bean, Ehrenfeucht, and McNulty (1979) and Zimin (1982). We briefly address the following Ramsey-theoretic question: For unavoidable word V and a fixed alphabet, what is the longest a word can be that avoids V? The density of V in W is the proportion of nonempty substrings of W that are instances of V. Since there are 45 substrings in Rorabaugh and 28 of them are instances of PhD, the density of PhD in Rorabaugh is 28/45. We establish a number of asymptotic results for word densities, including the expected density of a word in arbitrarily long, random words and the minimum density of an unavoidable word over arbitrarily long words. This is joint work with Joshua Cooper.Comment: 110 pages, dissertatio