17 problems

Let $M$ be an indecomposable, doubly stochastic matrix indexed by an alphabet of size $k$, satisfying … where $k+1\equiv1$. A word of length $N$ is generated by the Markov chain wi…

Let $W_{M,N}$ be the set of words of length $N$ on $M$ letters, equipped with the uniform distribution, and let the RSK correspondence map a word to a Young-diagram shape…

Let $n,k,m$ be positive integers with $k\ge2$, and let $R_{n,k,m}$ be the maximum power of length $m$ in a random word in $[k]^n$. The preceding bounds show that … for a universal…

Consider the square lattice ${\mathcal{G}}=({\mathbb{Z}}^2,{\mathbb{E}}_V\cup {\mathbb{E}}_H)$ with vertical edge probability $p_V=\varepsilon$ and horizontal connection probabilit…

Let $T_r$ denote the correction to the $r$th cumulant of $X^{(n)}_3$ arising from dependence among the pair indicators. Cumulant-correction conjecture. … The surrounding argument e…

Let $X^{(n)}_3$ denote the number of distinct adjacent pairs of distinct letters in a geometrically distributed word of length $n$. Gaussian-limit conjecture. The asymptotic distri…

Let $X^{(n)}_1$ be the number of distinct adjacent pairs of identical letters in a geometrically distributed word of length $n$, and let … where the independent random variables…

Let $w$ be a binary word, and call it a shuffle square if its positions can be partitioned into two sets whose induced subsequences are identical. Consider binary words of length…

Let $Z$ be the number of occurrences of a pattern $w$ as a subsequence in a random text of length $n$, and let $\binom{n}{m}$ be the number of possible positions for a subsequence…

Let $w$ be a pattern and let $Z$ denote its number of subsequence occurrences in a random text. For constant patterns, the limiting distribution is either asymptotically normal or…

Let $V$ and $W$ be independent random words of length $2n$, written as concatenations $V=V_1V_2$ and $W=W_1W_2$, where each component has length $n$. Define … and write…

Let $\Sigma$ be a finite alphabet, and let $V,W\sim\Sigma^n$ be independent random words of equal length $n$. Suppose $\operatorname{LCS}(V,W)=\ell$, and let … be a common subseque…

Let $LCI_n$ denote the length of the longest common and increasing subsequence in the random-word model, and let $p_{\max}$ and $k^*$ be as above. The two limiting random variables…

For the longest common subsequences of two random words, let $g$ be the shape function and let $\kappa$ be its curvature power at $(1,1)$. Curvature power conjecture. The curvature…

Let a random word be generated by an iid sequence on an ordered alphabet, with non-uniform letter probabilities. In the non-uniform case, the limiting law for the top row is descri…

Let $\varepsilon>0$ and set … Here $\mathbb{P}_p$ is the percolation probability, $\mathbb{V}$ is the vertex set, $\mu_\alpha$ is the word measure, $S_v(\omega)$ is the set of word…

Fluctuation conjecture. If $0<\alpha<\frac12$, then