自由群中的随机字、非交叉匹配与RNA二级结构
Random words in free groups, non-crossing matchings and RNA secondary structures
摘要：
考虑由$4$字母组成的字母表中的随机单词$X^n=(X_1,\ldots ,X_n)$，其中字母被视为$A$、$U$、$G$和$C$（即，RNA序列中的核苷酸）或$α$、$\barα$、$β$和$\barβ$（即，自由群$\langleα,β\rangle$及其逆的生成元）。我们表明，预期的分数$ρ(n)$的未配对的碱基在一个最佳的RNA二级结构（只有沃森-克里克债券，没有伪结）收敛到一个恒定的$λ_2$与$0<λ_2<1$为$n\to\infty$。因此，随机RNA链的正比例碱基不形成氢键。对于自由群，$ρ(n)$是由生成元及其逆的共轭组成的生成集中代表$X$的最短字的长度与$X$相对于标准生成元及其逆的字长之比。因此，对于一个典型的字，在由标准生成元的共轭组成的（无限）生成集中的字长随着标准生成元中的字长线性增长。事实上，我们证明了一个类似的结果适用于所有的非阿贝尔群生成的自由群$\langleα_1,\dots,α_k\rangle$，$k\geq 2$。
英文摘要：
Consider a random word $X^n=(X_1,\ldots ,X_n)$ in an alphabet consisting of $4$ letters, with the letters viewed either as $A$, $U$, $G$ and $C$ (i.e., nucleotides in an RNA sequence) or $α$, $\barα$, $β$ and $\barβ$ (i.e., generators of the free group $\langleα,β\rangle$ and their inverses). We show that the expected fraction $ρ(n)$ of unpaired bases in an optimal RNA secondary structure (with only Watson-Crick bonds and no pseudo-knots) converges to a constant $λ_2$ with $0<λ_2<1$ as $n\to\infty$. Thus, a positive proportion of the bases of a random RNA string do not form hydrogen bonds. We do not know the exact value of $λ_2$, but we derive upper and lower bounds for it. In terms of free groups, $ρ(n)$ is the ratio of the length of the shortest word representing $X$ in the generating set consisting of conjugates of generators and their inverses to the word length of $X$ with respect to the standard generators and their inverses. Thus for a typical word the word length in the (infinite) generating set consisting of the conjugates of standard generators grows linearly with the word length in the standard generators. In fact, we show that a similar result holds for all non-abelian finitely generated free groups $\langleα_1,\dots,α_k\rangle$, $k\geq 2$.
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