论文标题
自由组的随机单词，非交叉匹配和RNA二级结构
Random words in free groups, non-crossing matchings and RNA secondary structures
论文作者
论文摘要
考虑一个随机单词$ x^n =（x_1，\ ldots，x_n）$在一个由$ 4 $字母组成的字母中，将字母视为$ a $，$ a $，$ u $，$ g $和$ c $（即rna序列中的nucleotides）或$α$，$ \barα$，$，$β$（$ \β$ \ barβ$ \ barβ$（I.E.）。 $ \langleα，β\ rangle $及其倒置）。我们表明，在最佳RNA二级结构（只有Watson-Crick债券和没有伪结）中，预期的$ρ（n）$不成对基础的基础收敛到常数$λ_2$，$ 0 <λ_2<1 $ as $ n \ as $ n \ as $ n \ to \ infty $。因此，随机RNA串的碱基的正比例不形成氢键。我们不知道$λ_2$的确切值，但是我们为其得出了上限和下限。 就自由组而言，$ρ（n）$是代表$ x $的最短单词长度的比率，该生成集由发电机的偶联及其对倒置与标准生成器及其倒置相对于$ x $的单词长度的比率。因此，对于一个典型的单词，（无限）生成集中的单词长度由标准发电机的共轭物组成的（无限）生成集与标准发电机中的单词长度线性增长。实际上，我们表明，所有非亚洲有限生成的自由组$ \langleα_1，\ dots，α_k\ rangle $，$ k \ geq 2 $。
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 $\langleα_1,\dots,α_k\rangle$, $k\geq 2$.
