通过重复输入'a'同步随机自动机
Synchronizing random automata through repeated 'a' inputs
AI总结：
本研究改进了随机自动机同步字长度的已有结论，证明对任意ε>0，以1−ε概率存在长度为O(ε⁻¹√(n log n))的同步字，证明基于随机自动机的两项关键性质。
摘要：
Chapuy和Perarnau近期的一篇论文证明，在n个状态、2字母字母表的均匀随机自动机中，高概率存在长度为O(√n log n)的同步字。本注记改进了这一结果：对任意ε>0，以1−ε的概率存在长度为O(ε⁻¹√(n log n))的同步字。证明基于随机自动机的两个性质：第一，存在长度为O(√(n log n))的字ω，使得输入ω后自动机可能状态数的期望为O(√(n/log n))；第二，高概率下每对状态都可被长度为O(log n)的字同步。
英文摘要：
In a recent article by Chapuy and Perarnau, it was shown that a uniformly chosen automaton on $n$ states with a $2$-letter alphabet has a synchronizing word of length $O(\sqrt{n}\log n)$ with high probability. In this note, we improve this result by showing that, for any $\varepsilon>0$, there exists a synchronizing word of length $O(\varepsilon^{-1}\sqrt{n \log n})$ with probability $1-\varepsilon$. Our proof is based on two properties of random automata. First, there are words $ω$ of length $O(\sqrt{n \log n})$ such that the expected number of possible states for the automaton, after inputting $ω$, is $O(\sqrt{n/\log n})$. Second, with high probability, each pair of states can be synchronized by a word of length $O(\log n)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
