给定圈型随机置换生成的概率
Probability of generation by random permutations of given cycle type
摘要：
假设$π$和$π'$是具有约束循环类型的$S_n$的两个随机元素，使得$π$具有$x n^{1/2}$不动点和$yn/2$双循环，同样地，$π'$具有$x' n^{1/2}$不动点和$y'n/2$双循环。我们证明了$G = \langle π, π' \rangle$是传递的事件和$G \geq A_n$都有概率近似\[(1 - yy')^{1/2} \exp\left(- \frac{xx' + \frac12 x^2 y' + \frac12 {x'}^2 y}{1 - yy'}\right),\]，只要$(x, x')$不接近$(0, \infty)$或$(\infty, 0)$。这个公式是从作者最近的一篇论文（arXiv：1904.12180）中的一些初步结果导出的。作为应用，我们证明了$S_n$的一致随机共轭类的两个一致随机元以约51%的概率生成群。
英文摘要：
Suppose $π$ and $π'$ are two random elements of $S_n$ with constrained cycle types such that $π$ has $x n^{1/2}$ fixed points and $yn/2$ two-cycles, and likewise $π'$ has $x' n^{1/2}$ fixed points and $y'n/2$ two-cycles. We show that the events that $G = \langle π, π' \rangle$ is transitive and $G \geq A_n$ both have probability approximately \[(1 - yy')^{1/2} \exp\left(- \frac{xx' + \frac12 x^2 y' + \frac12 {x'}^2 y}{1 - yy'}\right),\] provided $(x, x')$ is not close to $(0, \infty)$ or $(\infty, 0)$. This formula is derived from some preliminary results in a recent paper (arXiv:1904.12180) of the authors. As an application, we show that two uniformly random elements of uniformly random conjugacy classes of $S_n$ generate the group with probability about 51%.
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。
