AI 中文总结
研究对称群\(\mathfrak{S}_n\)上由\(k\)-循环共轭类生成的随机游走的击中时间混合，结合固定时间近似与标记方案及奇偶兼容耦合，证明特定\(k\)范围结果并对所有\(2\leq k\leq n - 1\)提出猜想。
AI 中文摘要
本文研究由\(k\)-循环的共轭类生成的对称群\(\mathfrak{S}_n\)上的随机游走，其中\(2\leq k = o(n/(\log n)^4)\)。我们证明该游走呈现击中时间混合：在每张牌都被触碰的首次时刻，分布已接近平衡。对于奇数\(k\)，平衡测度是\(\mathfrak{A}_n\)上的均匀测度。对于偶数\(k\)，游走首先混合到由击中时间确定的奇偶混合，在此范围内该混合渐近为\(U_{\mathfrak{S}_n}\)。我们的论证结合了截止窗口附近随机\(k\)-循环游走的精细固定时间近似以及受Jain - Sawhney关于随机换位工作启发的辅助标记方案。主要新特性是一种奇偶兼容耦合，在统一框架中处理奇数和偶数\(k\)-循环。我们还证明了在相反情况\(k\geq n - o(n^{1/2})\)下的击中时间混合结果，并对所有\(2\leq k\leq n - 1\)提出一个猜想。
英文摘要
In this paper, we study the random walk on the symmetric group $\mathfrak{S}_n$ generated by the conjugacy class of $k$-cycles, where $2\le k=o(n/(\log n)^4)$. We prove that the walk exhibits hitting-time mixing: at the first time when every card has been touched, the distribution is already close to equilibrium. For odd $k$, the equilibrium measure is the uniform measure on $\mathfrak{A}_n$. For even $k$, the walk first mixes to the parity mixture determined by the hitting time, and in our range this mixture is asymptotically $U_{\mathfrak{S}_n}$. Our argument combines a refined fixed-time approximation for the random $k$-cycle walk near the cutoff window with an auxiliary marking scheme inspired by Jain-Sawhney's work (arXiv:2410.23944) on random transpositions. The main new feature is a parity-compatible coupling which handles both odd and even $k$-cycles in a unified framework. We also prove a hitting-time mixing result in the opposite regime $k\ge n-o(n^{1/2})$, and formulate a conjecture for all $2\le k\le n-1$.
Comments29 pages. Comments welcome!
