AI 中文总结
本文研究固定切割概率下翻手洗牌的全变差截止时间，证明其位于 $p^2/(2(1-p)\pi^2)n^2\log n$，并给出相对熵截止的窗口估计。
AI 中文摘要
我们研究固定切割概率 $p\in(0,1)$ 的翻手洗牌。在数据包反转约定下，每个间隙独立地以概率 $p$ 被切割，每个产生的数据包被原地反转。我们证明了全变差截止时间在 \\[ \frac{p^2}{2(1-p)\pi^2}\\,n^2\log n. \\] 下界使用精确的有限牌组余弦特征函数和一致条件方差估计。对于上界，一个合并耦合将换位差异简化为对称的杀死型双牌链。其主衰减率为 $(1-p)\pi^2/(p^2n^2)+o(n^{-2})$。一个边界通量恒等式限定了相邻差异概率之和，而不包含等于牌组大小的因子。剩余要素是排列上的双侧 $L^1$ 不等式，常数为至多 $\sqrt n\exp\{O(\sqrt{\log n\log\log(n+2)})\}$。其证明将固定的块表纤维分解为赋值词和独立的较小排列，并利用多元超几何不等式和显式路由比较闭合递归。我们还证明了相对熵截止，其窗口为 $O_p(n^2)$，围绕内在熵转变时间。其前导系数位于全变差系数和该值两倍之间；未获得精确的熵位置。证明陈述了其已建立函数不等式输入的归一化和假设。附录证明了中间静态界，并解释了从单尺度平均到递归纤维的转变。
英文摘要
We study the overhand shuffle with a fixed cut probability $p\in(0,1)$. In the packet-reversal convention, each gap is cut independently with probability $p$, and each resulting packet is reversed in place. We prove total-variation cutoff at \[ \frac{p^2}{2(1-p)π^2}\,n^2\log n. \] The lower bound uses an exact finite-deck cosine eigenfunction and a uniform conditional-variance estimate. For the upper bound, a coalescing coupling reduces a transposition discrepancy to a symmetric killed two-card chain. Its principal decay rate is $(1-p)π^2/(p^2n^2)+o(n^{-2})$. A boundary-flux identity bounds the sum of adjacent discrepancy probabilities without a factor equal to the deck size. The remaining ingredient is a two-sided $L^1$ inequality on permutations, with constant at most $\sqrt n\exp\{O(\sqrt{\log n\log\log(n+2)})\}$. Its proof decomposes a fixed block-table fiber into assignment words and independent smaller permutations, and closes a recursion using a multivariate hypergeometric inequality and an explicit routing comparison. We also prove relative-entropy cutoff with an $O_p(n^2)$ window about the intrinsic entropy transition time. Its leading coefficient lies between the total-variation coefficient and twice that value; a sharp entropy location is not obtained. The proof states the normalization and hypotheses of its established functional-inequality inputs. An appendix proves the intermediate static bounds and explains the change from one-scale averaging to recursive fibers.
