星置换洗牌的击中时间混合
Hitting-time mixing for the star transposition shuffle
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中文总结 AI 辅助
该研究针对对称群Sₙ上的星置换洗牌，证明其在非顶牌首次被全部选中的时刻τ渐近混合，总变分距离有上界，通过与随机置换洗牌对比的方法引入了适用于非共轭不变洗牌的技术。
中文摘要 AI 辅助
我们证明了对称群Sₙ上星置换洗牌的击中时间类似截止现象。设τ为每张非顶牌都被选中的首次时刻，我们证明该洗牌在时刻τ渐近混合，更精确地说，Y_τ的分布与Sₙ上均匀分布的总变分距离至多为exp(-(log n)^(1/2+o(1)))。我们的证明通过两个转移核的同时对角化将星置换洗牌与随机置换洗牌进行比较，随后采用Jain和Sawhney的击中时间策略，这引入了一种可应用于未必共轭不变的洗牌的技术。
英文摘要
We prove a hitting-time analogue of cutoff for the star transposition shuffle on the symmetric group S_n. Let tau be the first time at which every non-top card has been selected. We show that the shuffle is asymptotically mixed at time tau: more precisely, the total variation distance between the law of Y_tau and the uniform distribution on S_n is at most exp(-(log n)^(1/2+o(1))). Our proof compares the star transposition shuffle with the random transposition shuffle using simultaneous diagonalization of the two transition kernels, and then adapts the hitting-time strategy of Jain and Sawhney. This introduces a technique that can be applied to card shuffles that are not necessarily conjugacy invariant.
