A型Iwahori--Hecke代数中q-变形经典洗牌的截止现象
Cutoff for q-deformed classical card shuffles in the type A Iwahori--Hecke algebra
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中文总结 AI 辅助
本文对角化q-变形k-星换位洗牌，并利用三种q-变形洗牌的谱证明它们在固定q>1时具有全变差和l^2截止，且改进了短系统扫描的上下界。
中文摘要 AI 辅助
我们研究了$\u005cmathcal{H}_q(S_n)$上三种q-变形洗牌的混合行为：\u005ccite{DiaconisRam2000}中引入的短系统扫描，\u005ccite{AxelrodFreedBraunerChiangComminsLang2024}中引入的q-变形随机到随机洗牌，以及q-变形k-星换位洗牌，其q=1情形在\u005ccite{ArfaeeNestoridi}中被研究。前两条链分别在\u005ccite{DiaconisRam2000}和\u005ccite{AxelrodFreedBraunerChiangComminsLang2024}中被对角化。这里，我们对q-变形k-星换位洗牌进行对角化，并利用三条链的谱来研究它们的混合行为。对于固定的q>1，我们证明这三者都表现出全变差和$\u005cell^2$截止现象。对于短系统扫描，这些结果改进了Diaconis和Ram在\u005ccite{DiaconisRam2000}中的上下界。
英文摘要
We study the mixing behavior of three q-deformed card shuffles on $\mathcal{H}_q(S_n)$: the short systematic scan introduced in \cite{DiaconisRam2000}, the $q$--deformed random--to--random shuffle introduced in \cite{AxelrodFreedBraunerChiangComminsLang2024}, and the $q$--deformed $k$--star transposition shuffle, whose $q=1$ counterpart was studied in \cite{ArfaeeNestoridi}. The first two chains were diagonalized in \cite{DiaconisRam2000} and \cite{AxelrodFreedBraunerChiangComminsLang2024}, respectively. Here, we diagonalize the $q$--deformed $k$--star transposition shuffle and use the spectra of the three chains to study their mixing behavior. For fixed $q>1$, we prove that all three exhibit total variation and $\ell^2$ cutoff. For the short systematic scan, these improve both the upper and lower bounds of Diaconis and Ram \cite{DiaconisRam2000}.
发表机构
- Stony Brook University（石溪大学）
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