Year: 2026
Type: preprint
Abstract: 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}.
Source: arXiv (Cornell University)
Authors: Arfaee, Samira, Wong, Bryan
Institution: Stony Brook University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
Award: Mixing times and Cutoff
OpenAlex ID: W7213362286