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Cutoff for q-deformed classical card shuffles in the type A Iwahori--Hecke algebra

  • Samira Arfaee
  • Bryan Wong
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Abstract

We study the mixing behavior of three q-deformed card shuffles on ℋq(Sn): 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 ℓ2 cutoff. For the short systematic scan, these improve both the upper and lower bounds of Diaconis and Ram \cite{DiaconisRam2000}.

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Cite

BibTeXRIS
BibTeX
@misc{arfaee2026cutoff,
  title = {{Cutoff for q-deformed classical card shuffles in the type A Iwahori--Hecke algebra}},
  author = {Samira Arfaee and Bryan Wong},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13570},
  eprint = {2609.13570},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13570},
}
RIS
TY  - UNPB
TI  - Cutoff for q-deformed classical card shuffles in the type A Iwahori--Hecke algebra
AU  - Samira Arfaee
AU  - Bryan Wong
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.13570
UR  - https://arxiv.org/abs/2609.13570
AB  - We study the mixing behavior of three q-deformed card shuffles on 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 ℓ^2 cutoff. For the short systematic scan, these improve both the upper and lower bounds of Diaconis and Ram \cite{DiaconisRam2000}.
ER  -