arXiv subjects

Samira Arfaee

Publications and source records attributed to Samira Arfaee.

2 recordsLinked to original sources

arXiv2026

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

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}.

math.PR↗

arXiv2026

Shuffling via sums of Jucys--Murphy Elements

We consider a family of card shuffles of $n$ cards in which the allowed moves involve transpositions corresponding to the Jucys--Murphy elements of the symmetric group $\{S_m\}_{m \leq n}$. We determine the eigenvalues of the corresponding $n! \times n!$ transition matrices of these shuffles and study the mixing times for a special case, the $k$--star transpositions shuffle, a natural interpolation between the random transpositions shuffle, introduced and studied by Diaconis and Shahshahani, and the star transpositions shuffle, introduced and studied by Diaconis. We prove that the $k$--star transpositions shuffle exhibits total variation cutoff at $\frac{2n-(k+1)}{2(n-1)}n\log n$ with a window of $\frac{2n-(k+1)}{2(n-1)}n$. Furthermore, in the regimes $k/n \rightarrow 0$ or $k/n \rightarrow 1$, this shuffle has the same limit profile as random transpositions, which has been fully determined by Teyssier.

math.CO↗