The $k$-cycle shuffling with repeated cards

Abstract

We investigate the $k$-cycle shuffle on repeated cards, namely on a deck consisting of $l$ identical copies of each of $m$ card types, with total size $n=ml$. We establish asymptotic results for the total variation mixing of this shuffle, including cutoff and explicit limiting profiles. For fixed $l$, we show that the walk exhibits cutoff at time $\frac{n}{k}\log n$ with window of order $\frac{n}{k}$, and we identify the limiting profile in terms of the total variation distance between Poisson distributions arising from quotient fixed-point statistics. When $l\to\infty$ with sufficiently slow growth, more precisely when $l=o(\log n)$, we prove that the cutoff location shifts to $\frac{n}{k}\left(\log n-\frac 12\log l\right)$, again with window of order $\frac{n}{k}$, and that the limiting profile is asymptotically Gaussian, arising from a Poisson comparison after normal approximation. The proof is based on an approximation of the shuffling measure by an explicitly tractable auxiliary measure, generalizing the $k=2$ case from Jain and Sawhney (arXiv:2410.23944). The representation-theoretic framework underlying the analysis of this auxiliary measure follows from the work of Hough (arXiv:1605.00911) and Nestoridi and Olesker-Taylor (arXiv:2005.13437)