A deck of cards with many duplicates stays almost perfectly ordered until a 'magic number' of shuffles, where it suddenly becomes random.
March 31, 2026
Original Paper
The $k$-cycle shuffling with repeated cards
arXiv · 2603.27433
The Takeaway
While we assume shuffling is a gradual process, this research proves it behaves like a sudden phase transition. The deck remains structured for a long time and then becomes totally mixed almost instantly at a specific threshold, which the researchers have now precisely calculated.
From the 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 d