The $S_k$ shuffle block dynamics
Evita Nestoridi, Amanda Priestley, Dominik Schmid
TL;DR
The paper analyzes the $S_k$ shuffle, a block-dynamics generalization bridging local adjacent moves and global permutations, and its boundary-augmented variant. It develops an approximate spectral/Fourier framework and projection techniques, coupled with censoring and strong Rayleigh properties, to establish lower and upper bounds on the total-variation mixing time $t_{mix}(\\varepsilon)$ as $N\to\infty$ with $k=k(N)$. A key finding is a cutoff phenomenon for the boundary model in the regime $k=o(N^{1/6})$, with the leading mixing time obeying $\frac{ k(k^2-1) t_{mix}'(\\varepsilon)}{N^{2} \log N} \to \frac{6}{\pi^2}$, while the boundaryless variant exhibits pre-cutoff over a larger range of $k$ (up to $o(N^{2/3})$). The results illuminate how introducing non-local boundary updates accelerates or slows mixing, depending on the growth rate of $k$, and provide a detailed comparison between boundary and non-boundary dynamics using a combination of approximate eigenfunctions, coupling arguments, and Fourier analysis.
Abstract
We introduce and analyze the $S_k$ shuffle on $N$ cards, a natural generalization of the celebrated random adjacent transposition shuffle. In the $S_k$ shuffle, we choose uniformly at random a block of $k$ consecutive cards, and shuffle these cards according to a permutation chosen uniformly at random from the symmetric group on $k$ elements. We study the total-variation mixing time of the $S_k$ shuffle when the number of cards $N$ goes to infinity, allowing also $k=k(N)$ to grow with $N$. In particular, we show that the cutoff phenomenon occurs when $k=o(N^{\frac{1}{6}})$.