The $S_k$ shuffle block dynamics

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 $k=o(N^{\frac{1}{6}})$.