Hitting time mixing for random $k$-cycles

In this paper, we study the random walk on the symmetric group $\mathfrak{S}_n$ generated by the conjugacy class of $k$-cycles, where $2\le k=o(n/(\log n)^4)$.

We prove that the walk exhibits hitting-time mixing: at the first time when every card has been touched, the distribution is already close to equilibrium.

For odd $k$, the equilibrium measure is the uniform measure on $\mathfrak{A}_n$.

For even $k$, the walk first mixes to the parity mixture determined by the hitting time, and in our range this mixture is asymptotically $U_{\mathfrak{S}_n}$.

Our argument combines a refined fixed-time approximation for the random $k$-cycle walk near the cutoff window with an auxiliary marking scheme inspired by Jain-Sawhney's work (arXiv:2410.23944) on random transpositions.

The main new feature is a parity-compatible coupling which handles both odd and even $k$-cycles in a unified framework.

We also prove a hitting-time mixing result in the opposite regime $k\ge n-o(n^{1/2})$, and formulate a conjecture for all $2\le k\le n-1$.