2026/07/22 by Chen Shang, Jiahe Shen, Jiyue Zeng +1
#math.PR #math.CO #math.RT
In this paper, we study the random walk on the symmetric group \mathfrakSn generated by the conjugacy class of k-cycles, where 2≤ 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 \mathfrakAn. 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_\mathfrakSn. 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≥ n-o(n1/2), and formulate a conjecture for all 2≤ k≤ n-1.