Systematic scanning Glauber dynamics for the mean-field Ising model

We study the mixing time of systematic scan Glauber dynamics Ising model on the complete graph. On the complete graph $K_n$, at each time, $k\le n$ vertices are chosen uniformly random and are updated one by one according to the uniformly randomly chosen permutations over the $k$ vertices. We show that if $k=o(n^{1\mathrm{/}3})$, the high temperature regime $\beta <1$ exhibits cutoff phenomena. For critical temperature regime $\beta =1$, We prove that the mixing time is of order $n^{3\mathrm{/}2}{k}^{-1}$. For $\beta >1$, we prove the mixing time is of order $n{k}^{-1}\log n$ under the restricted dynamics.