A popular method for sampling from high-dimensional distributions is the \emph{Gibbs sampler}, which iteratively resamples sites from the conditional distribution of the desired measure given the values of the other coordinates. It is natural to ask to what extent does the order of site updates matter in the mixing time? Two natural choices are (i) standard, or \emph{random scan}, Glauber dynamics where the updated variable is chosen uniformly at random, and (ii) the \emph{systematic scan} dynamics where variables are updated in a fixed, cyclic order. We first show that for systems of dimension $n$, one round of the systematic scan dynamics has spectral gap at most a factor of order $n$ worse than the corresponding spectral gap of a single step of Glauber dynamics, tightening existing bounds in the literature by He, et al. [NeurIPS '16] and Chlebicka, {\L}atuszy\'nski, and Miasodejow [Ann. Appl. Probab. '25]. The corresponding bound on mixing times is sharp even for simple spin systems by an explicit example of Roberts and Rosenthal [Int. J. Statist. Prob. '15]. We complement this with a converse statement: if all, or even just one scan order rapidly mixes, the Glauber dynamics has a polynomially related mixing time, resolving a question of Chlebicka, {\L}atuszy\'nski, and Miasodejow.

翻译：从高维分布中采样的常用方法是 \emph{Gibbs 采样器}，该方法迭代地从给定其他坐标值的期望测度的条件分布中重新采样各个位置。一个自然的问题是：位置更新的顺序在多大程度上影响混合时间？两种自然的选择是：(i) 标准的或称为 \emph{随机扫描} 的 Glauber 动力学，其中被更新的变量是均匀随机选择的；以及 (ii) \emph{系统扫描} 动力学，其中变量按固定的循环顺序更新。我们首先证明，对于维度为 $n$ 的系统，一轮系统扫描动力学的谱隙至多比单步 Glauber 动力学对应的谱隙差一个 $n$ 阶因子，这改进了 He 等人 [NeurIPS '16] 以及 Chlebicka、{\L}atuszy\'nski 和 Miasodejow [Ann. Appl. Probab. '25] 在文献中已有的界。通过 Roberts 和 Rosenthal [Int. J. Statist. Prob. '15] 的一个显式例子，可以证明即使在简单自旋系统中，混合时间对应的这个界也是紧的。我们进一步给出了一个反向的结论：如果所有（甚至仅一个）扫描顺序都能快速混合，那么 Glauber 动力学的混合时间与其具有多项式关联，这解决了 Chlebicka、{\L}atuszy\'nski 和 Miasodejow 提出的一个问题。