It is well known that the spectral gap of the down-up walk over an $n$-partite simplicial complex (also known as Glauber dynamics) cannot be better than $O(1/n)$ due to natural obstructions such as coboundaries. We study an alternative random walk over partite simplicial complexes known as the sequential sweep or the systematic scan Glauber dynamics: Whereas the down-up walk at each step selects a random coordinate and updates it based on the remaining coordinates, the sequential sweep goes through each of the coordinates one by one in a deterministic order and applies the same update operation. It is natural, thus, to compare $n$-steps of the down-up walk with a single step of the sequential sweep. Interestingly, while the spectral gap of the $n$-th power of the down-up walk is still bounded from above by a constant, under a strong enough local spectral assumption (in the sense of Gur, Lifschitz, Liu, STOC 2022) we can show that the spectral gap of this walk can be arbitrarily close to 1. We also study other isoperimetric inequalities for these walks, and show that under the assumptions of local entropy contraction (related to the considerations of Gur, Lifschitz, Liu), these walks satisfy an entropy contraction inequality.

翻译：众所周知，由于余边界等自然障碍的存在，$n$ 分单纯复形上的下-上行走（也称为格劳伯动力学）的谱间隙不可能优于 $O(1/n)$。我们研究了一种替代的随机行走——分单纯复形上的序列扫描或系统扫描格劳伯动力学：下-上行走每一步随机选择一个坐标并根据其余坐标更新该坐标，而序列扫描则以确定顺序逐个遍历每个坐标并应用相同的更新操作。因此，比较下-上行走的 $n$ 步与序列扫描的单一步骤是自然的。有趣的是，尽管下-上行走 $n$ 次幂的谱间隙仍然被一个常数上界所限制，但在足够强的局部谱假设（遵循 Gur、Lifschitz、Liu，STOC 2022 的意义）下，我们可以证明该行走的谱间隙可以任意接近 1。我们还研究了这些行走的其他等周不等式，并表明在局部熵收缩假设（与 Gur、Lifschitz、Liu 的考量相关）下，这些行走满足熵收缩不等式。