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，由于上同调边界等自然障碍，$n$部单纯复形上的下-上行走（也称Glauber动力学）的谱间隙不可能优于$O(1/n)$。我们研究另一类在分部单纯复形上的随机行走，称为序贯扫描或系统扫描Glauber动力学：下-上行走每步随机选择一个坐标并基于其余坐标进行更新，而序贯扫描则按确定性顺序逐一遍历每个坐标并应用相同的更新操作。因此，自然可以将下-上行走的$n$步与序贯行走的单步进行比较。有趣的是，尽管下-上行走的$n$次幂的谱间隙仍受常数上界约束，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任意接近1。我们还研究了这些行走的其他等周不等式，并证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。