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. Concretely, we generalize a result of Lubetzky, Lubotzky, and Parzanchevski (Journal of the EMS) about the rapid mixing of sequential sweep in Ramanujan complexes to suitable high dimensional expanders.

翻译：众所周知，在n-部单纯复形（即格劳伯动力学）上的下行-上行游走的谱间隙由于上同调边界等自然障碍而不能优于O(1/n)。我们研究一种在分部单纯复形上的替代随机游走，称为序列扫描或系统扫描格劳伯动力学：下行-上行游走每一步随机选择一个坐标并基于其余坐标更新该坐标，而序列扫描则按确定性顺序逐一扫描每个坐标并应用相同的更新操作。因此，很自然地将下行-上行游走的n步与序列扫描的单步进行比较。有趣的是，尽管下行-上行游走的n次幂的谱间隙仍被正常数上界限制，但在足够强的局部谱假设下（依据Gur、Lifschitz、Liu，STOC 2022），我们可以证明该游走的谱间隙可以任意接近1。我们还研究了这些游走的其他等周不等式，并证明在局部熵收缩假设下（与Gur、Lifschitz、Liu的相关考虑相关），这些游走满足熵收缩不等式。具体而言，我们将Lubetzky、Lubotzky和Parzanchevski（EMS杂志）关于拉马努金复形中序列扫描快速混合的结果推广到合适的高维扩张子。