We study high order random walks in high dimensional expanders; namely, in complexes which are local spectral expanders. Recent works have studied the spectrum of high order walks and deduced fast mixing. However, the spectral gap of high order walks is inherently small, due to natural obstructions (called coboundaries) that do not happen for walks on expander graphs. In this work we go beyond spectral gap, and relate the shrinkage of a k -cochain by the walk operator, to its structure under the assumption of local spectral expansion. A simplicial complex is called a one-sided local spectral expander , if its links have large spectral gaps and a two-sided local spectral expander if its links have large two-sided spectral gaps. We show two Decomposition Theorems (one per one-sided/two-sided local spectral assumption): For every k -cochain ϕ defined on an n -dimensional local spectral expander, there exists a decomposition of ϕ into “orthogonal” parts that are, roughly speaking, the “projections” on the j -dimensional cochains for 0 ≤ j ≤ k . The random walk shrinks each of these parts by a factor of $$\frac{k+1-j}{k+2}$$ k + 1 − j k + 2 plus an error term that depends on the spectral expansion. When assuming one-sided local spectral gap, our Decomposition Theorem yields an optimal mixing for the high order random walk operator. Namely, negative eigenvalues of the links do not matter! This improves over [5] that assumed two-sided spectral gap in the links to get optimal mixing. This improvement is crucial in a recent breakthrough [1] proving a conjecture of Mihail and Vazirani. Additionally, we get an optimal mixing for high order random walks on Ramanujan complexes (whose links are one-sided local spectral expanders). When assuming two-sided local spectral gap, our Decomposition Theorem allows us to describe the whole spectrum of the random walk operator (up to an error term that is determined by the spectral gap) and give an explicit orthogonal decomposition of the spaces of cochains that approximates the decomposition to eigenspaces of the random walk operator.

中文翻译：

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高阶随机游走：超越光谱间隙

我们研究高维扩展器中的高阶随机游走；即，在作为局部频谱扩展器的复合物中。最近的工作研究了高阶游走的频谱并推导出了快速混合。然而，高阶游走的光谱间隙本质上很小，这是由于扩展图上的游走不会发生的自然障碍（称为共边界）。在这项工作中，我们超越了光谱间隙，并将步行算子对 ak-cochain 的收缩与其在局部光谱扩展假设下的结构联系起来。单纯复形被称为单边局部频谱扩展器，如果它的链接有大的频谱间隙，并且被称为双边局部频谱扩展器，如果它的链接有大的两侧频谱间隙。我们展示了两个分解定理（每个一侧/两侧局部谱假设一个）：对于定义在 n 维局部谱扩展器上的每个 k -cochain ϕ，存在 ϕ 分解为“正交”部分，粗略地说，即 0 ≤ j ≤ k 时 j 维 cochain 上的“投影”。随机游走以 $$\frac{k+1-j}{k+2}$$k + 1 − jk + 2 的因子加上一个取决于谱扩展的误差项来缩小这些部分中的每一个。当假设单边局部光谱间隙时，我们的分解定理为高阶随机游走算子产生了最佳混合。也就是说，链接的负特征值无关紧要！这比 [5] 改进了假设链路中的两侧光谱间隙以获得最佳混合。这一改进在最近的一项突破 [1] 中至关重要，该突破证明了 Mihail 和 Vazirani 的猜想。此外，我们在拉马努金复合体（其链接是单侧局部谱扩展器）上获得了高阶随机游走的最佳混合。当假设两侧局部谱间隙时，我们的分解定理允许我们描述随机游走算子的整个谱（直到由谱间隙确定的误差项），并给出协链空间的显式正交分解将分解近似为随机游走算子的特征空间。