高阶随机游动：超越谱隙
High Order Random Walks: Beyond Spectral Gap
摘要：
我们研究高阶随机游动在高维扩展，即在复杂的地方谱扩展。最近的工作研究了高阶游动的谱，并推导出快速混合。然而，高阶游走的谱隙本质上是小的，这是由于扩展图上的游走不会发生的自然障碍。在这项工作中，我们超越了谱隙，并联系到一个$k$ -上链的收缩的步行运营商，其结构的假设下，当地的频谱扩展。一个单纯复形，如果它的链路有大的谱隙，则称为单边局部谱扩展器，如果它的链路有大的双边谱隙，则称为双边局部谱扩展器。我们展示了两个分解定理（一个单侧/双侧局部谱假设）：对于每个定义在$n$维局部谱扩展器上的$k$ -上链$ϕ$，存在$ϕ$分解为“正交”部分，粗略地说，是$0 \leq j \leq k$的$j$维上链上的“投影”。随机游动通过$\frac{k+1-j}{k+2}$加上取决于频谱展开的误差项的因子来缩小这些部分中的每一个。我们的两个分解定理的不同之处在于它们对局部谱隙的假设--我们对单侧局部谱隙和双侧局部谱隙的情况导出了不同的分解定理。
英文摘要：
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 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 an 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 $0 \leq j \leq k$. The random walk shrinks each of these parts by a factor of $\frac{k+1-j}{k+2}$ plus an error term that depends on the spectral expansion. %Our two Decomposition Theorems differ in their assumptions on the local spectral gaps - we derive different Decomposition Theorems for the cases of one-sided local spectral gap and two-sided local spectral gap.
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