We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut–Kalai–Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only \(|X(k-1)|=O(n)\) points in contrast to \(\left( {\begin{array}{c}n\\ k\end{array}}\right) \) points in the (k)-slice (which consists of all n-bit strings with exactly k ones).

我们启动了高维展开器布尔函数分析的研究。我们给出了基于随机游走的高维扩展定义，这与早期的双边链接扩展器定义一致。使用这个定义，我们描述了单纯复形的布尔超立方体的傅里叶展开和傅里叶级的类比。我们的模拟是分解为与单纯复形相关的随机游走的近似特征空间。我们的随机游走定义和分解具有额外的优势，它们可以扩展到更一般的偏序集设置，包括高维扩展器和格拉斯曼偏序集，这出现在最近关于独特博弈猜想的工作中。然后，我们使用这种分解将 Friedgut-Kalai-Naor 定理扩展到高维扩展器。我们的结果表明，常度高维扩展器有时可以用作布尔切片或超立方体的稀疏模型，并且很可能可以将布尔函数分析的附加结果转移到该稀疏模型中。因此，该模型可以被视为布尔切片的去随机化，仅包含\(|X(k-1)|=O(n)\)点，与\(\left( {\begin{array}{ c}n\\ k\end{array}}\right) \)点位于 ( k ) 切片（由所有n位字符串组成，其中恰好有k个）。