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 $\binom{n}{k}$ points in the $(k)$-slice (which consists of all $n$-bit strings with exactly $k$ ones).

翻译：我们开启了在高维扩展子上进行布尔函数分析的研究。我们提出了基于随机游走的高维扩展定义，这与先前基于双边链接扩展子的定义一致。利用该定义，我们为单纯复形描述了布尔超立方体的傅里叶展开与傅里叶层次的类比。我们的类比是一种将随机游走分解为与单纯复形相关的近似特征空间的方法。我们的随机游走定义与分解具有额外优势：它们可推广至更一般的偏序集设定，涵盖高维扩展子与格拉斯曼偏序集（该偏序集出现在最近关于唯一博弈猜想的研究中）。随后，我们利用这种分解将Friedgut-Kalai-Naor定理推广至高维扩展子。我们的结果表明，常数度高维扩展子有时可作为布尔切片或超立方体的稀疏模型，并且布尔函数分析中的更多成果很可能可以移植到该稀疏模型中。因此，该模型可被视为布尔切片的一种去随机化，仅包含 $|X(k-1)|=O(n)$ 个点，而 $(k)$-切片（由所有恰好含 $k$ 个1的 $n$ 位比特串组成）则包含 $\binom{n}{k}$ 个点。