Boolean functions on high-dimensional expanders

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 analogue of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analogue is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. We then use this decomposition to extend the Friedgut-Kalai-Naor theorem to high-dimensional expanders. Our results demonstrate that a 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). Our random-walk definition and the decomposition has the additional advantage that they extend to the more general setting of posets, which include both high-dimensional expanders and the Grassmann poset, which appears in recent works on the unique games conjecture.