Boolean functions on high-dimensional expanders
Yotam Dikstein, Irit Dinur, Yuval Filmus, Prahladh Harsha
TL;DR
This work develops Boolean function analysis on high-dimensional expanders (HDX) by introducing a random-walk based expansion notion that is equivalent to two-sided link expansion in fixed dimensions. It constructs a Fourier-like decomposition of functions on faces of HDX into approximately orthogonal homogeneous components that are near-eigenvectors of natural Up/Down random walks, and it extends this framework to expanding posets (eposets), including the Grassmann poset. Using this decomposition, the authors prove a high-dimensional generalization of the Friedgut–Kalai–Naor degree-1 theorem (FKN) and demonstrate that constant-degree HDX can serve as a sparse, derandomized model for the Boolean slice, enabling transfer of Boolean-function insights to sparse combinatorial settings. The results unify spectral link-expansion and random-walk viewpoints, develop the theory of decomposable posets, and establish a versatile toolkit with potential implications for unique games, coding theory, and derandomization in Boolean-function analysis.
Abstract
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).