Improved Product-Based High-Dimensional Expanders
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High-dimensional expanders generalize the notion of expander graphs to higher-dimensional simplicial complexes. In contrast to expander graphs, only a handful of high-dimensional expander constructions have been proposed, and no elementary combinatorial construction with near-optimal expansion is known. In this paper, we introduce an improved combinatorial high-dimensional expander construction, by modifying a previous construction of Liu, Mohanty, and Yang (ITCS 2020), which is based on a high-dimensional variant of a tensor product. Our construction achieves a spectral gap of Ω(1/k^2) for random walks on the k-dimensional faces, which is only quadratically worse than the optimal bound of Θ(1/k). Previous combinatorial constructions, including that of Liu, Mohanty, and Yang, only achieved a spectral gap that is exponentially small in k. We also present reasoning that suggests our construction is optimal among similar product-based constructions.
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