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High-dimensional expanders generalize the notion of expander graphs tohigher-dimensional simplicial complexes. In contrast to expander graphs, only ahandful of high-dimensional expander constructions have been proposed, and noelementary combinatorial construction with near-optimal expansion is known. Inthis paper, we introduce an improved combinatorial high-dimensional expanderconstruction, 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 $\Omega(\frac{1}{k^2})$ for randomwalks on the $k$-dimensional faces, which is only quadratically worse than theoptimal bound of $\Theta(\frac{1}{k})$. Previous combinatorial constructions,including that of Liu, Mohanty, and Yang, only achieved a spectral gap that isexponentially small in $k$. We also present reasoning that suggests ourconstruction is optimal among similar product-based constructions.

Improved Product-Based High-Dimensional Expanders