Improved Product-Based High-Dimensional Expanders

TL;DR

This paper presents an improved combinatorial construction of high-dimensional expanders with near-optimal spectral gap, advancing the understanding and potential applications of these complex structures.

Contribution

The authors introduce a new high-dimensional expander construction based on a modified tensor product, achieving a spectral gap close to the optimal bound, improving upon previous combinatorial methods.

Findings

Spectral gap of $oldsymbol{ ilde{ extstylerac{1}{k^2}}}$ for the new construction.

Quadratic gap from the optimal $oldsymbol{rac{1}{k}}$ bound.

Evidence suggesting the construction's optimality among similar product-based expanders.

Abstract

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 $\Omega(\frac{1}{k^2})$ for random walks on the $k$-dimensional faces, which is only quadratically worse than the optimal bound of $\Theta(\frac{1}{k})$. 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.