Hypercontractivity on High Dimensional Expanders: Approximate Efron-Stein Decompositions for $\varepsilon$-Product Spaces

TL;DR

This paper establishes hypercontractive inequalities for high dimensional expanders, enabling new results like Fourier concentration and small-set expansion, through a novel approximate Efron-Stein decomposition technique.

Contribution

It introduces a new approximate Efron-Stein decomposition for high dimensional link expanders, extending hypercontractivity tools to this setting.

Findings

Proves hypercontractive inequalities for high dimensional expanders.

Derives Fourier concentration and small-set expansion results.

Develops a new decomposition technique for analysis.

Abstract

We prove hypercontractive inequalities on high dimensional expanders. As in the settings of the p-biased hypercube, the symmetric group, and the Grassmann scheme, our inequalities are effective for global functions, which are functions that are not significantly affected by a restriction of a small set of coordinates. As applications, we obtain Fourier concentration, small-set expansion, and Kruskal-Katona theorems for high dimensional expanders. Our techniques rely on a new approximate Efron-Stein decomposition for high dimensional link expanders.