Spectrum and combinatorics of two-dimensional Ramanujan complexes

TL;DR

This paper analyzes the spectral properties of two-dimensional Ramanujan complexes, revealing their combinatorial expansion and pseudo-randomness, and introduces new inequalities and mixing lemmas of independent interest.

Contribution

It computes the high-dimensional spectrum of Ramanujan complexes and establishes new combinatorial and pseudo-randomness properties with related inequalities.

Findings

Spectral analysis of Ramanujan complexes shows extremal properties.

Established a Cheeger-type inequality for these complexes.

Proved a mixing lemma indicating pseudo-randomness.

Abstract

Ramanujan graphs have extremal spectral properties, which imply a remarkable combinatorial behavior. In this paper we compute the high dimensional Hodge-Laplace spectrum of Ramanujan triangle complexes, and show that it implies a combinatorial expansion property, and a pseudo-randomness result. For this purpose we prove a Cheeger-type inequality and a mixing lemma of independent interest.