# Spectra of random regular hypergraphs

## TL;DR

This paper investigates the spectral properties of random regular hypergraphs, establishing analogs of classical spectral gap conjectures, and analyzing their implications for expansion, mixing, and spectral distribution convergence.

## Contribution

It introduces new spectral gap results for random regular hypergraphs and relates eigenvalues to expansion and mixing properties, extending previous work on non-backtracking operators.

## Key findings

- Proves spectral gap for random regular hypergraphs.
- Relates second eigenvalues to expansion and mixing.
- Shows convergence and local laws for spectral distribution.

## Abstract

In this paper, we study the spectra of regular hypergraphs following the definitions from Feng and Li (1996). Our main result is an analog of Alon's conjecture for the spectral gap of the random regular hypergraphs. We then relate the second eigenvalues to both its expansion property and the mixing rate of the non-backtracking random walk on regular hypergraphs. We also prove the spectral gap for the non-backtracking operator of a random regular hypergraph introduced in Angelini et al. (2015). Finally, we obtain the convergence of the empirical spectral distribution (ESD) for random regular hypergraphs in different regimes. Under certain conditions, we can show a local law for the ESD.