Eigenvalue gaps of the Laplacian of random graphs

TL;DR

This paper proves that the Laplacian of a random graph typically has a simple spectrum with quantifiable spectral gaps, and introduces new results on eigenvector delocalization and eigenvalue distribution.

Contribution

It provides the first high-probability proof of simple spectrum for random graph Laplacians and offers effective estimates of spectral gaps, along with novel delocalization results.

Findings

Laplacian spectra are simple with high probability

Quantitative estimates of spectral gaps are established

Eigenvector delocalization properties are demonstrated

Abstract

We show that, with very high probability, the random graph Laplacian has simple spectrum. Our method provides a quantitatively effective estimate of the spectral gaps. Along the way, we establish results on affine no-gaps delocalization, no-structure delocalization, overcrowding and small entries of the eigenvectors for the Laplacian model. These findings are of independent interest.