Eigenvalue gaps of the Laplacian of random graphs

TL;DR

The paper establishes that the Laplacian of a random graph $G(n,p)$ has a simple spectrum with high probability and provides quantitative bounds on spectral gaps. It introduces and proves affine no-gaps delocalization and no-structure delocalization for Laplacian eigenvectors, along with an overcrowding estimate and small-coordinate bounds, using a centered Laplacian framework and LCD-based anti-concentration techniques. A novel neighbor-switching argument injects controlled randomness, enabling robust small-ball estimates and transfer of results from the centered model to the original Laplacian. The findings have implications for quantum walks, spectral algorithms, and graph-based learning, offering rigorous foundations for spectral stability in random graphs and connections to random matrix theory.

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.

Figures (5)

• Figure 1: Reshuffling neighbors.
• Figure 2: $|I_s|>|I_m|$
• Figure 4: $|I_s|>|I_m|$
• Figure 6: Finding non-structured subvectors.
• Figure 7: Decomposition into i.i.d. parts.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2: Eigenvalue Gaps of Laplacian
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3: Affine no-gaps delocalization of eigenvectors of $\overline{L}'_{n-1}$
• Definition 2.3
• Theorem 2.4: No-structure delocalization of Laplacian eigenvectors
• Theorem 2.5
• Theorem 2.6: Overcrowding for spectrum of Laplacian matrices
• Theorem 2.7: Small coordinates of eigenvectors