Spectral gaps of random graphs and applications

TL;DR

This paper analyzes the spectral gap of Erdős–Rényi random graphs near the connectivity threshold, establishing concentration results for eigenvalues and demonstrating optimality, with applications to stochastic topology and geometric group theory.

Contribution

It provides precise spectral gap estimates for Erdős–Rényi graphs around the connectivity threshold and shows the optimality of the 1/2 threshold, with applications to high-dimensional expanders.

Findings

Spectral gap concentrates around 1 for p ≥ (1/2 + δ) log n / n.

The 1/2 threshold for p is proven to be optimal.

Applications include spectral geometry in high-dimensional simplicial complexes.

Abstract

We study the spectral gap of the Erd\H{o}s--R\'enyi random graph through the connectivity threshold. In particular, we show that for any fixed $\delta > 0$ if $$p \ge \frac{(1/2 + \delta) \log n}{n},$$ then the normalized graph Laplacian of an Erd\H{o}s--R\'enyi graph has all of its nonzero eigenvalues tightly concentrated around $1$. We estimate both the decay rate of the spectral gap to $1$ and the failure probability, up to a constant factor. We also show that the $1/2$ in the above is optimal, and that if $p = \frac{c \log n}{n}$ for $c < 1/2,$ then there are eigenvalues of the Laplacian restricted to the giant component that are separated from $1.$ We then describe several applications of our spectral gap results to stochastic topology and geometric group theory. These all depend on Garland's "p-adic curvature" method, a kind of spectral geometry for simplicial complexes. These can all be considered to be high-dimensional expander properties.