The spectral gap of random regular graphs

TL;DR

This paper establishes precise bounds on the second eigenvalue of random regular graphs across a wide degree range using innovative Fourier analysis techniques, advancing understanding of their spectral properties.

Contribution

It introduces a novel Fourier analysis approach to bound eigenvalues of random regular graphs, extending results to dense regimes previously unexplored.

Findings

For degrees $d$ with $ ext{log}^{10} n \,\ll\, d \leq c n$, the second eigenvalue $\,\lambda(G_{n, d})$ asymptotically equals $(2 + o(1)) \sqrt{d(n - d)/n}$.

The method combines Fourier analysis with existing tools, providing a comprehensive understanding of spectral gaps in random regular graphs.

Results fully determine the asymptotic second eigenvalue for all degrees up to a linear fraction of $n$.

Abstract

We bound the second eigenvalue of random $d$-regular graphs, for a wide range of degrees $d$, using a novel approach based on Fourier analysis. Let $G_{n, d}$ be a uniform random $d$-regular graph on $n$ vertices, and let $\lambda (G_{n, d})$ be its second largest eigenvalue by absolute value. For some constant $c > 0$ and any degree $d$ with $\log^{10} n \ll d \leq c n$, we show that $\lambda (G_{n, d}) = (2 + o(1)) \sqrt{d (n - d) / n}$ asymptotically almost surely. Combined with earlier results that cover the case of sparse random graphs, this fully determines the asymptotic value of $\lambda (G_{n, d})$ for all $d \leq c n$. To achieve this, we introduce new methods that use mechanisms from discrete Fourier analysis, and combine them with existing tools and estimates on $d$-regular random graphs - especially those of Liebenau and Wormald.