Size biased couplings and the spectral gap for random regular graphs

TL;DR

This paper extends the spectral gap bounds for random regular graphs to higher degrees using size biased couplings, progressing towards a conjecture that the bound holds for all degrees up to half the number of vertices.

Contribution

It introduces new concentration of measure results via size biased couplings, enabling analysis of spectral properties for larger degrees in random regular graphs.

Findings

Proves $oldsymbol{ ext{λ=O}( ext{√d})}$ for degrees up to $oldsymbol{O(n^{2/3})}$ with high probability.

Breaks previous barrier at $d=o(n^{1/2})$ for spectral gap bounds.

Develops Bennett-type tail estimates for unbounded size biased couplings.

Abstract

Let $\lambda$ be the second largest eigenvalue in absolute value of a uniform random $d$-regular graph on $n$ vertices. It was famously conjectured by Alon and proved by Friedman that if $d$ is fixed independent of $n$, then $\lambda=2\sqrt{d-1} +o(1)$ with high probability. In the present work we show that $\lambda=O(\sqrt{d})$ continues to hold with high probability as long as $d=O(n^{2/3})$, making progress towards a conjecture of Vu that the bound holds for all $1\le d\le n/2$. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at $d=o(n^{1/2})$. We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on $d$-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.