The Annals of Probability

Size biased couplings and the spectral gap for random regular graphs

Nicholas Cook, Larry Goldstein, and Tobias Johnson

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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 toward 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.

Article information

Ann. Probab., Volume 46, Number 1 (2018), 72-125.

Received: January 2016
Revised: February 2017
First available in Project Euclid: 5 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60E15: Inequalities; stochastic orderings

Second eigenvalue random regular graph Alon’s conjecture size biased coupling Stein’s method concentration


Cook, Nicholas; Goldstein, Larry; Johnson, Tobias. Size biased couplings and the spectral gap for random regular graphs. Ann. Probab. 46 (2018), no. 1, 72--125. doi:10.1214/17-AOP1180.

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