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January 2018 Size biased couplings and the spectral gap for random regular graphs
Nicholas Cook, Larry Goldstein, Tobias Johnson
Ann. Probab. 46(1): 72-125 (January 2018). DOI: 10.1214/17-AOP1180


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.


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Nicholas Cook. Larry Goldstein. Tobias Johnson. "Size biased couplings and the spectral gap for random regular graphs." Ann. Probab. 46 (1) 72 - 125, January 2018.


Received: 1 January 2016; Revised: 1 February 2017; Published: January 2018
First available in Project Euclid: 5 February 2018

zbMATH: 06865119
MathSciNet: MR3758727
Digital Object Identifier: 10.1214/17-AOP1180

Primary: 05C80
Secondary: 60B20 , 60E15

Keywords: Alon’s conjecture , Concentration , random regular graph , second Eigenvalue , size biased coupling , Stein’s method

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 1 • January 2018
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