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May 2019 On the almost eigenvectors of random regular graphs
Ágnes Backhausz, Balázs Szegedy
Ann. Probab. 47(3): 1677-1725 (May 2019). DOI: 10.1214/18-AOP1294

Abstract

Let $d\geq3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector of $G$ (with entry sum 0 and normalized to have length $\sqrt{n}$) is close to some Gaussian distribution $N(0,\sigma)$ in the weak topology where $0\leq\sigma\leq1$. Our theorem holds even in the stronger sense when many entries are looked at simultaneously in small random neighborhoods of the graph. Furthermore, we also get the Gaussianity of the joint distribution of several almost eigenvectors if the corresponding eigenvalues are close. Our proof uses graph limits and information theory. Our results have consequences for factor of i.i.d. processes on the infinite regular tree.

In particular, we obtain that if an invariant eigenvector process on the infinite $d$-regular tree is in the weak closure of factor of i.i.d. processes then it has Gaussian distribution.

Citation

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Ágnes Backhausz. Balázs Szegedy. "On the almost eigenvectors of random regular graphs." Ann. Probab. 47 (3) 1677 - 1725, May 2019. https://doi.org/10.1214/18-AOP1294

Information

Received: 1 February 2017; Revised: 1 May 2018; Published: May 2019
First available in Project Euclid: 2 May 2019

zbMATH: 07067280
MathSciNet: MR3945757
Digital Object Identifier: 10.1214/18-AOP1294

Subjects:
Primary: 05C80
Secondary: 60B20

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 3 • May 2019
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