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May 2016 The measurable Kesten theorem
Miklós Abért, Yair Glasner, Bálint Virág
Ann. Probab. 44(3): 1601-1646 (May 2016). DOI: 10.1214/14-AOP937


We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite $d$-regular graphs. It follows that the a finite $d$-regular Ramanujan graph $G$ contains a negligible number of cycles of size less than $c\log\log\vert G\vert$.

We prove that infinite $d$-regular Ramanujan unimodular random graphs are trees. Through Benjamini–Schramm convergence this leads to the following rigidity result. If most eigenvalues of a $d$-regular finite graph $G$ fall in the Alon–Boppana region, then the eigenvalue distribution of $G$ is close to the spectral measure of the $d$-regular tree. In particular, $G$ contains few short cycles.

In contrast, we show that $d$-regular unimodular random graphs with maximal growth are not necessarily trees.


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Miklós Abért. Yair Glasner. Bálint Virág. "The measurable Kesten theorem." Ann. Probab. 44 (3) 1601 - 1646, May 2016.


Received: 1 February 2012; Revised: 1 May 2014; Published: May 2016
First available in Project Euclid: 16 May 2016

zbMATH: 1339.05365
MathSciNet: MR3502591
Digital Object Identifier: 10.1214/14-AOP937

Primary: 05C81 , 60G50
Secondary: 82C41

Keywords: Girth , mass transport principal , Ramanujan graphs , spectral radius , unimodular random graphs

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 3 • May 2016
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