## The Annals of Probability

### The measurable Kesten theorem

#### Abstract

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.

#### Article information

Source
Ann. Probab. Volume 44, Number 3 (2016), 1601-1646.

Dates
Revised: May 2014
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.aop/1463410029

Digital Object Identifier
doi:10.1214/14-AOP937

Mathematical Reviews number (MathSciNet)
MR3502591

Zentralblatt MATH identifier
1339.05365

#### Citation

Abért, Miklós; Glasner, Yair; Virág, Bálint. The measurable Kesten theorem. Ann. Probab. 44 (2016), no. 3, 1601--1646. doi:10.1214/14-AOP937. https://projecteuclid.org/euclid.aop/1463410029

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