Abért, Miklós and Glasner, Yair and Virág, Bálint (2016) The measurable kesten theorem. Annals of Probability, 44 (3). pp. 16011646. ISSN 00911798

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Abstract
We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite dregular graphs. It follows that the a finite dregular Ramanujan graph G contains a negligible number of cycles of size less than c log log G. We prove that infinite dregular Ramanujan unimodular random graphs are trees. Through BenjaminiSchramm convergence this leads to the following rigidity result. If most eigenvalues of a dregular finite graph G fall in the AlonBoppana region, then the eigenvalue distribution of G is close to the spectral measure of the dregular tree. In particular, G contains few short cycles. In contrast, we show that dregular unimodular random graphs with maximal growth are not necessarily trees. © Institute of Mathematical Statistics, 2016.
Item Type:  Article 

Additional Information:  N1 Funding Details: 1107263, NSF, National Science Foundation N1 Funding Details: 1107367, NSF, National Science Foundation N1 Funding Details: DMS1107452, NSF, National Science Foundation 
Uncontrolled Keywords:  Unimodular random graphs; Spectral radius; RAMANUJAN GRAPHS; Mass transport principal; Girth; Eigenvalue 
Subjects:  Q Science / természettudomány > QA Mathematics / matematika 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  03 Jan 2017 08:02 
Last Modified:  03 Jan 2017 08:02 
URI:  http://real.mtak.hu/id/eprint/44165 
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