The measurable Kesten theorem

Details The measurable Kesten theorem The measurable Kesten theorem

2011-11-08

arxiv.org/abs/1111.2080v2

Mathematics

Probability

The previous, longer version 1 has been split in two parts: the present paper, and a more group-theoretic one with the title "

Scientific paper

We give explicit estimates between the spectral radius and the densities of short cycles for finite d-regular graphs. This allows us to show that the essential girth of a finite d-regular Ramanujan graph G is at least c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Using Benjamini-Schramm convergence this leads to a rigidity result saying that 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. Kesten showed that if a Cayley graph has the same spectral radius as its universal cover, then it must be a tree. We generalize this to unimodular random graphs.

Affiliated with

Also associated with

No associations

Rating

The measurable Kesten theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The measurable Kesten theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The measurable Kesten theorem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-41529