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The measurable Kesten theorem | | Miklos Abert
; Yair Glasner
; Balint Virag
; | | Date: |
9 Nov 2011 | | Abstract: | 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.
We also show that for a nonamenable invariant random subgroup H, the limiting
exponent of the probability of return to H is greater than the exponent of the
probability of return to 1. This generalizes a theorem of Kesten who proved
this for normal subgroups. | | Source: | arXiv, 1111.2080 | | Services: | Forum | Review | PDF | Favorites | |
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