The measurable Kesten theorem

By Miklós Abért, Yair Glasner, B'alint Vir'ag. 2011

47 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/38a0d08c9afd807eeda7be327f96915bc2b03773

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

On the growth of Betti numbers of locally symmetric spaces (2011, background): “ SubG is compact and G acts on SubG continuously by conjugation. Deﬁnition 2.1 An invariant random subgroup (IRS) of G is a G-invariant probability measure on SubG. This notion has been introduced in [2]. It follows from the compactness of SubG that the space of invariant random subgroups of G equip”
Kesten's theorem for Invariant Random Subgroups (2012, methodology): “entially large girth. The result follows from the original Kesten theorem in the case when the Hn are normal in Γ, but for arbitrary subgroups, one needs Theorem 3. Note that in the forthcoming paper [2], which focuses on graph theory and is not assuming the graphs to be Schreier graphs, we prove mo”
On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, background): “All the material here is well known; for more information, we refer the reader to [7], [10], [26].”