## The measurable Kesten theorem.(English)Zbl 1339.05365

Summary: 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 $$c\log\log| G|$$.
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.

### MSC:

 05C80 Random graphs (graph-theoretic aspects) 05C05 Trees 05C38 Paths and cycles 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 60G50 Sums of independent random variables; random walks 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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### References:

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