Word maps and spectra of random graph lifts

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Word maps and spectra of random graph lifts
We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ1 and ρ are the spectral radii of G and T respectively, then, as shown by Friedman [Fri03], in almost every n-lift H of G, all “new” eigenvalues of H are ≤ O λ 1/2 1 ρ1/2 . Here we improve this bound to O λ 1/3 1 ρ2/3 . It is conjectured in [Fri03] that the statement holds with the bound ρ + o(1) which, if true, is tight by [Gre95]. For G a bouquet with d/2 loops, our arguments yield a simple proof that almost every d-regular graph has second eigenvalue O(d2/3 ). For the bouquet, Friedman [Fri] has famously proved the (nearly?) optimal bound of 2 √ d − 1 + o(1). Central to our work is a new analysis of formal words. Let w be a formal word in letters g±1 1 , . . . , g±1 k . The word map associated with w maps the permutations σ1, . . ., σk ∈ Sn to the permutation obtained by replacing for each i, every occurrence ...
Nati Linial, Doron Puder
Added 30 Jan 2011
Updated 30 Jan 2011
Type Journal
Year 2010
Where RSA
Authors Nati Linial, Doron Puder
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