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 ...

• Type: Journal
• Year: 2010
• Where: RSA
• Authors: Nati Linial, Doron Puder
• Keywords: Formal Words, Permutation, RSA 2010, Simple Proof