Word maps and spectra of random graph lifts

• Authors: Nati Linial;Doron Puder
• Venue: Random Structures & Algorithms
• Year: 2010

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 (Graphs Duke Math J 118 (2003), 19–35), in almost every n-lift H of G, all “new” eigenvalues of H are ≤ O(λ 11-2 ρ1-2). Here we improve this bound to O(λ 11-3 ρ2-3). It is conjectured in (Friedman, Graphs Duke Math J 118 (2003) 19–35) that the statement holds with the bound ρ + o(1) which, if true, is tight by (Greenberg, PhD thesis, 1995). 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 (2008). has famously proved the (nearly?) optimal bound of $ 2\sqrt{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 k±1. The word map associated with w maps the permutations σ1,…,σk ∈ Sn to the permutation obtained by replacing for each i, every occurrence of gi in w by σi. We investigate the random variable X w(n) that counts the fixed points in this permutation when the σi are selected uniformly at random. The analysis of the expectation 𝔼(X w(n)) suggests a categorization of formal words which considerably extends the dichotomy of primitive vs. imprimitive words. A major ingredient of a our work is a second categorization of formal words with the same property. We establish some results and make a few conjectures about the relation between the two categorizations. These conjectures suggest a possible approach to (a slightly weaker version of) Friedman's conjecture. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica (Nica, Random Struct Algorithms 5 (1994), 703–730), which determines, for every fixed w, the limit distribution (as n →∞) of X w(n). A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w = ud for some word u. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010