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 (Graphs Duke Math J 118 (2003), 19-35), in almost every n-lift H of G, all "new" eigenvalues of H are ≤ O(λ^1/2₁ ρ ^1/2). Here we improve this bound to O(λ^1/3₁ ρ^2/3). 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(d^2/3). For the bouquet, Friedman (2008). 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},...,g^{± 1}. The word map associated with w maps the permutations σ₁,...,σ_k ∈ S_n to the permutation obtained by replacing for each i, every occurrence of g_i in w by σ_i. We investigate the random variable Xⁿ_w that counts the fixed points in this permutation when the σ_i are selected uniformly at random. The analysis of the expectation E(Xⁿ_w) 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 Xⁿ_w. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w = u^d for some word u.