Spectra of lifted Ramanujan graphs

A random n-lift of a base-graph G is its cover graph H on the vertices [n]×V(G), where for each edge uv in G there is an independent uniform bijection Π, and H has all edges of the form (i,u),(Π(i),v). A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan.Let G be a graph with largest eigenvalue λ1 and let ρ be the spectral radius of its universal cover. Friedman (2003) [12] proved that every "new" eigenvalue of a random lift of G is O(ρ^1/2λ₁^1/2) with high probability, and conjectured a bound of Π+o(1), which would be tight by results of Lubotzky and Greenberg (1995) [15]. Linial and Puder (2010) [17] improved FriedmanΠs bound to O(Π^2/3λ₁^1/3). For d-regular graphs, where ρ1=d and d-1, this translates to a bound of O(d2/3), compared to the conjectured 2√d-1. Here we analyze the spectrum of a random n-lift of a d-regular graph whose nontrivial eigenvalues are all at most λ in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is O((λVρ)logρ). This result is tight up to a logarithmic factor, and for λ≤d^2/3-ε it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical n-lift of a Ramanujan graph is nearly Ramanujan.