Local Law and Eigenvector Delocalization for Random Lifts

We establish a quantitative local law for random N-lifts of a fixed finite connected simple graph with minimum degree at least three. Away from the singular energies of the universal-cover spectrum, we obtain high-probability entrywise Green's function estimates down to spectral scales (log N)C/N, uniformly in the bulk and near regular edges and cusps. As a consequence, we establish complete eigenvector delocalization: every normalized eigenvector u whose eigenvalue stays a small but fixed positive distance away from the singular energies satisfies \|u\|_∞≤(log N)C/√N with high probability.