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 $\|u\|_\infty\leq(\log N)^C/\sqrt{N}$ with high probability.