Conjectured optimal eigenvector sup‑norm bound

Prove that the sup‑norm of eigenvectors satisfies ||u_i||_∞ ≲ √(log n)/√n (up to absolute constants) in the setting discussed around complete eigenvector delocalization, such as for the adjacency matrix of the Erdős–Rényi random graph G(n,p), with high probability.

Background

In the application to eigenvector delocalization, the authors establish a bound of order (log n)/√n under sparsity conditions. They remark that the conjectured optimal bound is of order √(log n)/√n. While this has been proved for bulk eigenvectors of Wigner matrices with bounded entries and, for all eigenvectors, for Wigner matrices with sub‑exponential entries, achieving this bound in other sparse matrix models (e.g., adjacency matrices of Erdős–Rényi graphs) remains a central challenge.

This conjectured rate reflects the expected optimal delocalization in the sup‑norm and has been a benchmark in random matrix theory and random graphs.