DYNAMICAL LOCALIZATION FOR POLYNOMIAL LONG-RANGE HOPPING RANDOM OPERATORS ON Zd

The paper establishes a power-law (strong) dynamical localization statement for the long-range Anderson model with polynomial hopping at large disorder via a multiscale analysis anchored in Shi’s power-law localization and a SULE-type polynomial bound on eigenfunctions, then a counting argument for localization centers to deduce uniform-in-time polynomial moment bounds (Theorem 1.2) . The candidate solution obtains the same end result under essentially the same hypotheses, but by a different route: a fractional-moment method (FMM) adapted to polynomial hopping (rs>d), producing polynomially decaying fractional moments of Green functions, eigenfunction-correlator bounds, and, finally, polynomial moment bounds for the dynamics. The model’s Step 5 needs a minor correction (use Cauchy–Schwarz with Q^2≤Q to bound expectations), but the approach is sound. Thus both are correct, with substantially different proofs.