LOCALIZATION FOR RANDOM OPERATORS ON Zd WITH THE LONG-RANGE HOPPING

The paper proves localization for the long-range random operator with ρ-logarithmic hopping via a Green-function multiscale analysis, establishing a uniform energy-window result (Theorems 3.4–3.5) and an explicit eigenfunction decay bound |ψ(x)| ≤ exp(−(κ∞/(2αρ))·log_ρ(1+||x||)) for ||x||≫1, with all key ingredients (initial Neumann-series step, Wegner estimate, induction on scales, coupling lemma, and Shnol’s route to eigenfunctions) internally consistent and documented (e.g., Theorem 3.4, Theorem 3.5, Lemma 5.1, Section 8) . By contrast, the candidate solution’s core induction geometry and gain estimates diverge in ways that are not supported and, in places, untenable: it assumes a tiling by L-boxes whose mutual separations are ≥ L^ρ (claimed “possible since ρ>1”), which generally cannot be arranged unless one additionally imposes α>ρ; no such constraint is in the paper and, for typical choices (α in (5/4, 2p/(p+2d))), the packing becomes impossible when ρ≥α. This unsupported geometric separation is used to justify a per-crossing “gain” L^{−γρ/ln ρ}, which does not appear in the paper’s coupling analysis (where gains come from quasi-metric inequalities and resolvent identities, yielding different scale-dependent estimates) . The candidate also proposes an unsubstantiated recursion for κk and claims L2-summability can be forced “by starting the iteration at a large enough scale,” which does not change the decay exponent. Hence, while the model’s final statement matches the paper’s theorem, its proof outline contains critical gaps/incorrect steps.