Diffusion bound for the nonlinear Anderson model

The uploaded paper (Cong–Shi, arXiv:2008.10171v1, dated Aug 24–25, 2020) proves precisely the power-law diffusion bound D(t) = ∑j j^2|qj(t)|^2 ≤ t^κ for the 1D discrete nonlinear Anderson model with constant nonlinearity δ>0 and small ϵ+δ, removing the decaying λj assumption of Bourgain–Wang and resolving Bourgain’s 2006 problem (see the statement around Theorem 1.2 and remarks) . Their proof introduces a new tame “norm” (Definition 2.1) to run a finite-step Birkhoff normal form near barriers of width ∼ ln j0, obtaining a remainder estimate |||R̃||| ≤ j0^{-3/κ} and, consequently, a uniform-in-time differential inequality for the exterior mass that yields the desired t^κ bound (Sections 3 and 5) . The candidate solution asserts the result was likely open as of the cutoff and outlines only a finite-window barrier scheme; it misses the paper’s core tame-norm mechanism and almost-sure multiscale measure control that make the argument global in time. Hence the paper is correct on the main claim, while the model’s status assessment is incorrect.