Spectral and dynamical localization at large disorder for the nonlocal random operators on Zd

In this paper, we study the nonlocal random operators Hω = ɛSϕ + Vω(n) acting on l2(Zd), whose free part Sϕ is a translation invariant operator with off diagonal elements decaying as e−logρ(‖n−m‖+1) with ρ 1 and n,m∈Zd. Under the assumption that the probability distribution of Vω(n) is absolutely continuous, we prove that at large disorder (0 ɛ ≪ 1), the spectrum of Hω is pure point for typical ω, the corresponding eigenfunctions exhibit the same decay rate as the hopping term. This provides a partial resolution to a conjecture proposed by Yeung and Oono, Europhys. Lett. 4(9), 1061–1065 (1987). Moreover, the subexponential dynamical localization for Hω is also established, which implies the absence of quantum transport. Our proof is based on the fractional moment method (FMM) and Simon–Wolff criterion.