We show that a modification of the proof of our paper Coquille et al. (J. Stat. Phys. 172(5), 1210–1222 (2018)), in the spirit of Fröhlich and Pfister (Commun. Math. Phys. 81, 277–298 (1981)), shows delocalisation in the long-range Discrete Gaussian Chain, and generalisations thereof, for any decay power \(\alpha >2\) and at all temperatures. The argument proceeds by contradiction: any shift-invariant and localised measure (in the \(L^1\) sense), is a convex combination of ergodic localised measures. But the latter cannot exist: on one hand, by the ergodic theorem, the average of the field over growing boxes would be almost surely bounded ; on the other hand the measure would be absolutely continuous with respect to its height-shifted translates, as a simple relative entropy computation shows. This leads to a contradiction and answers, in a non-quantitative way, an open question stated in a recent paper of C. Garban (Invisibility of the integers for the discrete Gaussian Chain via a caffarelli-silvestre extension of the discrete fractional laplacian. Preprint arXiv:2312.04536v2, (2023)).
