Absence of shift-invariant Gibbs states (delocalisation) for one-dimensional $\mathbb Z$-valued fields with Long-Range interactions

We show that a modification of the proof of our paper [CvELNR18], in the spirit of [FP81], 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 [G23] of C.Garban.