Delocalisation and scaling limit for the disordered long-range Discrete Gaussian Chain

RESUMO

In this paper, we study the effect of an external random field on the localisation / delocalisation of the long-range Discrete Gaussian Chain (DGC), recently studied by Garban arXiv:2312.04536 and Coquille \emph{et al.} arXiv:2412.15782 . We prove that the model is delocalised at every inverse temperature for any polynomial decay $\alpha > 3/2$. This behaviour differs qualitatively from the one of the disorder-free long-range DGC, for which delocalisation at every temperature occurs only for $\alpha > 2$, similarly to what happens for long-range Ising models as already proved by Aizenman and Wehr. Additionally, we provide a more detailed description of the delocalised phase by identifying the growth exponent and scaling limit of the height at the origin for any decay exponent $\alpha > 3/2$.