Quantitative delocalisation for the Gaussian and q -SOS long-range chain

The goal of this article is to study quantitatively the localisation/delocalisation properties of the integer-valued Gaussian chain with long-range interactions. Specifically, we consider the integer-valued Gaussian chain of length N , with Dirichlet boundary condition, range exponent α ∈ ( 1 , ∞ ) and inverse temperature β ∈ ( 0 , ∞ ) , and show that: for α ∈ ( 2 , 3 ) and β ∈ ( 0 , ∞ ) , the fluctuations of the chain are at least of order N 1 2 ( α - 2 ) ; for α = 3 and β ∈ ( 0 , ∞ ) , the fluctuations of the chain are of order N / ln N (sharp upper and lower bounds up to multiplicative constants are derived). Combined with the results of Kjaer–Hilhorst [KH82], Fröhlich–Zegarlinski [FZ91] and Garban [Gar23], these estimates provide an (almost) complete picture for the localisation/delocalisation of the integer-valued Gaussian chain. The proofs are based on graph surgery techniques which have been recently developed by van Engelenburg–Lis [EL23, EL25] and Aizenman–Harel Peled–Shapiro [AHPS22] to study the phase transitions of two dimensional integer-valued height functions (and of their dual spin systems). Additionally, by combining the previous strategy with a technique introduced by Sellke [Sel24], we are able extend the method to study the q -SOS long-range chain with exponent q ∈ ( 0 , 2 ) and show that, for any inverse temperature β ∈ ( 0 , ∞ ) and any range exponent α ∈ ( 1 , ∞ ) : the fluctuations of the chain are at least of order N 1 q ( α - 2 ) ∧ 1 2 ; the fluctuations of the chain are at most of order