Quantitative delocalisation for the Gaussian and $q$-SOS long-range chains
The goal of this article is to study quantitatively the localisation/delocalisation properties of the discrete Gaussian chain with long-range interactions. Specifically, we consider the discrete Gaussian chain of length $N$, with Dirichlet boundary condition, range exponent $α\in (1 , \infty)$ and inverse temperature $β\in (0,\infty)$, and show that: - For $α\in (2 ,3)$ and $β\in (0 , \infty)$, the fluctuations of the chain are at least of order $N^{\frac{1}{2}(α- 2)}$; - For $α= 3$ and $β\in (0 , \infty)$, the fluctuations of the chain are of order $\sqrt{N / \ln N}$ (sharp upper and lower bounds up to multiplicative constants are derived). Combined with the results of Kjaer-Hilhorst, Fröhlich-Zegarlinski and Garban, these estimates provide an (almost) complete picture for the localisation/delocalisation of the discrete Gaussian chain. The proofs are based on graph surgery techniques which have been recently developed by van Engelenburg-Lis and Aizenman-Harel-Peled-Shapiro 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, we are able extend the method to study the $q$-SOS long-range chain with exponent $q \in (0 , 2)$ and show that, for any inverse temperature $β\in (0, \infty)$ and any range exponent $α\in (1 , \infty)$: - The fluctuations of the chain are at least of order $N^{\frac{1}{q}(α-2) \wedge \frac{1}{2}}$; - The fluctuations of the chain are at most of order $N^{\left( \frac{1}{q}α- 1 \right) \wedge \frac 12}$.