Quantitative delocalisation for the Gaussian and $q$-SOS long-range chains

Abstract

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 $\hat{I}\pm \in (1,\infty )$ and inverse temperature ${\hat{I}}^2\in (0,\infty )$, and show that: - For $\hat{I}\pm \in (2,3)$ and ${\hat{I}}^2\in (0,\infty )$, the fluctuations of the chain are at least of order $N^{\frac{1}{2}(\hat{I}\pm -2)}$; - For $\hat{I}\pm =3$ and ${\hat{I}}^2\in (0,\infty )$, 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 ${\hat{I}}^2\in (0,\infty )$ and any range exponent $\hat{I}\pm \in (1,\infty )$: - The fluctuations of the chain are at least of order $N^{\frac{1}{q}(\hat{I}\pm -2)\wedge \frac{1}{2}}$; - The fluctuations of the chain are at most of order $N^{\left(\frac{1}{q}\hat{I}\pm -1\right)\wedge \frac{1}{2}}$.

45 pages, 25 Figures. Revised version, accepted for publication in Annales Henri Lebesgue