Central Limit Theorem for Multi-Point Functions of the 2D Discrete Gaussian Model at high temperature

TL;DR

This paper proves a central limit theorem for the two-point function of the 2D Discrete Gaussian model at high temperature, using renormalisation group techniques to analyze multi-point functions and their asymptotic behavior.

Contribution

It introduces a novel application of renormalisation group methods to establish the CLT for multi-point functions in the 2D Discrete Gaussian model at high temperature.

Findings

Proves the CLT for the two-point function at high temperature.

Extends the method to multi-point functions and sine-Gordon models.

Connects the analysis with multi-scale polymer expansion techniques.

Abstract

We study microscopic observables of the Discrete Gaussian model (i.e., the Gaussian free field restricted to take integer values) at high temperature using the renormalisation group method. In particular, we show the central limit theorem for the two-point function of the Discrete Gaussian model by computing the asymptotic of the moment generating function $\langle e^{z (\sigma (0) - \sigma (y))} \rangle_{\beta, \mathbb{Z}^2}^{\dg}$ for $z \in \mathbb{C}$ sufficiently small. The method we use has direct connection with the multi-scale polymer expansion used in \cite{dgauss1, dgauss2}, where it was used to study the scaling limit of the Discrete Gaussian model. The method also applies to multi-point functions and lattice models of sine-Gordon type studied in \cite{MR634447}.