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We study microscopic observables of the Discrete Gaussian model (i.e., theGaussian free field restricted to take integer values) at high temperatureusing the renormalisation group method. In particular, we show the centrallimit theorem for the two-point function of the Discrete Gaussian model bycomputing 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 withthe multi-scale polymer expansion used in \cite{dgauss1, dgauss2}, where it wasused to study the scaling limit of the Discrete Gaussian model. The method alsoapplies to multi-point functions and lattice models of sine-Gordon type studiedin \cite{MR634447}.

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