The Discrete Gaussian model is a Gaussian free field on lattice restricted to take integer values. In dimension two, it was proved by the seminal work of Fröhlich-Spencer that the Discrete Gaussian model exhibits localisation-delocalisation phase transition. The phase transition is ubiquitous in two-dimensional statistical physics models, intriguing the need for a unified framework for studying these phenomena.

The goal of this thesis is to apply rigorous renormalisation group method to study the two-dimensional discrete Gaussian model in the delocalised phase, thereby obtaining central limit theorems in long-distance limit—in physics literature, the renormalisation group is a standard apparatus used to study scaling phenomena, in particular computing critical exponents and proving scaling limits and universality.

We study the central limit theorem in three different regimes, first on macroscopic scale, second on mesoscopic scale and the third on microscopic scale. The first two amount to studying the scaling limits of the spin model under different limit regimes, while the final one discusses both pointwise and limit results. The final results have in particular prolific by-products, producing analogues of a number of results proved for different interface models.

The entire thesis is devoted to solving these problems, but the strategy of the proof we develop is expected to have general applicability. Indeed, we develop renormalisation technology in the first half (Chapter 2–4) that only has weak requirements on the model. Then in the rest of the thesis, we develop an analysis specific to our model to prove the main theorems.