Statistical reconstruction of the Gaussian free field and KT transition

In this paper, we focus on the following question. Assume $\phi$ is a discrete Gaussian free field (GFF) on $\Lambda \subset \frac 1 n \mathbb{Z}^2$ and that we are given $e^{iT \phi}$, or equivalently $\phi \pmod{\frac {2\pi} T}$. Can we recover the macroscopic observables of $\phi$ up to $o(1)$ precision? We prove that this statistical reconstruction problem undergoes the following Kosterlitz-Thouless type phase transition: -) If $TT_{rec}^+$, it is impossible to fully recover the field $\phi$ from the knowledge of $\phi \pmod{\frac {2\pi} T}$. To prove this result, we generalise the delocalisation theorem by Fr\"ohlich-Spencer to the case of integer-valued GFF in an inhomogeneous medium. This delocalisation result is of independent interest and an interesting connection with Riemann-theta functions is drawn along the proof. This statistical reconstruction problem is motivated by the two-dimensional XY and Villain models. Indeed, at low-temperature $T$, the large scale fluctuations of these continuous spin systems are conjectured to be governed by a Gaussian free field. It is then natural to ask if one can recover the underlying macroscopic GFF from the observation of the spins of the XY or Villain model. Another motivation for this work is that it provides us with an "integrable model" (the GFF) subjected to a KT transition.