# Statistical reconstruction of the Gaussian free field and KT transition

@article{Garban2020StatisticalRO, title={Statistical reconstruction of the Gaussian free field and KT transition}, author={Christophe Garban and Avelio Sep'ulveda}, journal={arXiv: Probability}, year={2020} }

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 $T<T_{rec}^-$ , one can fully recover $\phi$ from the…

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