Statistical reconstruction of the GFF and KT transition
Christophe Garban, Avelio Sepúlveda · Journal of the European Mathematical Society · 2023
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{ {2\pi} T} . Can we recover the macroscopic observables of \phi with o(1) precision? We prove that this statistical reconstruction problem undergoes the following Kosterlitz–Thouless type phase transition: 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) that undergoes a KT transition.