[Paper Review] Statistical reconstruction of the Gaussian free field and KT transition

Christophe Garban, Avelio Sepúlveda|arXiv (Cornell University)|Feb 27, 2020

Stochastic processes and statistical mechanics28 references11 citations

TL;DR

This paper investigates the statistical reconstruction of a 2D Gaussian free field (GFF) from noisy, modulo-2π observations of its phase, showing a Kosterlitz-Thouless-type phase transition: reconstruction is possible when temperature $ T < T_{\text{rec}}^{-} $ via a deterministic reconstruction function, but impossible when $ T > T_{\text{rec}}^{+} $, due to delocalization. The key contribution is a rigorous phase transition in reconstruction feasibility tied to the GFF's macroscopic behavior.

ABSTRACT

In this paper, we focus on the following question. Assume $ϕ$ is a discrete Gaussian free field (GFF) on $Λ\subset \frac 1 n \mathbb{Z}^2$ and that we are given $e^{iT ϕ}$, or equivalently $ϕ\pmod{\frac {2π} T}$. Can we recover the macroscopic observables of $ϕ$ 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 $ϕ$ from the knowledge of $ϕ\pmod{\frac {2π} T}$. To prove this result, we generalise the delocalisation theorem by Fröhlich-Spencer to the case of integer-valued GFF in an inhomogeneous medium. This delocalisation result is of independent interest and we give an application of our techniques to the {\em random-phase Sine-Gordon model} in Appendix B. Also, 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) that undergoes a KT transition.

Motivation & Objective

• To determine under what conditions the macroscopic observables of a 2D Gaussian free field (GFF) can be reconstructed from noisy, modulo-2π observations of its phase.
• To establish a Kosterlitz-Thouless-type phase transition in the statistical reconstruction problem of the GFF.
• To prove that reconstruction is possible when $ T < T_{\text{rec}}^{-} $, using a novel annealed Peierls argument to handle unknown quenched ground states.
• To show that reconstruction fails when $ T > T_{\text{rec}}^{+} $, by generalizing Fröhlich-Spencer's delocalization theorem to integer-valued GFFs in inhomogeneous media.
• To connect the reconstruction problem to the random-phase Sine-Gordon model and establish a lower bound on fluctuations in the high-temperature regime.

Proposed method

• The authors introduce an annealed Peierls argument to control the quenched ground state in the low-temperature regime, enabling reconstruction when $ T < T_{\text{rec}}^{-} $.
• They generalize the Fröhlich-Spencer delocalization theorem to integer-valued GFFs in inhomogeneous media to prove the impossibility of reconstruction when $ T > T_{\text{rec}}^{+} $.
• The reconstruction function $ F_T $ is constructed deterministically to recover $ \phi_n $ from $ \exp(iT\phi_n) $, with uniform $ L^2 $ and exponential decay bounds on reconstruction error.
• The proof leverages the connection to Riemann-theta functions and periodic functions in the phase space, particularly in analyzing the random-phase Sine-Gordon model.
• The authors extend their delocalization result to the random-phase Sine-Gordon model by treating the disorder $ \mathbf{a} $ as quenched and showing that the variance of $ \phi(0) $ grows at least as $ \Omega(1)\log n $ for $ \beta < \beta_0 $.
• The analysis is carried out for both Dirichlet and free boundary conditions, with special care taken in the free case due to non-centered conditional laws.

Experimental results

Research questions

• RQ1 Can the macroscopic observables of a 2D Gaussian free field be reconstructed from $ \phi \mod \frac{2\pi}{T} $, and if so, under what conditions on $ T $?
• RQ2 Does the statistical reconstruction problem of the GFF exhibit a phase transition analogous to the Kosterlitz-Thouless transition?
• RQ3 What is the role of the quenched ground state in the reconstruction problem, and how can it be controlled in the low-temperature regime?
• RQ4 How does the delocalization of the GFF in an inhomogeneous medium affect the feasibility of reconstruction?
• RQ5 What lower bounds can be established for the fluctuations of the random-phase Sine-Gordon model in the high-temperature regime?

Key findings

• For $ T < T_{\text{rec}}^{-} $, there exists a deterministic reconstruction function $ F_T $ such that the $ L^2 $ reconstruction error is uniformly bounded in $ n $, and the error decays exponentially in distance.
• For $ T > T_{\text{rec}}^{+} $, no deterministic reconstruction function can recover the GFF macroscopically, as the $ L^2 $ error grows at least as $ c(T,x)\log n $ for any $ x \in (-1,1)^2 $.
• The reconstruction failure at high $ T $ is due to delocalization, proven by generalizing the Fröhlich-Spencer delocalization theorem to integer-valued GFFs in inhomogeneous media.
• The paper establishes a lower bound of $ \Omega(1)\log n $ on the variance of $ \phi(0) $ in the random-phase Sine-Gordon model for $ \beta < \beta_0 $, uniformly in the quenched disorder $ \mathbf{a} $.
• The phase transition in reconstruction feasibility mirrors the Kosterlitz-Thouless transition, with $ T_{\text{rec}}^{-} $ and $ T_{\text{rec}}^{+} $ marking the critical temperatures.
• The results apply to both Dirichlet and free boundary conditions, though the exponential decay of error (1.4) does not hold in the free case due to non-centered conditional laws.