论文标题
高斯自由场和KT过渡的统计重建
Statistical reconstruction of the Gaussian free field and KT transition
论文作者
论文摘要
在本文中，我们关注以下问题。假设$ ϕ $是$λ\ subset \ frac 1 n \ mathbb {z}^2 $上的离散高斯免费字段（gff），并且我们得到了$ e^{IT ϕ} $，或等效地$ n ϕ \ pmod {\ pmod {\ frac {\ frac {2π} t} $。我们可以恢复$（1）$ precision的$ ϕ $的宏观可观察物吗？我们证明，这个统计重建问题经历了以下kosterlitz-无尽的类型相变： - ）如果$ t <t_ {rec}^ - $，一个人可以从$ ϕ \ pmod {\ frac {2π} t} $的知识中完全恢复$ ϕ $。在这个制度中，我们的证明依赖于一种新型的PEIERLS参数，我们称之为退火的Peierls参数，它使我们能够处理未知的淬火地面。 - ）如果$ t> t_ {rec}^+$，则不可能从$ ϕ \ pmod {\ frac {2π} t} $的知识中完全恢复字段$ ϕ $。为了证明这一结果，我们将Fröhlich-Spencer推广到不均匀培养基中的整数值GFF的情况。这种定位结果引起了独立的兴趣，我们将技术应用于附录B中的{\ em随机阶段正弦模型}。此外，与Riemann-Theta函数的有趣联系也沿证明借鉴。 这种统计重建问题是由二维XY和反派模型激发的。实际上，在低温$ t $的情况下，这些连续自旋系统的大规模波动猜想是由高斯自由场控制的。很自然地询问是否可以从XY或小人模型的旋转观察中恢复基础的宏观GFF。 这项工作的另一个动机是，它为我们提供了经历KT过渡的``集成模型''（GFF）。
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 $T<T_{rec}^-$ , one can fully recover $ϕ$ from the knowledge of $ϕ\pmod{\frac {2π} T}$. In this regime our proof relies on a new type of Peierls argument which we call annealed Peierls argument and which allows us to deal with an unknown quenched groundstate. -) If $T>T_{rec}^+$, 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.
