Scaling limits and extreme values of Gibbs measures

近年来，在概率领域，随机系统发挥着越来越重要的作用，在空间结构中观察到了随机性。定义在晶格上的随机系统被引入为离散模型，描述了从多孔介质中的液体到疾病传播等各种现象的相变。在过去的几十年里，我们对其中的一些模型，如渗流和伊辛模型的理解有了很大的提高，围绕它的工作已经导致了2006和2010年的菲尔兹奖。这项拟议的研究的目的是为晶格上的随机系统中的几个长期悬而未决的问题开辟新的方向。其中一圈问题涉及梯度吉布斯测度，这是一种随机表面模型，由BrasCamp，Lebowitz和Lieb在20世纪70年代引入，作为晶体界面的模型。一个由来已久的普适性猜想指出，这些随机表面的大尺度统计性质表现为一个高斯自由场。纳达夫和斯宾塞(以及其他人)的工作部分证实了这一点。PI的目的是通过量化现有的涨落定理，解决20年来关于表面张力(描述具有整体倾斜的表面轮廓的能量)的猜想，并建立一些关于对数关联场极值的普适性猜想，从而提高对梯度吉布斯测量的理解。第二轮问题涉及XY和Villain模型，它们是液晶、液氦和超导体的数学模型。围绕它的研究导致了2016年的诺贝尔物理学奖(科斯特利茨和索利斯)。物理学家预测，在低温下，这些模型的大尺度特性与高斯自由场密切相关。这就是众所周知的高斯自旋波猜想。在20世纪70年代和80年代初，围绕着弗罗利希、西蒙和斯宾塞的作品，这一猜想在数学上取得了一些进展。然而，这些文献中发展的方法(红外界和库仑气体重整化)不足以完成这一猜想的证明。PI旨在解决XY模型和反派模型在三维和更高维度上的长期存在的高斯自旋波猜想。通过这样做，PI将开发一个稳健的框架来研究一大类Gibbs度量的标度极限、涨落和大偏差。概率、统计力学和数学分析之间将建立新的桥梁。

In the area of probability, an increasingly important role has been played in recent years by random systems in which the randomness is observed in the spatial structure. Random systems defined on lattices have been introduced as discrete models that describe phase transitions for various phenomena, ranging from liquid in porous media to the spread of disease. Our understanding of some of these models, such as percolation and Ising model, has been improved greatly in the last decades, and works around it have led to Fields medals in 2006 and 2010.The aim of the proposed research is to open new directions for several long standing open questions in random systems on lattices. One circle of the questions concern the gradient Gibbs measures, which is a model of random surface introduced in the 1970s by Brascamp, Lebowitz and Lieb as a model for crystal interfaces. A long standing universality conjecture states that the large scale statistical properties of these random surfaces behave like a Gaussian free field. This has been partially confirmed by the work of Naddaf and Spencer (and others). The PI intends to improve the understanding of the gradient Gibbs measures, by quantifying the existing fluctuation theorems, settling the 20-year-old conjectures in surface tension (that describes the energy of a surface profile with a global tilt), and to establish some universality conjectures of the extremes of log-correlated fields.The second circle of questions concern the XY and the Villain models, which are mathematical models of liquid crystals, liquid helium and superconductors. Works around it have led to the Nobel Prize in Physics (Kosterlitz and Thouless) in 2016. Physicists predict that at low temperature the large scale property of these models are closely related to the Gaussian free field. This is known as the Gaussian spin wave conjecture. Some mathematical progress was made towards the conjecture in the 1970s and the early 1980s, building around the works of Frohlich, Simon and Spencer. However, methods developed in these papers (infrared bounds and Coulomb gas renormalization) were not sufficient to complete the proof of this conjecture. The PI intends to resolve this long-standing Gaussian spin wave conjecture for the XY and the Villain models in dimension three and higher.In doing so, the PI will develop a robust framework to study the scaling limits, fluctuations and large deviations of a large class of Gibbs measures. New bridges will be built between probability, statistical mechanics and mathematical analysis.