Scaling limits and extreme values of Gibbs measures

吉布斯测度的尺度限制和极值

• 批准号: EP/T00472X/1
• 负责人: Wei Wu
• 金额: $ 25.06万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FT00472X%2F1
• 关键词: Scaling; limits; extreme; values; Gibbs

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.

近年来，在概率领域，随机系统发挥了越来越重要的作用，在随机系统中，在空间结构中观察到随机性。定义在格子上的随机系统已经被引入作为描述各种现象的相变的离散模型，从多孔介质中的液体到疾病的传播。在过去的几十年里，我们对其中的一些模型，如渗流模型和Ising模型的理解有了很大的提高，并且围绕它的工作在2006年和2010年获得了菲尔兹奖。其中一个问题涉及梯度吉布斯测度，这是Brascamp、Lebowitz和Lieb在20世纪70年代引入的随机表面模型，作为晶体界面的模型。一个长期存在的普遍性猜想指出，这些随机表面的大尺度统计特性表现得像一个高斯自由场。Naddaf和Spencer（以及其他人）的工作部分证实了这一点。PI旨在通过量化现有的波动定理，解决表面张力20年的问题，提高对梯度吉布斯测度的理解。（描述具有全局倾斜的表面轮廓的能量），并建立对数相关场的极端的一些普适性模型。第二圈问题涉及XY和Villain模型，它们是液晶、液氦和超导体的数学模型。围绕它的工作导致了2016年的诺贝尔物理学奖（Kosterlitz和Kosterless）。物理学家预测，在低温下，这些模型的大尺度性质与高斯自由场密切相关。这被称为高斯自旋波猜想。在1970年代和1980年代初，围绕着Frohlich、Simon和Spencer的作品，在数学上取得了一些进展。然而，在这些论文中开发的方法（红外边界和库仑气体重整化）不足以完成这一猜想的证明。PI的目标是解决XY和Villain模型在三维及更高维度上的高斯自旋波猜想。在此过程中，PI将开发一个强大的框架来研究一大类Gibbs测度的标度极限，波动和大偏差。在概率论、统计力学和数学分析之间将建立新的桥梁。