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.

在概率领域，近年来，随机系统在空间结构中观察到随机性，近年来已经发挥了越来越重要的作用。在晶格上定义的随机系统被引入为离散模型，描述了各种现象的相变，从多孔培养基中的液体到疾病的传播。在过去的几十年中，我们对其中一些模型的理解，例如渗透和伊辛模型，已经得到了很大的改进，并且在2006年和2010年围绕它的工作导致了田野奖牌。拟议的研究的目的是为Lattices的随机系统中的几个长期开放问题打开新的方向。这些问题的一个圆圈涉及梯度吉布斯的测量，这是Brascamp，Lebowitz和Lieb在1970年代引入的随机表面模型，作为晶体接口的模型。长期存在的普遍性猜想指出，这些随机表面的大规模统计特性的行为就像高斯自由场。纳达夫和斯宾塞（其他）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，在表面张力中解决20年前的猜想（描述具有全球倾斜度的表面能量），并建立一些与固定型机器的液体模型的液体模型，以下是固定型圆周的圆周，这是XY的液体模型，这是，这是XY的液体模型，这是，这是XY的模型，这是，该模型是，该模型是，该模型是，该模型是，该模型是，该模型是，该模型和固定型模型是，该模型是，这是XY和XY的多个模型，这是，该模型是，该模型是，该模型是，该模型是，该模型是，该模型是，该模型是，该模型是，该模型和XY的模型是，该模型是，该模型和数字是，该模型是，该模型和数字是，该模型是超导体。围绕它的作品导致了2016年诺贝尔物理学（Kosterlitz and Thouless）。物理学家预测，在低温下，这些模型的大规模特性与高斯自由领域密切相关。这被称为高斯旋转波。在1970年代和1980年代初，对猜想取得了一些数学进步，围绕弗洛希里奇，西蒙和斯宾塞的作品进行了发展。但是，在这些论文中开发的方法（红外边界和库仑气体重新归化）不足以完成该猜想的证明。 PI打算在第三和更高的维度上解决XY的长期高斯自旋波和小人模型的猜想。这样做，PI将开发出强大的框架来研究缩放限制，波动和巨大的大型Gibbs度量。新的桥梁将建立在概率，统计力学和数学分析之间。