Large Scale Phenomena in Models of Statistical Mechanics

统计力学模型中的大尺度现象

• 批准号: 1712632
• 负责人: Marek Biskup
• 金额: $ 27万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712632&HistoricalAwards=false
• 关键词: Large; Scale; Phenomena; Models; Statistical

The quantitative description of many natural phenomena, as well as modeling and forecasting, is invariably rooted in rigorous mathematics. Probability turns out to be a particularly useful tool as it has, by its very nature, the means to deal with systems consisting of a large number of individual constituents. The present project addresses questions that arise at the interface of probability and statistical mechanics whose common feature is a simple definition, interesting physical phenomena and a substantive challenge for mathematics. The specific problems start with the consideration of complex microscopic systems such as interacting random fields, random walks, local times and percolation. Their analysis aims to demonstrate that, under the conditions when the system size tends to infinity, a new structure, or a new degree of regularity, naturally emerges. The resulting object, often referred to as the scaling limit, is then studied using the methods of analysis. A crucial point is that many microscopic systems lead to the same scaling limit. This fact, sometimes referred to as universality, is one of the cornerstones of our understanding of large systems. The project also includes education and training of students.The project is naturally divided into three specific subject areas. The first of these concerns the extrema of logarithmically-correlated spatial processes. The goal here is to establish a level of universality of the recent findings for the Gaussian Free Field by extending them to other processes such as the local time of random walks and gradient fields. Gradient fields are then to be studied also in their own right in a number of specific contexts. The unifying theme here is to find ways to get around the restriction of the existing theory to strictly convex interactions. The third area focuses on more traditional subjects of bootstrap percolation, where the goal is to use scaling-limit ideas to establish a sharp threshold in a disordered model, and positional long-range order, where the aim is to prove crystallization in a specific model of an interacting gas. The stated problems draw on recent advances in the field, some of which are due to PI, and they bear strong connections to the theory of disordered systems, extreme-order statistics, random geometry, homogenization, etc. Each of the subareas has clearly delineated objectives of varied level of difficulty. Many of these have been designed with the aim to include graduate students and postdocs in research.

对许多自然现象的定量描述，以及建模和预测，总是植根于严格的数学。事实证明，概率是一个特别有用的工具，因为就其本质而言，它具有处理由大量单个成分组成的系统的手段。本项目解决的问题，出现在概率和统计力学的共同特点是一个简单的定义，有趣的物理现象和数学的实质性挑战的接口。具体的问题开始考虑复杂的微观系统，如相互作用的随机场，随机行走，当地时间和渗流。他们的分析旨在证明，在系统规模趋于无穷大的条件下，一种新的结构或新的规律性程度自然会出现。由此产生的对象，通常被称为缩放限制，然后使用分析的方法进行研究。关键的一点是，许多微观系统导致相同的标度极限。这个事实，有时被称为普遍性，是我们理解大系统的基石之一。该项目还包括对学生的教育和培训，该项目自然分为三个具体的主题领域。其中第一个涉及的极值的空间相关的过程。这里的目标是建立一个普遍性的水平，最近发现的高斯自由场，将它们扩展到其他过程，如本地时间的随机游动和梯度场。梯度场，然后也被研究在自己的权利，在一些特定的情况下。这里的统一主题是找到方法来绕过现有理论对严格凸相互作用的限制。第三个领域侧重于更传统的自举渗滤的主题，其目标是使用缩放限制的想法来建立一个无序模型中的尖锐阈值，以及位置长程有序，其目的是证明相互作用气体的特定模型中的结晶。所述问题借鉴了该领域的最新进展，其中一些是由于PI，它们与无序系统理论，极端顺序统计，随机几何，均匀化等有着密切的联系。每个子领域都有明确的目标，不同程度的难度。其中许多都是为了包括研究生和博士后的研究而设计的。

项目成果

An invariance principle for one-dimensional random walks among dynamical random conductances

动态随机电导中一维随机游走的不变性原理

• DOI: 10.1214/19-ejp348
• 发表时间: 2019
• 期刊: Electronic Journal of Probability
• 影响因子: 1.4
• 作者: Biskup, Marek

Exceptional points of two-dimensional random walks at multiples of the cover time

覆盖时间倍数处二维随机游走的异常点

• DOI: 10.1007/s00440-022-01113-4
• 发表时间: 2022
• 期刊: Probability Theory and Related Fields
• 影响因子: 2
• 作者: Abe, Yoshihiro;Biskup, Marek
• 通讯作者: Biskup, Marek