Large Scale Phenomena in Models of Statistical Mechanics

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。概率被证明是一个特别有用的工具，因为它本身就具有处理由大量单独成分组成的系统的手段。本项目涉及概率和统计力学交界处出现的问题，它们的共同特点是定义简单、有趣的物理现象和对数学的实质性挑战。具体问题首先考虑复杂的微观系统，如相互作用的随机场、随机游动、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，自然会出现一种新的结构，或一种新的规律性。然后使用分析方法来研究所产生的对象，通常被称为比例极限。关键的一点是，许多微观系统导致了相同的比例限制。这一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学科领域。第一个问题涉及对数相关的空间过程的极值。这里的目标是通过将高斯自由场的最新发现推广到其他过程，如随机游动和梯度场的当地时间，来建立一个普适性的水平。然后，还将在一些特定的环境中研究梯度场。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。第三个领域专注于更传统的自举渗流主题，其中的目标是使用比例限制思想在无序模型中建立一个尖锐的阈值，以及位置长程有序，其中的目标是证明相互作用气体的特定模型中的结晶。所陈述的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、齐化等理论有很强的联系。每个子领域都清楚地描绘了不同难度的目标。其中许多都是为了将研究生和博士后纳入研究而设计的。

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.