Large Scale Phenomena in Models of Statistical Mechanics

点击翻译按钮获取中文摘要

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.