Large scale phenomena in models of statistical mechanics

该项目解决了概率论和统计力学边缘的各种数学问题。它们的共同特征是相对简单的表述，耐人寻味的潜在现象和与物理学科的深刻联系。第一轮问题与随机电导模型有关。其中包括无序介质中随机游动标度极限的推导、随机拉普拉斯谱性质的研究、有效电导率涨落的分析、热核衰变的详细控制等。这些问题在物理和应用科学中无处不在，但由于无法克服潜在相互作用的非凸性，它们的数学理解目前陷入停滞。我们提供了如何在技术上克服这一障碍的具体方法和途径。第三类问题涉及无序系统领域。感兴趣的具体主题是随机场自旋模型和抛物线安德森模型所支配的动力学。文中概述了精确的数学方法，这些方法可以显著提高对这些问题的数学理解。许多列出的问题旨在为对概率论和数学物理感兴趣的研究生和博士后提供培训和研究。硬科学在准确描述、建模甚至预测复杂自然现象方面取得的巨大成功，在很大程度上源于它们严格的数学基础。在固体物理和材料科学的背景下，最常用的数学方法是概率和/或统计学方法。这并不令人惊讶，因为这些学科的设计恰好是为了处理涉及大量个人选民的系统。本项目研究大系统概率论中的三类特定问题，它们起源于材料物理。我们问的最中肯的一般问题是，材料的结构细节及其各种内在的不规则性如何准确地在它们的宏观性质中表现出来。通常，我们寻求一个包罗万象的原则，或“物理定律”，从细节中以定量或定性的形式提取本质特征。人们希望，发展这类系统的理论基础最终将在工程应用方面取得重大进展。对数学本身也有立竿见影的影响：对复杂现象的分析不可避免地涉及一些单独的数学分支学科，因此将导致它们之间富有成效的思想交流。该项目自然提供了一个机会，将研究生和博士后纳入美国顶尖研究型大学的研究环境。

The project addresses a variety of mathematical problems at the borderline of probability theory and statistical mechanics. The common feature of these is a relatively simple formulation, intriguing underlying phenomena and deep connections to physical disciplines. The first round of problems pertains to the Random Conductance Model. These include derivation of scaling limits of random walks in disordered media, study of spectral properties of random Laplacians, analysis of fluctuations of effective conductivity, detailed control of heat-kernel decay, etc. The second area proposed to investigate is that of Gradient Fields. These problems are ubiquitous in physical and applied sciences but their mathematical understanding is currently stalled because of the inability to overcome non-convexity of the underlying interaction. We offer specific methods and approaches how this obstacle may technically be overcome. The third class of problems addresses the area of Disordered Systems. The specific subjects of interest are random-field spin models and the dynamics governed by the parabolic Anderson model. Precise mathematical approaches are outlined that could lead to significant improvements in the mathematical understanding of these problems. A good many of listed problems are devised with the intention to provide training, and inclusion in research, of graduate students and postdocs who have interest in Probability Theory and Mathematical Physics.The tremendous success of hard sciences in accurate description, modeling and even forecasting complex natural phenomena derives, in large part, from their foundation in rigorous mathematics. In the context of solid-state physics and material sciences, the mathematical methods most commonly used are those of probability and/or statistics. This comes as no surprise as these disciplines have been designed precisely to deal with systems involving large numbers of individual constituents. The present project studies three specific classes of problems in probability theory of large systems whose origin is rooted in physics of materials. The most pertinent general question we ask is how the structural details of materials, and their various inherent irregularities, exactly express themselves in their macroscopic properties. As a rule, we seek an enveloping principle, or a "physical law", that extracts, in quantitative or qualitative form, the essential features from the specifics. It is hoped that developing the theoretical foundations of such systems will eventually lead to significant advances in engineering applications. There is also an immediate impact for mathematics itself: analysis of complex phenomena inevitably involves a number of separate sub-disciplines of mathematics and will thus lead to fruitful exchange of ideas among them. The project naturally offers an opportunity to include graduate students and postdocs into the research environment at a top research US university.