Large Scale Phenomena in Models of Statistical Mechanics

概率论的主要任务之一是解释为什么不规则性或随机性和无序性在小尺度上占主导地位的系统在大尺度上出现规律性或决定论和秩序。这在理论上很重要，因为它有助于在小规模现象不相关的情况下验证现象学理论的使用，但也有实际的后果，因为人们还需要计算进入控制现象学方程的各种材料常数的配方。目前的建议在某种程度上通过分析特定的系统来实现这些目标，在这些系统中，大规模现象产生了新的规律性程度。一般来说，这些系统对材料理论和统计力学都很有意义。拟研究的具体问题可分为五个具体的子领域：(1)与长程渗流相关的线上随机度量结构；(2)随机电阻网络中的涨落理论与均匀化；(3)二维离散高斯自由场的极值点；(4)平稳扩散有限聚集；(5)随机环境中的等周Wulff问题。这些问题的一个共同特征是，当与大小有关的自然参数趋于无穷时，就会出现一个极限结构。这种结构可以是确定性的，例如问题(5)中的极限形状，也可以是随机的，例如上面问题(3)中具有随机分形强度测量的泊松过程。要研究的问题涉及概率和统计力学的几个主题领域，例如，无序系统，增长过程，极序统计，随机几何。方法和工具也将从数学的其他部分借鉴，如谐波分析、微分方程、均匀化理论和几何测量理论。

Among the principal tasks of probability theory is to explain why regularity, or determinism and order, arise at large scales in systems where irregularity, or randomness and disorder, dominate at small scales. This is important for theoretical reasons as it helps validate the use of phenomenological theories whenever small-scale phenomena are not expected to be of relevance, but has also practical consequences as one also needs a recipe for computing various material constants that enter the governing phenomenological equations. The present proposal goes some way towards these goals by analyzing specific systems where large scale phenomena give rise to a new degree of regularity. These systems are of interest for material theory and statistical mechanics in general.The specific problems proposed to be studied divide into five specific subareas: (1) Random metric structures on the line associated with long-range percolation, (2) Fluctuation theory and homogenization in random resistor networks, (3) Extreme points of the two-dimensional discrete Gaussian Free Field, (4) Stationary Diffusion-Limited Aggregation, (5) Isoperimetric Wulff problems in random environment. A common feature of these problems is that, in the limit when a natural parameter related to size tends to infinity, a limiting structure emerges. This structure can be deterministic, e.g., a limit shape in question (5), or random, e.g., a Poisson process with random fractal intensity measure in question (3) above. The questions to be studied have bearing on several subject areas of probability and statistical mechanics, e.g., disordered systems, growth processes, extreme-order statistics, random geometry. Methods and tools will be borrowed from other parts of mathematics as well, e.g., harmonic analysis, differential equations, homogenization theory and geometric measure theory.