Large Scale Phenomena in Models of Statistical Mechanics

概率论的主要任务之一是解释为什么在小尺度上不规则性、随机性和无序性占主导地位的系统中，规则性、决定论和有序性会在大尺度上出现。这是重要的理论原因，因为它有助于验证使用的现象学理论时，小规模的现象预计不会是相关的，但也有实际的后果，因为人们还需要一个食谱计算各种材料常数进入治理现象学方程。目前的建议通过分析特定的系统来实现这些目标，在这些系统中，大规模的现象产生了新的规律性。这些系统一般对材料理论和统计力学都很有意义。建议研究的具体问题分为五个具体的子领域：（1）与长程逾渗相关的线上的随机度量结构，（2）随机电阻网络中的涨落理论和均匀化，（3）二维离散高斯自由场的极值点，（4）稳态扩散限制聚集，（5）随机环境中的等周Wulff问题。这些问题的一个共同特征是，在极限中，当与尺寸相关的自然参数趋于无穷大时，出现了极限结构。该结构可以是确定性的，例如，所讨论的极限形状（5），或随机形状，例如，在上述问题（3）中具有随机分形强度测度的泊松过程。要研究的问题与概率和统计力学的几个学科领域有关，例如，无序系统，增长过程，极端顺序统计，随机几何。方法和工具也将从数学的其他部分借用，例如，调和分析、微分方程、均匀化理论和几何测度理论。

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.