Large Scale Phenomena in Models of Statistical Mechanics

概率论的主要任务之一是解释为什么规律性或决定论和秩序在规律性或随机性和无序在小尺度上占主导地位的系统中大规模出现。这从理论上来说很重要，因为它有助于验证现象学理论的使用，无论何时预期的小规模现象不具有相关性，但也具有实际结果，因为人们还需要计算进入主导现象学方程的各种材料常数的配方。目前的建议通过分析大规模现象产生新的规律性程度的具体系统，在一定程度上实现了这些目标。这些系统对材料理论和统计力学具有普遍意义。拟研究的具体问题可分为五个子领域：(1)与长程渗流有关的线上随机度规结构，(2)随机电阻网络中的涨落理论和齐次化，(3)二维离散高斯自由场的极点，(4)定常扩散-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，在极限中，当与大小有关的自然参数趋于无穷大时，出现极限结构。该结构可以是确定性的，例如，问题(5)中的极限形状，或者是随机的，例如，具有上述问题(3)中的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学科领域，例如无序系统、增长过程、极序统计量、随机几何。方法和工具也将借鉴数学的其他部分，如调和分析，微分方程，齐次化理论和几何测度论。

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.