Correlations and Scaling in Disordered and Critical Stochastic Models

在许多物理系统中，大而有序的结构是在系统的遥远部分没有直接通信的情况下形成的。理解这些结构是概率论中一个长期存在的挑战；本项目着重于这些结构的数学研究。其中一类，渗透，已经兴起，用于研究各种随机网络：例如，携带石油的地下岩层，城市交通网络和聚合物链。另一类网络，阿贝尔网络，是用来描述能量在微观上积累，直到达到某个阈值，然后导致大规模“雪崩”的情况，并已被用于地震模型。这个项目下的工作将涉及的一个主要问题是，随机结构中相隔多远的部分（在空间或时间上）是如何相互依赖的：例如，雪崩中早期的倾倒是如何影响后来发生的倾倒的？这些问题与其他重要的物理系统以及计算机科学中的优化问题有关。该奖项下的研究将研究无序系统（第一通道渗透和相关无序磁体）和临界系统（渗透和Bak-Tang-Wiesenfeld模型，也称为沙堆）的相关性和波动。首道渗流的一个主要焦点是随机度规的波动顺序及其与测地线性质的关系。在临界渗流中，工作将集中在大型有限簇中的化学或图距离的缩放上，特别是在二维上限定化学距离指数。最后的重点将是阿贝尔沙堆模型中雪崩的结构特性，包括沙堆在二维中的稳定性问题。

In many physical systems, large and orderly structures are formed without distant parts of the system communicating directly. Understanding these structures is a long-standing challenge in probability theory; this project focuses on the mathematical study of these structures. One class of these, percolation, has arisen to study various random networks: for instance, underground rock formations carrying oil, traffic networks in cities, and polymer chains. Another class of these, Abelian networks, arose to describe situations where energy builds up microscopically until it reaches a certain threshold and then causes a large "avalanche," and has been used to model earthquakes. A major issue the work under this project will touch on is how widely-separated (in space or time) parts of the random structure depend on each other: for instance, how do early topplings in an avalanche influence those occurring much later? These questions have connections to other important physical systems as well as to optimization problems from computer science.The research under this award will study correlations and fluctuations in disordered systems (first-passage percolation and related disordered magnets) and critical systems (percolation, and Bak-Tang-Wiesenfeld models also known as sandpiles). A primary focus in first-passage percolation will be the order of fluctuations of the random metric and its relationship to properties of geodesics. In critical percolation, the work will focus on the scaling of the chemical or graph distance in large finite clusters -- in particular, bounding the chemical distance exponent in two dimensions. A final focus will be structural properties of avalanches in the Abelian sandpile model, including the question of stabilizability of sandpiles in two dimensions.