CAREER: Halting Problems In Statistical Mechanics

滑坡、地震、雪崩、野火和金融危机都是复杂系统中灾难事件的例子。本研究的目的是证明这些系统的数学模型定理，有三个特定的目标：(1)量化大尺度观测对小细节的依赖性。哪些细节真正重要？(2)开发有效的算法来确定给定系统是否会稳定以及稳定需要多长时间。(3)检查大灾难事件发生前的阈值状态，以及触发灾难的因素。虽然仿真是复杂系统建模的重要工具，但本研究的重点是数学证明。模拟揭示了系统的行为方式，而证明揭示了为什么。许多相互作用的粒子系统在一种最终稳定的状态和一种活动永远持续的状态之间有一个相变。这些系统具有第二层次的动力学，在较慢的时间尺度上运行，并使系统变得更不稳定，最终导致活动永远持续的阈值状态。直到最近，这个阈值状态仍然超出了严格证明的范围，但现在它被理解为在一个特殊的情况下，当初始条件趋于负无穷时，极限中的阿贝尔沙堆。莱文博士将利用这一认识，在更广泛的模型类别中定位活动相变。这个项目也有一个教育部分，其目标是鼓励和使聪明的年轻人追求科学和技术事业，为他们提供在这些事业中取得成功所需的概率工作知识。

Landslides, earthquakes, avalanches, wildfires, and financial crises are examples of disaster events in complex systems. The aim of this research is to prove theorems about mathematical models of these systems, with three particular goals:(1) Quantify the dependence of large-scale observations on small details. Which details actually matter?(2) Develop efficient algorithms to determine whether a given system will stabilize and how long it will take.(3) Examine the threshold state that precedes a large disaster event, and what triggers the disaster.While simulation is an important tool for modeling complex systems, the focus of this research is on mathematical proof. A simulation reveals how a system behaves, but a proof reveals why.Many interacting particle systems have a phase transition between a state that eventually stabilizes and a state in which activity persists forever. These systems have a second level of dynamics, operating on a slower time scale and driving the system toward greater instability, leading ultimately to a threshold state in which activity persists forever. Until recently this threshold state remained beyond the realm of rigorous proof, but it is now understood in a particular case, the abelian sandpile in the limit as the initial condition tends to negative infinity. Dr. Levine will leverage this understanding to locate the activity phase transition in a wider class of models. This project also has an educational component, whose goal is to encourage and enable bright young people to pursue science and technology careers by providing them with the working knowledge of probability they will need to succeed in those careers.