CAREER: Halting Problems In Statistical Mechanics

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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.