Two-Dimensional KPZ Evolution, Fluctuation Lower Bounds, and Ultrametricity

这个项目涉及概率论中的几个问题。一类问题是根据KPZ方程解决随机曲面的演化，以其发现者Kardar，Parisi和Zhang命名。随机曲面在概率论中引起了广泛的关注，但仍有许多问题没有得到解答。本项目旨在回答其中一些问题。第二类问题涉及扩展和发展理论的下限波动的随机变量。了解随机变量的波动是概率论的基本目标之一，但有许多重要的问题，现有的方法没有得到理想的结果。PI旨在通过提供一套新的工具在这一领域取得一些进展。最后，第三类问题集中在理解超度量空间，这些空间是在研究统计力学模型时出现的。同时处理概率论不同领域问题的策略有可能发现新的联系。关于KPZ演化的问题主要是关于在2D中产生方程的解。这将是一个重要的突破，因为到目前为止，构造2D KPZ方程的任何解的任务仍然是棘手的。关于涨落下界的结果给出了平面增长模型当前最佳下界的最优证明条件。 以前，这种下限需要限制性条件。所提出的解决方案的方法是基于一种新的耦合，这可能是独立的利益。对超度规性的研究将有助于理解自旋玻璃模型中态的等级组织，特别是对于完全复制对称性破缺的模型。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns several problems in probability theory. One class of problems addresses the evolution of random surfaces according to the KPZ equation, named after its discoverers, Kardar, Parisi, and Zhang. Random surfaces have attracted a lot of recent attention in probability theory, and there are many unanswered questions. This project will aim to answer some of these questions. A second class of problems involves extending and developing a theory of lower bounds on fluctuations of random variables. Understanding fluctuations of random variables is one of the basic goals of probability theory, but there are many important problems where existing methods do not give desirable results. The PI aims to make some progress in this area by providing a new set of tools. Finally, a third class of problems centers around understanding ultrametric spaces that arise in the study of models from statistical mechanics. The strategy of working on problems in varied areas of probability theory at the same time has the potential of uncovering new connections.The problems concerning the KPZ evolution are mainly about producing a solution of the equation in 2D. This would be an important breakthrough because the task of constructing any solution for the 2D KPZ equation has remained intractable so far. The results about fluctuation lower bounds would give the optimal conditions under which the current best lower bounds can be proved for planar growth models. Previously, such lower bounds required restrictive conditions. The proposed method of solution is based on a novel coupling, which may be of independent interest. The research on ultrametricity will shed light on the hierarchical organization of states in spin glass models, especially for models with full replica symmetry breaking. It will also introduce a novel connection between the study of these models and tools from graph theory such as Szemeredi's regularity lemma.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.