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

这个项目涉及概率论中的几个问题。一类问题根据KPZ方程解决随机曲面的演化问题，KPZ方程以其发现者Kardar、Parisi和Zhang的名字命名。随机曲面最近在概率论中引起了很多关注，并且有许多未解之谜。本项目旨在回答其中的一些问题。第二类问题涉及扩展和发展随机变量波动的下界理论。理解随机变量的波动是概率论的基本目标之一，但在许多重要的问题上，现有的方法不能给出理想的结果。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统计力学模型研究中出现的超度量空间。同时研究概率论不同领域的问题的策略有可能发现新的联系。关于KPZ演化的问题主要是产生二维方程的解。这将是一个重要的突破，因为构建二维KPZ方程的任何解的任务到目前为止仍然是棘手的。关于波动下界的结果给出了平面生长模型当前最佳下界的最优条件。以前，这样的下界需要限制性条件。所提出的求解方法是基于一种新的耦合，这可能是独立的兴趣。超对称性的研究将有助于揭示自旋玻璃模型中状态的层次组织，特别是对于具有全复制对称性破缺的模型。它还将介绍这些模型的研究与图论工具（如Szemeredi的正则引理）之间的新联系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.