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

二维 KPZ 演化、波动下界和超计量性

• 批准号: 1855484
• 负责人: Sourav Chatterjee
• 金额: $ 30万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855484&HistoricalAwards=false
• 关键词: Two; Dimensional; KPZ; Evolution; Fluctuation

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.

该项目涉及概率论中的几个问题。一类问题根据 KPZ 方程解决随机表面的演化问题，KPZ 方程以其发现者卡达尔 (Kardar)、帕里西 (Parisi) 和张 (Zhang) 的名字命名。随机曲面最近在概率论中引起了很多关注，并且有许多未解答的问题。该项目旨在回答其中一些问题。第二类问题涉及扩展和发展随机变量波动下界理论。理解随机变量的波动是概率论的基本目标之一，但现有方法在许多重要问题上无法给出理想的结果。 PI 旨在通过提供一套新工具在该领域取得一些进展。最后，第三类问题集中于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问题的策略具有发现新联系的潜力。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，因为到目前为止，构建二维 KPZ 方程的任何解的任务仍然很棘手。关于波动下界的结果将给出可以证明平面增长模型的当前最佳下界的最佳条件。 此前，这样的下限需要限制性条件。所提出的解决方法基于一种新颖的耦合，这可能具有独立的意义。超计量性的研究将揭示自旋玻璃模型中状态的层次结构，特别是对于完全复制对称性破缺的模型。它还将引入这些模型的研究与图论工具（例如 Szemeredi 正则引理）之间的新颖联系。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。

项目成果

Convergence of Deterministic Growth Models

确定性增长模型的收敛

• DOI: 10.1007/s00205-022-01798-w
• 发表时间: 2022
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Chatterjee, Sourav;Souganidis, Panagiotis E.
• 通讯作者: Souganidis, Panagiotis E.

Average Gromov hyperbolicity and the Parisi ansatz

平均格罗莫夫双曲性和帕里西 ansatz

• DOI: 10.1016/j.aim.2020.107417
• 发表时间: 2021
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Chatterjee, Sourav;Sloman, Leila
• 通讯作者: Sloman, Leila

A Deterministic Theory of Low Rank Matrix Completion

• DOI: 10.1109/tit.2020.3019569
• 发表时间: 2020-12-01
• 期刊: IEEE TRANSACTIONS ON INFORMATION THEORY
• 影响因子: 2.5
• 作者: Chatterjee, Sourav

Constructing a solution of the $(2+1)$-dimensional KPZ equation

构造 $(2 1)$ 维 KPZ 方程的解

• DOI: 10.1214/19-aop1382
• 发表时间: 2020
• 期刊: Annals of Probability
• 影响因子: 2.3
• 作者: Chatterjee, Sourav;Dunlap, Alexander
• 通讯作者: Dunlap, Alexander

A SIMPLE MEASURE OF CONDITIONAL DEPENDENCE

• DOI: 10.1214/21-aos2073
• 发表时间: 2021-12-01
• 期刊: ANNALS OF STATISTICS
• 影响因子: 4.5
• 作者: Azadkia, Mona;Chatterjee, Sourav
• 通讯作者: Chatterjee, Sourav