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Limit shapes for square ice and tails of the KPZ equation

KPZ 方程的方冰和尾部的极限形状

基本信息

批准号：
EP/T013893/1
负责人：
Thomas Bothner
金额：
$ 34.04万
依托单位：
King's College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT013893%2F1
关键词：
Limit shapes square ice tails

项目摘要

This research proposal is pointing at two fundamental problems in the theory of exactly solvable lattice models in statistical physics and the field of integrable probability, investigating various conjectures on scaling limits and universality behaviors for correlation and distribution functions. It aims at: (a) the exact description of limit shapes in the domain wall six-vertex model in its different phase regions; and (b) the derivation of tail expansions and large deviation principles for the Kardar-Parisi-Zhang equation. The central goal of this proposal is to discover a problem specific Riemann-Hilbert approach for both projects (a) and (b) and complete them through the development of novel nonlinear steepest descent techniques in combination with original ideas and techniques coming from random matrix theory and integrable systems.The six-vertex model is the prototypical vertex integrable model for two-dimensional crystals with hydrogen bonding. It was introduced by Pauling as model for a flat H2O crystal and famously analyzed by Lieb and Sutherland for periodic boundary conditions. Subject to domain wall boundary conditions, the six-vertex model generalizes the dimer model on the Aztec diamond as well as ensembles of enumerated alternating sign matrices. Yet, the integrability in the six-vertex model with domain wall boundary conditions is fundamentally different from the determinantal or Pfaffian structures encountered in tiling or dimer models. In turn, almost nothing has been rigorously established about the six-vertex model's general geometry and its limit shapes. This fact identifies strand (a) as a central problem in mathematical statistical mechanics.The celebrated Kardar-Parisi-Zhang (KPZ) equation has become the quintessential model for random surface growth processes with numerous remarkable connections to a number of different physical phenomena. Despite several impressive results in recent KPZ literature there is still a substantial lack of fine solution properties, for instance rigorous lower tail expansions are poorly understood. I propose to derive such estimates by developing a nonlinear steepest descent method for operator-valued Riemann-Hilbert problems. This is an analytical approach to a problem in integrable probability and stochastic analysis which was previously inaccessible from either field. This fact identifies strand (b) as a current important problem in integrable probability which will be solved through the development of novel integrable systems techniques. Thus, firmly placing one of the most celebrated stochastic PDEs in the realm of integrable systems.The results of this proposal will resolve long-standing conjectures in statistical physics and integrable probability that have attracted considerable interest over the past 15 years but which were previously inaccessible by rigorous methods. Alongside the solution of strands (a) and (b), the proposed approach to both projects develops powerful mathematical techniques for the analysis of scaling and universality behaviors in mathematical physics and will thus have broad impact in other areas of mathematics and science. To be precise, I fully expect that the proposal's interdisciplinary character and mathematical results will impact the following physical problems: the theory of critical phenomena and phase separations, random growth models, combinatorial asymptotics in quantum gravity, lattice models in statistical physics, interacting particle systems, and others.

本研究计划针对统计物理学中精确可解格点模型理论和可积概率领域中的两个基本问题，研究相关函数和分布函数的标度极限和普适性行为的各种理论。其目的是：（a）畴壁六顶点模型在不同相区极限形状的精确描述，（B）Kardar-Parisi-Zhang方程尾展开式和大偏差原理的推导。该方案的核心目标是为方案（a）和（B）找到一个问题特定的Riemann-Hilbert方法，并通过发展新的非线性最速下降技术，结合随机矩阵理论和可积系统的原始思想和技术来完成方案（a）和（b）。它是由Pauling引入作为平面H2O晶体的模型，并由Lieb和Sutherland在周期性边界条件下进行了著名的分析。受畴壁边界条件，六顶点模型概括了二聚体模型的阿兹特克钻石以及合奏枚举交替符号矩阵。然而，在六顶点模型与畴壁边界条件的可积性是从根本上不同的瓷砖或二聚体模型中遇到的行列式或Pfidian结构。反过来，几乎没有任何关于六顶点模型的一般几何形状和极限形状的严格规定。这一事实表明strand（a）是数理统计力学中的一个中心问题，著名的Kardar-Parisi-Zhang（KPZ）方程已经成为随机表面生长过程的典型模型，它与许多不同的物理现象有着许多显著的联系。尽管在最近的KPZ文献中有几个令人印象深刻的结果，但仍然大量缺乏精细的解性质，例如严格的下尾展开式知之甚少。我建议得到这样的估计，通过发展一个非线性的最速下降法算子值黎曼-希尔伯特问题。这是一个分析方法的问题，在可积概率和随机分析，这是以前无法从任何领域。这一事实确定链（B）作为当前的重要问题，可积概率，将通过发展新的可积系统技术来解决。因此，坚定地把最著名的随机偏微分方程领域的可积systems. Results的这一建议将解决长期以来在统计物理学和可积概率，吸引了相当大的兴趣，在过去的15年，但以前无法通过严格的方法。除了链（a）和（B）的解决方案，这两个项目的拟议方法开发了强大的数学技术，用于分析数学物理中的标度和普适性行为，因此将在数学和科学的其他领域产生广泛的影响。准确地说，我完全期望该提案的跨学科性质和数学结果将影响以下物理问题：临界现象和相分离理论，随机增长模型，量子引力中的组合渐近，统计物理中的晶格模型，相互作用粒子系统等。