Critical systems in random geometry

随机几何中的关键系统

• 批准号: MR/W008513/1
• 负责人: Ellen Powell
• 金额: $ 103.47万
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=MR%2FW008513%2F1
• 关键词: Critical; systems; random; geometry

Random planar geometry is the study of canonical random geometrical structures arising as scaling limits from 2D statistical physics models. It aims to gain insight into the behaviour of and connections between: random curves (limits of interfaces); random fields (limits of "height functions"); and random metric measure spaces (limits of "random planar maps"). Such objects have been subjects of intense study by physicists for decades. Broadly speaking, it is conjectured that the limits of many discrete models should be essentially independent of their small-scale behaviour; hence, understanding these limits provides scope for describing entire classes of discrete systems simultaneously. On the other hand, proving such universality statements is notoriously challenging. Celebrated results include the identification of Schramm--Loewner evolution curves as scaling limits of percolation interfaces and loop erased random walks, and more recently, the "Brownian map" as the limit of random triangulations of the plane. This proposal targets similar results in another setting, where the P.I. has recently developed several novel and exciting techniques. This setting corresponds to a particular "universality class" of statistical physics models which display notably different behaviour. This causes standard analytical techniques to break down, meaning that developing a rigorous mathematical theory presents unique challenges. As such, this regime is much less well understood. On the other hand, it is especially relevant from both a physical and mathematical perspective. For example, it is expected to describe universal extreme value behaviour associated with many models; ranging from random matrices to the Riemann-zeta function, a central object in number theory. Developing a deep understanding of the picture here is the focus of this ambitious proposal.The broad goals of the research are: to rigorously establish conjectural properties of the main mathematical objects; to discover connections between them; and to identify scaling limits. Such results will have direct and significant consequences for open problems in several related fields. As a result, they will provide an exciting platform for the initiation of interdisciplinary collaborations between probability and other mathematical areas (such as complex analysis, number theory) as well as other subjects (such as theoretical physics and computing). Creating a strong collaborative environment between disciplines such as these has been consistently recognised as an area of key strategic importance.In the longer term, this work will serve to exhibit the United Kingdom as a world-leading centre for research in random geometry. The subsequent expansion of a specialised group in Durham will provide a unique capability for fundamental research in this area, underpinning the UK's ability to develop novel and ground-breaking techniques in the physical sciences, and ultimately, in industry.

随机平面几何是对作为二维统计物理模型的缩放限制而产生的规范随机几何结构的研究。它的目的是深入了解以下行为和之间的联系： 随机曲线（界面的限制）；随机场（“高度函数”的限制）；和随机度量测量空间（“随机平面地图”的限制）。几十年来，此类物体一直是物理学家深入研究的主题。从广义上讲，人们推测许多离散模型的极限本质上应该独立于它们的小规模行为；因此，了解这些限制为同时描述整个类别的离散系统提供了范围。另一方面，证明这种普遍性陈述是出了名的具有挑战性。著名的成果包括将 Schramm-Loewner 演化曲线识别为渗流界面和循环擦除随机游走的缩放限制，以及最近将“布朗图”识别为平面随机三角剖分的限制。该提案的目标是在另一个环境中获得类似的结果，其中 P.I.最近开发了几种新颖且令人兴奋的技术。此设置对应于统计物理模型的特定“普遍性类别”，其表现出明显不同的行为。这导致标准分析技术崩溃，这意味着开发严格的数学理论提出了独特的挑战。因此，这一制度不太为人所知。另一方面，从物理和数学的角度来看，它都特别相关。例如，期望描述与许多模型相关的普遍极值行为；范围从随机矩阵到黎曼-zeta 函数（数论的核心对象）。深入理解这里的情况是这个雄心勃勃的提案的重点。该研究的总体目标是：严格建立主要数学对象的猜想属性；发现它们之间的联系；并确定缩放限制。这些结果将对几个相关领域的开放问题产生直接和重大的影响。因此，它们将为概率和其他数学领域（例如复分析、数论）以及其他学科（例如理论物理和计算）之间的跨学科合作提供一个令人兴奋的平台。在诸如此类的学科之间创建强大的协作环境一直被认为是具有关键战略重要性的领域。从长远来看，这项工作将有助于展示英国作为世界领先的随机几何研究中心的地位。达勒姆专业小组的后续扩张将为该领域的基础研究提供独特的能力，巩固英国在物理科学领域以及最终在工业领域开发新颖和突破性技术的能力。

项目成果

Many-to-few for non-local branching Markov process

非局部分支马尔可夫过程的多对少

• DOI: http://dx.10.1214/24-ejp1098
• 发表时间: 2024
• 期刊: Electronic Journal of Probability
• 影响因子: 1.4
• 作者: Harris S

Brownian half-plane excursion and critical Liouville quantum gravity

布朗半平面偏移和临界刘维尔量子引力

• DOI: http://dx.10.1112/jlms.12689
• 发表时间: 2022
• 期刊: Journal of the London Mathematical Society
• 作者: Aru J

Thick points of the planar GFF are totally disconnected for all ??0

平面 GFF 的粗点对于所有 ??0 来说都是完全断开的

• DOI: http://dx.10.1214/23-ejp975
• 发表时间: 2023
• 期刊: Electronic Journal of Probability
• 影响因子: 1.4
• 作者: Aru J

A characterisation of the continuum Gaussian free field in arbitrary dimensions

任意维度连续高斯自由场的表征

• DOI: http://dx.10.5802/jep.201
• 发表时间: 2022
• 期刊: Journal de l'École polytechnique - Mathématiques
• 作者: Aru J