Critical systems in random geometry

随机平面几何学是研究从二维统计物理模型中产生的规范随机几何结构。它的目的是深入了解的行为和之间的联系：随机曲线（界面的限制）;随机场（限制的“高度函数”）;和随机度量测量空间（限制的“随机平面地图”）。几十年来，这些物体一直是物理学家们深入研究的对象。广义上讲，许多离散模型的极限应该基本上独立于它们的小尺度行为;因此，理解这些极限为同时描述整个离散系统类提供了范围。另一方面，证明这种普遍性陈述是众所周知的挑战。著名的成果包括识别Schramm-Loewner演化曲线的尺度限制的渗流接口和循环擦除随机行走，最近，“布朗地图”的限制随机三角形的平面。这项建议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而令人兴奋的技术。这种设置对应于一个特殊的“普适类”的统计物理模型，显示出显着不同的行为。这导致标准的分析技术崩溃，这意味着发展严格的数学理论提出了独特的挑战。因此，对这一制度的了解要少得多。另一方面，从物理和数学的角度来看，它特别相关。例如，它预计将描述与许多模型相关的普遍极值行为;从随机矩阵到黎曼zeta函数，数论中的中心对象。深入理解这幅图是这项雄心勃勃的计划的重点。研究的广泛目标是：严格建立主要数学对象的几何性质;发现它们之间的联系;并确定缩放限制。这样的结果将对几个相关领域的未决问题产生直接和重大的影响。因此，它们将为概率与其他数学领域（如复分析，数论）以及其他学科（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。在这些学科之间创造一个强有力的合作环境一直被认为是一个具有关键战略意义的领域。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。随后在达勒姆的一个专业小组的扩展将为这一领域的基础研究提供独特的能力，巩固英国在物理科学和最终在工业中开发新颖和突破性技术的能力。

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.