Critical systems in random geometry

随机平面几何是研究由二维统计物理模型作为尺度限制而产生的典型随机几何结构。它的目的是洞察行为和连接之间：随机曲线（界面的限制）；随机场（“高度函数”的极限）；随机度量度量空间（“随机平面图”的极限）。几十年来，物理学家一直在对这些天体进行深入研究。广义地说，据推测，许多离散模型的极限基本上与它们的小尺度行为无关；因此，理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。著名的结果包括Schramm- Loewner进化曲线作为渗透界面和循环删除随机游走的尺度限制的识别，以及最近的“布朗图”作为平面随机三角剖分的限制。这项提议的目标是在另一种情况下取得类似的结果，私家侦探最近开发了几种新颖而令人兴奋的技术。这种设置对应于统计物理模型的特定“普适性类”，这些模型显示出明显不同的行为。这导致标准的分析技术崩溃，这意味着发展一个严格的数学理论提出了独特的挑战。因此，人们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，它是特别相关的。例如，期望描述与许多模型相关的普遍极值行为；从随机矩阵到Riemann-zeta函数，数论的中心对象。深入了解这里的情况是这个雄心勃勃的提议的重点。研究的主要目标是：严格地建立主要数学对象的猜想性质；发现它们之间的联系；并确定缩放极限。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。随后在达勒姆的一个专业小组的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开发新颖和突破性技术的能力，并最终在工业领域。

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.