Critical systems in random geometry

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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.