Critical Systems in Random Geometry
Lead Research Organisation:
DURHAM UNIVERSITY
Abstract
Understanding complex random systems is a central challenge in modern mathematics and theoretical physics. Many striking examples, such as random surfaces in conformal field theory, the statistics of the Riemann zeta function, and models for turbulence, share a common feature: they are governed by logarithmically correlated fluctuations. This proposal builds on the belief, now supported by rigorous progress, that there is a universal mathematical structure describing the extremes and geometry of such systems. My research aims to construct and analyse this structure, and apply it across probability, mathematical physics, and related fields.
Two themes drive this programme: the geometry of random surfaces known as Liouville quantum gravity (LQG) and the theory of multiplicative chaos, both considered in their critical, or threshold, regimes. LQG provides a probabilistic model for random two-dimensional geometries, rooted in string theory and conformal field theory, and is believed to capture the large-scale behaviour of discrete random surfaces known as random planar maps (RPMs). Multiplicative chaos is the rigorous mathematical framework for constructing areas associated with these surfaces, and more broadly, for building measures from log-correlated fields, initiated by Mandelbrot to model energy dissipation in intermittent turbulence.
In both areas, the most challenging and least understood regimes are the critical and supercritical phases, where classical theory breaks down and measures become dominated by exceptional “spikes”. It is widely believed, and now known in several cases, that the extremes of systems with logarithmic correlations such as the planar Gaussian free field, branching random walks, and random matrix characteristic polynomials are governed by critical multiplicative chaos. Establishing this universality rigorously and understanding the associated geometry in the (super)critical regime is a key aim of this proposal.
During the first phase of my UKRI FLF, I developed a mathematical framework for critical LQG, established connections with Conformal Loop Ensembles (CLE), and solved a major open problem by proving the first scaling limit for RPMs in the critical universality class. In parallel, I contributed foundational ideas to the theory of critical multiplicative chaos, developed new methods for analysing extremes of log-correlated fields, and co-authored a major book clarifying the area.
The renewal builds on these breakthroughs. Work Package A develops the geometry of random surfaces in the (super)critical regime. This includes: proving convergence of embedded RPMs to critical LQG; constructing the LQG metric on the CLE(4) gasket and linking it to newly constructed scaling limits of maps with large faces; developing a general theory of supercritical LQG surfaces; and deriving the correlation functions of Liouville conformal field theory in the (super)critical phase.
Work Package B focuses on the universality of critical multiplicative chaos and extreme-value phenomena. This includes: identifying the full extremal process for planar Brownian local times; establishing sharp results on the multifractal structure of critical chaos; and developing a direct construction of critical multiplicative chaos that applies to non-Gaussian systems, including those in random matrix theory and number theory.
The proposal is ambitious but timely. The field is undergoing rapid development, and new tools, many of which I helped establish, place these long-standing challenges within reach. My track record in the fellowship so far demonstrates strong mathematical progress and ability to lead collaborative, high-impact research. The renewal will allow me to expand my team, deepen collaborations, and firmly establish the UK as a centre for random geometry.
Two themes drive this programme: the geometry of random surfaces known as Liouville quantum gravity (LQG) and the theory of multiplicative chaos, both considered in their critical, or threshold, regimes. LQG provides a probabilistic model for random two-dimensional geometries, rooted in string theory and conformal field theory, and is believed to capture the large-scale behaviour of discrete random surfaces known as random planar maps (RPMs). Multiplicative chaos is the rigorous mathematical framework for constructing areas associated with these surfaces, and more broadly, for building measures from log-correlated fields, initiated by Mandelbrot to model energy dissipation in intermittent turbulence.
In both areas, the most challenging and least understood regimes are the critical and supercritical phases, where classical theory breaks down and measures become dominated by exceptional “spikes”. It is widely believed, and now known in several cases, that the extremes of systems with logarithmic correlations such as the planar Gaussian free field, branching random walks, and random matrix characteristic polynomials are governed by critical multiplicative chaos. Establishing this universality rigorously and understanding the associated geometry in the (super)critical regime is a key aim of this proposal.
During the first phase of my UKRI FLF, I developed a mathematical framework for critical LQG, established connections with Conformal Loop Ensembles (CLE), and solved a major open problem by proving the first scaling limit for RPMs in the critical universality class. In parallel, I contributed foundational ideas to the theory of critical multiplicative chaos, developed new methods for analysing extremes of log-correlated fields, and co-authored a major book clarifying the area.
The renewal builds on these breakthroughs. Work Package A develops the geometry of random surfaces in the (super)critical regime. This includes: proving convergence of embedded RPMs to critical LQG; constructing the LQG metric on the CLE(4) gasket and linking it to newly constructed scaling limits of maps with large faces; developing a general theory of supercritical LQG surfaces; and deriving the correlation functions of Liouville conformal field theory in the (super)critical phase.
Work Package B focuses on the universality of critical multiplicative chaos and extreme-value phenomena. This includes: identifying the full extremal process for planar Brownian local times; establishing sharp results on the multifractal structure of critical chaos; and developing a direct construction of critical multiplicative chaos that applies to non-Gaussian systems, including those in random matrix theory and number theory.
The proposal is ambitious but timely. The field is undergoing rapid development, and new tools, many of which I helped establish, place these long-standing challenges within reach. My track record in the fellowship so far demonstrates strong mathematical progress and ability to lead collaborative, high-impact research. The renewal will allow me to expand my team, deepen collaborations, and firmly establish the UK as a centre for random geometry.