Scaling limits and extreme values of Gibbs measures

In the area of probability, an increasingly important role has been played in recent years by random systems in which the randomness is observed in the spatial structure. Random systems defined on lattices have been introduced as discrete models that describe phase transitions for various phenomena, ranging from liquid in porous media to the spread of disease. Our understanding of some of these models, such as percolation and Ising model, has been improved greatly in the last decades, and works around it have led to Fields medals in 2006 and 2010.

The aim of the proposed research is to open new directions for several long standing open questions in random systems on lattices. One circle of the questions concern the gradient Gibbs measures, which is a model of random surface introduced in the 1970s by Brascamp, Lebowitz and Lieb as a model for crystal interfaces. A long standing universality conjecture states that the large scale statistical properties of these random surfaces behave like a Gaussian free field. This has been partially confirmed by the work of Naddaf and Spencer (and others). The PI intends to improve the understanding of the gradient Gibbs measures, by quantifying the existing fluctuation theorems, settling the 20-year-old conjectures in surface tension (that describes the energy of a surface profile with a global tilt), and to establish some universality conjectures of the extremes of log-correlated fields.

The second circle of questions concern the XY and the Villain models, which are mathematical models of liquid crystals, liquid helium and superconductors. Works around it have led to the Nobel Prize in Physics (Kosterlitz and Thouless) in 2016. Physicists predict that at low temperature the large scale property of these models are closely related to the Gaussian free field. This is known as the Gaussian spin wave conjecture. Some mathematical progress was made towards the conjecture in the 1970s and the early 1980s, building around the works of Frohlich, Simon and Spencer. However, methods developed in these papers (infrared bounds and Coulomb gas renormalization) were not sufficient to complete the proof of this conjecture. The PI intends to resolve this long-standing Gaussian spin wave conjecture for the XY and the Villain models in dimension three and higher.

In doing so, the PI will develop a robust framework to study the scaling limits, fluctuations and large deviations of a large class of Gibbs measures. New bridges will be built between probability, statistical mechanics and mathematical analysis.

The outcome of the proposed research has the potential to benefit a wider set of practitioners. For example, the Kosterlitz-Thouless transition has been applied to superconductor-insulator transitions as well as the quantum Hall effect. So in the long run, the work can be useful to scientists working at finding new generation materials. The two recently funded CDTs MathSys and HETSYS at Warwick with their broad supervisor pool from a number of departments provide a natural basis for interaction with the physics community, to discuss concrete applications of the conceptual insights from my work. Both of these CDTs also provide great opportunities to supervise postgraduate students that are interested in physical and real world applications.

The proposed project can also be useful to statistical machine learning, which has emerged as a powerful tool for a wide variety of tasks such as computer vision, social network filtering, medical diagnosis, and so on. A central theme in machine learning is to study optimization in high dimensional landscapes, and it is usually studied using stochastic gradient decent. The tools developed in this project can be used to improve and quantify these optimization methods. I plan to discuss with the Turing Fellows at Warwick concrete applications in machine learning and neural networks, and to reach out to the people from the artificial intelligence industry (such as DeepMind) through the events at the Turing Institute, in order to identify potential industry applications in the long run.

Besides, the quantitative scaling limit theorems in this proposal will help understand the precise convergence rate for the Markov Chain Monte Carlo (MCMC) algorithms. We have a leading research group in MCMC at Warwick, with extensive research activities range from statistical methodology to real world applications (e.g., genetic data interpretation, or inference of disease risk). I will discuss with members of the group about potential methodological applications, so that in the long run, it may eventually be useful to the biomedical applications.

In order to disseminate my results among the widest possible audiences, I will do the following:

- Continue publishing my work in top mathematical journals (such as Communications in Pure and Applied Mathematics, Annals of Probability, Communications in Mathematical Physics, Probability Theory and Related Field).

- Write expository papers aiming at inform, stimulate and enlighten a wider audience. For example I have recently finished an expository paper and submitted to Bulletin of the American Mathematical Society.

- Present my work at relevant international conferences (Conference on Stochastic Processes and Applications, EquaDiff,
International Congress on Mathematical Physics, etc.), specialised workshops and seminars, as well as give talks at other
departments and at the university open days

- Organise an international workshop on lattice models and Gaussian Free Field at Warwick. I will reserve slots and funds to allow young researchers to participate and present their work.

- Continue co-organising the Warwick Probability Seminar. Invite established researchers to give seminars and mini-courses, open to all graduate students working in the broad area of probability.

In order to promote the interest of new generations, I start to get in touch with the Further Mathematics Centre at Warwick, which provides exciting range of activities to bring mathematics closer to GCSE and A-level students students.