Walks in Random Media, Stochastic Growth and Pinning Effects.

A cornerstone of probability theory has been the establishment of the Law of Large Numbers and the Central Limit Theorem, both of them having impact beyond mathematical sciences. Roughly speaking, the sum of n independent, identically distributed variables with finite second moment is macroscopically of order n and has fluctuations of order n^{1/2}, obeying the Gaussian distribution. Underlying the Gaussian fluctuations is the linear dependence of the sum on the collection of the independent variables. However, most phenomena in nature exhibit a nonlinear dependence on the inherent randomness and the challenge is to (i) understand the nonlinear structure that propagates the randomness and (ii) reveal the universal features of this mechanism.

Random walks in random media are widely used to model such phenomena in statistical physics. Two such instances that are receiving increasingly high attention are (A) stochastic growth models and (B) pinning models on defect lines.

In case (A) one deals with a randomly growing interface. The non rigorous work of Kardar-Parisi-Zhang (KPZ) in the mid 80's set the framework of what is currently known as the KPZ universality class, by predicting that this class of models exhibits t^{1/3} fluctuations. More recent mathematical works have related, in special cases, the fluctuations of such systems to those coming from the theory of random matrices. Our goal is to build a rigorous mathematical theory that will explain the nature of these fluctuations by looking into the exactly solvable nature of these models, connect it to other mathematical fields and eventually perturb it in order to reveal universal phenomena.

In case (B) one deals with a random walk in the vicinity of a defect line. The goal is to understand phase transitions related to localization and delocalization phenomena. Techniques related to large deviations and coarse graining have been used recently to study the phase diagrams of such phenomena. While progress has been made a number of important questions remain unresolved.

We propose to provide a new path in the field through the construction of continuum limits of such models. In this way we aim to resolve the open questions and also make deep and novel connections to KPZ phenomena.

The impact of my work will be primarily and directly felt by probabilists. The problems with which I will engage lie in the centre of attention in modern probability. I plan to contribute novel methodologies that bring together ideas from different directions within probability. Furthermore, my proposed methods promise to bring into probability ideas from other areas of mathematics such as random matrix theory, tropical combinatorics, representation theory and number theory. Vice versa it is my aim to develop these external ideas within probability and the framework of my work and eventually return them to those areas. In this way the impact of my work is likely to be felt by mathematicians outside probability.

My work is in Applied probability, motivated by physical phenomena such as growth models and pinning phenomena. Although my primary goal is to use this motivation as a springboard to develop novel mathematics, experience has shown that high quality mathematics will eventually find their way into applied sciences. Twenty-five years after the non-rigorous, theoretical physics work of Kardar-Parisi-Zhang (KPZ) on the fluctuations of randomly growing surfaces, experimentalists were motivated by the rigorous follow up work by mathematicians to search for and detect the precise type of fluctuations in liquid crystals. It is therefore possible that, in the very long term, my work could be of interest to more applied scientists. So, although any impact outside mathematics is only likely to occur well after the project has finished, nevertheless, I intend to initiate pathways which would allow me to explore possibilities. Steps that I will take are

(a) a workshop that will bring together scientists from different fields,

(b) publish in high quality journals and participate in international conferences, which go beyond the scope of my core mathematics.

(c) enhance my engagement with the Complexity Science Centre at Warwick and the EPSRC funded Doctoral Training Centre embedded within it, which provides opportunities to establish new contacts in other disciplines,
for example through the seminar programme,

(d) broaden my work in nurturing young talent through supervising multidisciplinary MSc/PhD students both through the DTC and through my overseas contacts.