Continuous gradient interfaces with disorder

Continuous gradient models are natural generalizations to higher d-dimensional time of the standard random walk and have drawn a lot of attention lately. Partly, this is due to the fact that the contour lines of their interface height converge in d=2 to Schramm's SLE - a family of random planar curves shown to be the universal scaling limit of many important two-dimensional lattice models in probability and statistical mechanics (2006 Fields Medal for Werner). Moreover, gradient models are connected to random interlacements, a novel probability area pioneered by Sznitman, to reinforced random walks, and to Liouville quantum gravity.

Informally, the random interface is given by highly-dependent real-valued random variables whose distribution is a function of the nearest-neighbour interactions V of the interface. In the case with V a quadratic function, this distribution is a Gaussian measure - the Gaussian Free Field (GFF) - the d-dimensional time analog of Brownian motion.

The classic gradient model assumes a smooth medium, i.e. without disorder. However, most phenomena in nature exhibit some disorder due to impurities entering the systems or to materials which have defects or inhomogeneities. In this proposal, we will mainly explore the effects of disorder on continuous gradient models which is an almost unchartered territory mathematically. I will seek to answer questions such as whether the addition of a small amount of disorder modifies the nature of the phase transitions of the underlying homogeneous gradient model, i.e. if disorder is relevant, I will aim to identify non-standard phase transitions, to find new instances of universality behaviour, and to create connections between gradients and other models with disorder by taking questions from d=1 (polymers) to the next level d>1 (gradients), e.g. quenched vs annealed free energy.

Applied Probability has always been driven by practical problems: ruin theory, introduced by Swedish actuary Filip Lundberg, uses probabilistic models to describe an insurer's vulnerability to insolvency/ruin, and statistical physics uses methods of probability theory and statistics to deal with large populations and approximations.
Modern challenges to probability and statistics are just as motivated by practical concerns such as the current need for statisticians to develop methods that are scalable to huge datasets, yet also flexible enough
to utilize data of different types and from different sources. In recognition of such challenges, the
EPSRC has identified Statistics and Applied Probability as an area that it wishes to grow further.
The UK has historically been at the forefront of statistics and probabilistic research, from Thomas Bayes, UCL's Karl Pearson, Geoffrey Grimmett etc. The study of interface models and of models with disorder is strongly represented in the UK, with among others, names such as Stefan Adams, David Croydon, Andrew Wade and Nikolaous Zygouras, experts whose research was recently funded by EPSRC.

The
development of tools for gradient models with disorder will not only give rise to novel interesting mathematical behaviour, but will also ensure that the UK remains a leader in this field. Moreover, the programme reaches across other areas of probability, such as random polymer models, random walks in random environments, and has the potential of developing tools useful for models with disorder in general.