Random Fields of Gradients

Random fields of gradients are a class of model systems arising in the studies of random interfaces, random geometry, field theory, and elasticity theory. These random objects pose challenging problems for probabilists as even an a priori distribution involves strong correlations. The ultimate aim of the proposal is to make significant progress for the open problem of non-convex interactions and to explore new connections; e.g., the connection between the level lines of the Gaussian Free Field and Schramm's SLE, and the mathematical justification of the Cauchy-Born rule in elasticity theory. Random fields of gradients are families of highly correlated random variables arising in the studies of e.g. random surfaces and discrete Gaussian Free Fields (harmonic crystal). Recently their study has attained a lot of attention. There are several reasons for that. On one hand, these are approximations of critical systems and natural models for a macroscopic description of elastic systems as well as, in a different setting, for fluctuating phase interfaces. In addition, over continuum, the level lines of the Gaussian Free Field are connected to Schramm's SLE (an active field of modern mathematics---Fields Medal in 2006--for understanding critical phenomena) and the fields are natural space-time analog of Brownian motions and as such a simple random object of widespread application and great intrinsic beauty. Gradient fields are likely to be an universal class of models combining probability, analysis and physics in the study of critical phenomena, and these mass-less fields are also a starting point for many constructions in field theory. The random fields of gradients emerge in the following areas, in models for effective random interfaces and critical phenomena, in stochastic geometry and Gaussian Free Field, and in elasticity theory (Cauchy-Born rule).The proposal focus on the open problems in these areas when the interaction functions of the gradient models are non-convex. This includes the study of free energy limits, uniqueness and (non-) uniqueness of gradient Gibbs measures, scaling limits to Gaussian Free Fields and possible non-Gaussian Free Fields, and proof of the Cauchy-Born rule at positive temperature.

The primary beneficiary of the proposed high quality basic research is the probability community. The area of the proposed research has been of interest to both probabilists and mathematical physicists throughout the last decades, and more recently also to analysts. In mathematics, these models have been studied in three disciplines - probability theory, mathematical physics and analysis - with rather disjointed goals and technical means. The proposed research activities will strengthen the role probaility (random walk representations, scaling limits, large deviations, random geometry) has in our basic understanding of critical phenomena of systems of high complexity. The increased importance is recognised by the Field Medal in 2006 devoted to Schramm's SLE, a new conceptual framework for understanding critical phenomena and an area close to the proposed research objectives. The random fields of gradients are mathematical models arising also in a variety of applied contexts. Hence, a further impact will be - at a later stage after the period of funding - applications of the anticipated results of the proposed research. One is theoretical physics, where they are believed to capture the large-scale behavior of microscopic fluctuating interfaces or magnetic materials at the so-called Curie temperature point. Another use of these models comes from variational analysis (material science), where they describe the displacement of microscopic constituents of a piece of material that is deformed by stress/shear forces on its boundary. Direct beneficiary is the applicant as the grant will allow him to make significant steps in his development, to explore a new area of research, and to enhance his skills and knowledge and as well as his visibility and recognition towards a leading role in the new research field of the proposal. This in turn will improve the visibility and leading role of basic UK research. The engagement to target the anticipated impacts consists in publishing original research papers in leading international journals, organising international workshops, providing preprints on international preprint servers, delivering national and international talks, and presentations at workshops, conferences, as well as publishing proceedings. The BIRS (Banff Interantional Research Station) workshop on 'Gradient Random Fields' in May 2011 and the workshop 'Random Fields of Gradients' at Warwick in May 2012 will make the results widely known among probabilists, mathematical physicists and mathematicians interestested in the field and foster interactions between these groups. The latter workshop at Warwick will allow the results of the proposed research to be disseminated. Furthermore, these meetings will showcase UK research in these fields. A lasting impact are the proceedings for the BIRS workshop the applicant plans to publish with a major university press. The communities of probability, random geometry, and renormalisation group techniques will benefit from the two international workshops. The applicant has a world-wide network of collaborators in topics close to the project ensuring that a critical mass of researchers is reached and guarantees a quick dissemination of the results. Furthermore, these collaborations are of utmost importance for the success of the proposed research as well the reach to the wider mathematical community. This proposal will enable the applicant and henceforth the UK research community to assume a leading role in that field. Finally, this will lead to further collaborations, and to further networks, be they formal (EPSRC-funded) or informal, and to UK-lead EU-applications.