# Random Fields of Gradients

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

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.

### Planned Impact

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.

## People |
## ORCID iD |

Stefan Adams (Principal Investigator) |

### Publications

Adams S
(2011)

*From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage*in Communications in Mathematical Physics
Adams S
(2011)

*A variational formula for the free energy of an interacting many-particle system*in The Annals of Probability
Adams S
(2016)

*Representation and Poly-time Approximation for Pressure of $$\mathbb {Z}^2$$ Lattice Models in the Non-uniqueness Region*in Journal of Statistical Physics
Adams S
(2013)

*Large deviations and gradient flows.*in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Adams S
(2016)

*Sample path large deviations for Laplacian models in $(1+1)$-dimensions*in Electronic Journal of Probability
Adams S
(2015)

*Phase Transitions in Delaunay Potts Models*in Journal of Statistical Physics
Adams S
(2013)

*Finite range decomposition for families of gradient Gaussian measures*in Journal of Functional Analysis
Stefan Adams
(2010)

*Gradient Models with Non-Convex Interactions*Description | The project focused on the mathematical challenge to study highly dependent random variables when their distributions are given in terms of non-convex interactions. During the period novel mathematical techniques have been developed to address the objectives of the project. The idea of these techniques, called finite range decompositions, is basically to slice the underlying distributions in more tractable sub-distributions. Expectations of families of random-variables amount to perform the integrations in several sufficiently controlled integration steps. This achievement of the finite range decomposition has been published recently and has put the whole project on track concerning non-convex interactions with its connections to scaling limits to Gaussian Free Fields, correlations, Cauchy-Born rule and level lines. Using this achievement in conjunction with a multi-scale analysis lead to the following results. It turned out that the free energy and the correlations have Gaussian behavior despite the fact that the interactions are non-convex. The Gaussian decay has been proven by the PI and are an important step to fully understand the Cauchy-Born regime for materials. The latter problem has been addressed in the scalar-valued case, and the outcomes of the project show that the vector case can be achieved in the near future. The strict convexity of the free energy has been shown for a class of non-convex interactions and has been submitted to 'Memoirs of the American Mathematical Society'. The natural extension to the vector-valued case is currently an ongoing work to be finalized within the next months. The third major focus of the project are the connections of the random field of gradients with the continuum Gaussian Free Field and their level-lines. A first result has been obtained here in showing that the distributions with non-convex interactions scale to the Gaussian Free Field. This has long been expected but the proof in the non-convex interaction regime required completely different techniques and methods as the ones used to prove the same for convex interactions. The study of these scaling limits has opened new insights and ideas which have been initialized during the summer school in Brazil 2010 studying results on conformal invariance by Smirnov (field medal 2010). Recently the PI has observed connections of the project to random interlacements - a novel research line going back to Alain Snitzman. The outcomes of this project provide input in that research as well as the new concepts will help in studying the scaling limits of the level lines and the percolation of the level sets of the random field of gradients under non-convex interaction distributions. Another new research outcome is the idea to obtain an infinite version of the Laplace integral method. In this method integrals are basically given by solving certain variational problems of the exponent function in the integrand. This new research has recently begun during the PI's visit to UBC Vancouver and is geared to tackle open problems of the Cauchy-Born rule, e.g., the question of the uniqueness of Gibbs measures. This kind of research is the seed of a larger project to be continued next year when the PI spends his sabbatical year at UBC Vancouver. In this project the PI is collaborating with Brydges and Slade from UBC and their student Bauerschmidt. The latter one is a strong PhD student which the PI aims to get over to the UK as a Postdoctoral researcher at Warwick. The research activity of the project has initialized a couple of new collaborations (UBC in Vancouver; IMPA in Rio de Janeiro; MPI in Leipzig; TU-Berlin) and new ideas and follow-up projects (scaling to Gaussian Free Field, percolation of level sets; and random interlacements). In summary the project has delivered first important steps towards understanding the role of non-convex interactions and has initiated highly promising follow-up projects and further ideas to be worked on. |

Exploitation Route | At this stage there is no direct potential use in non-academic context. However, it is expected that a further development can be used later in material sciences. |

Sectors | Education,Other |

Description | EPSRC CDT MASDOC |

Amount | £4,500 (GBP) |

Organisation | University of Warwick |

Sector | Academic/University |

Country | United Kingdom |

Start | 04/2015 |

End | 05/2015 |

Description | EPSRC platform grant |

Amount | £15,000 (GBP) |

Organisation | University of Warwick |

Sector | Academic/University |

Country | United Kingdom |

Start | 04/2016 |

End | 05/2016 |

Description | Focused Research Group |

Amount | £7,450 (GBP) |

Organisation | Heilbronn Institute for Mathematical Research |

Sector | Academic/University |

Country | United Kingdom |

Start | 02/2017 |

End | 01/2018 |

Description | International Exchange Scheme |

Amount | £5,970 (GBP) |

Organisation | The Royal Society |

Sector | Charity/Non Profit |

Country | United Kingdom |

Start | 08/2013 |

End | 08/2014 |

Description | International Partnership fund |

Amount | £3,275 (GBP) |

Organisation | University of Warwick |

Sector | Academic/University |

Country | United Kingdom |

Start | 06/2013 |

End | 07/2014 |

Description | Sabbatical year grant |

Amount | $10,000 (CAD) |

Funding ID | Canada Research grant Prof David Brydges |

Organisation | University of British Columbia |

Sector | Academic/University |

Country | Canada |

Start | 06/2013 |

End | 08/2014 |

Description | University of British Columbia (UBC) |

Amount | £2,500 (GBP) |

Funding ID | Prof David Brydges |

Organisation | University of British Columbia |

Sector | Academic/University |

Country | Canada |

Start | 04/2012 |

End | 07/2012 |

Title | Finite range decomposition |

Description | We have developed a novel finite range decomposition for Gaussian measures which can be used for any future renormalisation group approach to critical systems in probability. In particular we have established the tool for dealing with any random field of gradient models. Our approach is general and works in all dimensions and for the vector-valued case. |

Type Of Material | Improvements to research infrastructure |

Year Produced | 2014 |

Provided To Others? | Yes |

Impact | We have presented our tools at UBC Vancouver where a group of probabilists is dealing with critical phenomena and they will use our methods in their future renormalisation group approaches to the self-avoiding random walk and other related models. |

URL | http://dx.doi.org/10.1016/j.jfa.2012.10.006 |