Objective The main purpose of this proposal is to explore the canonical models of planar random geometry that have been introduced in the recent years. We call this theory Brownian geometry because one of the central objects, the Brownian map, arises as the universal scaling limit of many discrete models of large random graphs embedded in the plane, in a way very similar to Brownian motion, which is the continuous limit of many different classes of random paths. The preceding scaling limit holds for the Gromov-Hausdorff distance on compact metric spaces. Furthermore, recent developments show that, in addition to its metric structure, the Brownian map can be equipped with a conformal structure.Our objectives will be to combine the different approaches to develop a systematic study of the Brownian map and its variants called the Brownian disk and the Brownian plane, as well as of the associated discrete models, which are finite graphs embedded in the plane or infinite random lattices such as the uniform infinite planar triangulation. We will also study random phenomena in random geometry, starting with random walks on infinite random lattices, with the ultimate goal of constructing Brownian motion on our continuous models. A question of importance in mathematical physics is to understand the behavior of statistical physics models in random geometry. Another fundamental question is to connect the conformal structure of the Brownian map with the conformal embeddings that are known to exist for discrete planar maps.The field of random geometry gives rise to exceptionally fruitful interactions between specialists of probability theory, theoretical physicists and mathematicians coming from other areas, in particular from combinatorics. To ensure the best chances of success for the proposed research, we will rely on the expertise of several members of the Laboratoire de Mathématiques d'Orsay, and on the unique environment of Université Paris-Sud and neighboring institutions. Fields of science natural sciencesmathematicsapplied mathematicsmathematical physicsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsdiscrete mathematicsgraph theorynatural sciencesmathematicspure mathematicsdiscrete mathematicscombinatoricsnatural sciencesmathematicsapplied mathematicsstatistics and probability Keywords random geometry random planar map triangulation Brownian map conformal structure scaling limit random walk statistical physics model Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2016-ADG - ERC Advanced Grant Call for proposal ERC-2016-ADG See other projects for this call Funding Scheme ERC-ADG - Advanced Grant Host institution UNIVERSITE PARIS-SACLAY Net EU contribution € 1 263 607,00 Address BATIMENT BREGUET - 3 RUE JOLIOT CURIE 91190 Gif-Sur-Yvette France See on map Region Ile-de-France Ile-de-France Essonne Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 263 607,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all UNIVERSITE PARIS-SACLAY France Net EU contribution € 1 263 607,00 Address BATIMENT BREGUET - 3 RUE JOLIOT CURIE 91190 Gif-Sur-Yvette See on map Region Ile-de-France Ile-de-France Essonne Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 263 607,00
