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Brownian geometry: at the interface between probability theory, combinatorics and mathematical physics.

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.

Call for proposal

ERC-2016-ADG
See other projects for this call

Host institution

UNIVERSITE PARIS-SACLAY
Address
Batiment Breguet - 3 Rue Joliot Curie
91190 Gif-sur-yvette
France
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 263 607

Beneficiaries (1)

UNIVERSITE PARIS-SACLAY
France
EU contribution
€ 1 263 607
Address
Batiment Breguet - 3 Rue Joliot Curie
91190 Gif-sur-yvette
Activity type
Higher or Secondary Education Establishments