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

Total Cost €


EC-Contrib. €






Project "GeoBrown" data sheet

The following table provides information about the project.


There are not information about this coordinator. Please contact Fabio for more information, thanks.

 Coordinator Country France [FR]
 Total cost 1˙263˙607 €
 EC max contribution 1˙263˙607 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2016-ADG
 Funding Scheme ERC-ADG
 Starting year 2017
 Duration (year-month-day) from 2017-05-01   to  2022-04-30


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    UNIVERSITE PARIS-SACLAY FR (SAINT AUBIN) coordinator 1˙263˙607.00


 Project 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.


year authors and title journal last update
List of publications.
2020 Sébastien Martineau, Franco Severo
Strict monotonicity of percolation thresholds under covering maps
published pages: , ISSN: 0091-1798, DOI:
The Annals of Probability 2020-03-11
2020 Nicolas Curien, Tom Hutchcroft, Asaf Nachmias
Geometric and spectral properties of causal maps
published pages: , ISSN: 1435-9855, DOI:
Journal of the European Mathematical Society 2020-03-11
2018 Nicolas Curien, Cyril Marzouk
How fast planar maps get swallowed by a peeling process
published pages: , ISSN: 1083-589X, DOI: 10.1214/18-ecp123
Electronic Communications in Probability 23 2020-03-11
2020 Jean-François Le Gall, Armand Riera
Some explicit distributions for Brownian motion indexed by the Brownian tree
published pages: , ISSN: 1024-2953, DOI:
Markov Processes and Related Fields 2020-03-11
2019 Thomas Budzinski
Supercritical causal maps: geodesics and simple random walk
published pages: , ISSN: 1083-6489, DOI: 10.1214/19-EJP341
Electronic Journal of Probability 24/0 2020-03-11
2019 Thomas Budzinski, Nicolas Curien, Bram Petri
Universality for random surfaces in unconstrained genus
published pages: , ISSN: 1077-8926, DOI:
The Electronic Journal of Combinatorics 26(4) 2020-03-11
2019 Jean-François Le Gall
Brownian geometry
published pages: 135-174, ISSN: 0289-2316, DOI: 10.1007/s11537-019-1821-7
Japanese Journal of Mathematics 14/2 2020-03-11
2020 Nicolas Curien, Loïc Richier
Duality of random planar maps via percolation
published pages: , ISSN: 1777-5310, DOI:
Annales de l\'Institut Fourier 2020-03-11
2020 Jean-François Le Gall, Armand Riera
Growth-fragmentation processes in Brownian motion indexed by the Brownian tree
published pages: , ISSN: 0091-1798, DOI:
The Annals of Probability 2020-03-11
2018 Jean-François Le Gall
Subordination of trees and the Brownian map
published pages: 819-864, ISSN: 0178-8051, DOI: 10.1007/s00440-017-0794-9
Probability Theory and Related Fields 171/3-4 2020-03-11
2019 Jean-François Le Gall
Brownian disks and the Brownian snake
published pages: 237-313, ISSN: 0246-0203, DOI: 10.1214/18-aihp882
Annales de l\'Institut Henri Poincaré, Probabilités et Statistiques 55/1 2020-03-11

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