The general goal of the project is to get a better understanding of random geometry in two dimensions. This area includes both the discrete models called planar maps, which are graphs embedded in the plane or in the two-dimensional sphere, and the continuous models which are scaling limits of the discrete objects and form what may be called Brownian geometry. This domain is at the interface between probability theory and combinatorics and is also related to the areas of physics called 2-dimensional quantum gravity and Liouville quantum gravity. In physics, the case of two dimensions corresponds to a toy model, but having a good understanding of this case should help to tackle the higher dimensions, in particular the cases of 3 and 4 dimensions which are more relevant to quantum gravity but also much more difficult to study from the mathematical point of view. In mathematics, the study of random geometry is very natural in probability theory: In the same way as Brownian motion provides a universal model for a long random path consisting of individual steps chosen independently at random, one finds that the properties of large graphs drawn at random in the plane are represented by a universal model called the Brownian sphere. These questions are also important for specialists of combinatorics who have been interested in planar maps since the work of Tutte in the 1960s in connection with the famous four color theorem.
A major objective of the project is to study the basic objects of Brownian geometry, which are the Brownian sphere, the Brownian disk and the Brownian plane, as well as the relations existing between these objects. The construction of these models from the stochastic process called Brownian motion indexed by the Brownian tree indicates that a number of calculations, concerning for instance the distribution of the volume of balls or the length of separating cycles, should be feasible and lead to explicit formulas. Another objective is to extend as much as possible the universality class of the Brownian map, that is, the class of all discrete models whose asymptotic properties are described by the Brownian map. It is also of interest to study discrete models that lie outside this universality class, e.g. random planar maps with very large faces (these models have attracted the interest of the physics community).
Overall, the work of the project team has led to a much better understanding of the properties of these fascinating models of random geometry. In particular, surprising relations have been discovered between the different compact or non-compact models of Brownian surfaces. Asymptotic properties of large random planar maps (graphs drawn on surfaces) have also been investigated in connection with Brownian surfaces.
