Periodic Reporting for period 4 - GeoBrown (Brownian geometry: at the interface between probability theory, combinatorics and mathematical physics.)
Berichtszeitraum: 2021-11-01 bis 2023-04-30
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.
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.
Several problems that were raised in the description of the action at the beginning of the project have been tackled successfully. In particular, a new construction of the Brownian disk, which gives better insight into the properties of this model, has been developed from the excursion theory for Brownian motion indexed by the Brownian tree. This construction made it possible to show that various subsets of the Brownian map, such as connected components of the complement of metric balls, are Brownian disks, thus providing a complete solution to a problem raised in the project. In the direction of extending the universality class of the Brownian map, it has been shown that the scaling limit of large random quadrangulations or of large random general planar maps equipped with a first-passage percolation distance is again the Brownian map. A by-product of the method is the fact that the classical Tutte bijection between quadrangulations and general planar maps is almost isometric in large scales. In the area of statistical physics models on planar maps, percolation on Boltzmann models of planar maps depending on a real parameter has been studied: for these models, the percolation cluster of the origin has been shown to be itself a random planar map with a well-identified parameter different from the one of the underlying graph. Random geometry in higher genus has also been studied by several members of the team. In particular, one of the major open questions raised in the project has been solved by showing that the local limits of large triangulations in high genus are the so-called planar stochastic hyperbolic triangulations. The team has also made some important progress in the study of non-compact Brownian surfaces: a unified approach involving a spine decomposition has been developed for the three main models of non-compact Brownian surfaces. On the other hand, the volume measure on these (compact or non-compact) Brownian surfaces has been shown to coincide with a suitable Hausdorff measure, thus showing that the volume measure is completely determined by the volume measure. Exceptional points of Brownian surfaces have also been studied, and in particular the Hausdorff dimension of the so-called geodesic stars (terminal points of disjoint geodesics) has been computed.
All these results have been presented by the different members of the team in international conferences around the world. In the direction of young researchers, the PI has given a course on Brownian geometry at the 2022 Vancouver Summer School in Probability, and the senior scientist of the team (Nicolas Curien) has given a series of lectures on random planar maps at the famous Saint-Flour Summer School in Probability in 2019.
All these results have been presented by the different members of the team in international conferences around the world. In the direction of young researchers, the PI has given a course on Brownian geometry at the 2022 Vancouver Summer School in Probability, and the senior scientist of the team (Nicolas Curien) has given a series of lectures on random planar maps at the famous Saint-Flour Summer School in Probability in 2019.
The work performed from the beginning of the project goes beyond the state of the art in several directions. Among these, the new construction of the Brownian disk provides a powerful tool to study this random metric space and yields means of investigation that were not available in earlier approaches. Several versions of the so-called spatial Markov property in Brownian geometry have been derived and have found applications to isoperimetric inequalities. Detailed information about almost sure properties of the Brownian sphere has been obtained, and in particular the Hausdorff dimension of the exceptional points called geodesic stars has been computed. A surprising Markov property has been derived for the pair consisiting of the local time of Brownian motion indexed by the Brownian tree and its derivative, with immediate applications to the evolution of the size of spheres in Brownian geometry. In the direction of universality, the Brownian sphere limit has been extended to first-passage percolation distances (meaning that edges have a random length) on quadrangulations or general planar maps, thus enlarghing significantly the universality class of the Brownian sphere. Concerning planar maps in high genus, the explicit derivation of the local limit is an achievement of primary importance in the area.