Skip to main content

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

Periodic Reporting for period 2 - GeoBrown (Brownian geometry: at the interface between probability theory, combinatorics and mathematical physics.)

Reporting period: 2018-11-01 to 2020-04-30

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 map. 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 map, 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). A third objective is to study random processes in random geometry, and in particular random walks on large random planar maps: It has been proved that these random walks have a subdiffusive behavior and thus behave very differently from random walks on deterministic lattices, but many related questions are still open. Finally, a last objective, of major interest in mathematical physics, is to study statistical physics models, such as percolation or the Ising model, in random geometry.
Several problems that were raised in the description of the action at the beginning of the project have been tackeld 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. A subsequent article relying on similar methods has studied a remarkable growth-fragmentation process that arises when looking at the boundary sizes of the connected components of the set of points of a Brownian disk whose distance to the boundary is greater than a fixed quantity. 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. Concerning random walks on random planar maps, the subdiffusivity property that was known to hold in the universality class of the Brownian map has been extended to random planar maps with very large faces. On the other hand, random walk on certain hyperbolic models of random geometry has been shown to exhibit a ballistic behavior. 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. Other aspects of random surfaces have been studied, and in particular surfaces produced from an arbitrary sequence of polygons by pairing sides uniformly at random have been shown to exhibit universal properties.
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. Generalizing the Brownian map limit to first-passage percolation distances on quadrangulations or general planar maps has significantly enlarged the universality class of the Brownian map. Similarly the subdiffusivity property of random walk on random planar maps is a natural and important question, and extending this property is a question of major interest. Concerning planar maps in high genus, the explicit derivation of the local limit is an achievement of primary importance in the area.
In the time remaining until the end of the project, the team will continue along the same lines of research. A problem that seems feasible is to extend the new construction of the Brownian disk to the so-called Brownian half-plane, which is also a very natural model of random geometry. One expects that a number of explicit distributions can be computed for this model in a more tractable way than for the Brownian disk. Still in the area of continuous models, a fundamental question on which one may expect significant progress is to prove the uniqueness of the so-called stable maps, which are scaling limits of random planar maps with large faces. It took a long time to arrive at the uniqueness of the Brownian map, and one believes that the case of stable maps will be even more difficult, but still not intractable. Certain members of the team are working on a multi-vertex extension of the peeling process, which is a powerful tool allowing to discover a random planar map one face after another starting from a given root vertex. Such an extension would presumably give access to new interesting properties of discrete models of random geometry. The team is also interested in the general area of hyperbolic random surfaces in the spirit of the work of the Fields medallist Maryam Mirzakhani. Finally, a last topic on which some progress could be expected is the study of random graphs associated with models of random geometry in higher dimensions: such models have already been introduced in the physics literature but until now they have not led to interesting scaling limits.
conformal embedding of a triangulation
simulation of random walk on a stable planar map
simulation of a large triangulation
simulation of percolation on a large quadrangulation