Random Geometry, 2022
The area of random geometry has known several important developments in the last ten years. This area lies at the boundary between probability theory, combinatorics and theoretical physics, and gives rise to fruitful interactions between researchers coming from these different fields. In particular, the study of planar maps, which are graphs drawn on the sphere, has been of interest for long both in combinatorics, in connection with the celebrated four color theorem, and in theoretical physics, in the setting of two-dimensional quantum gravity. The continuous model called the Brownian map has been shown to be the universal scaling limit of large planar maps chosen uniformly at random in a suitable combinatorial class such as the class of triangulations. A program connecting the Brownian map with Liouville quantum gravity has been developed in a series of papers of Miller and Sheffield. More recently, the work of several authors has shown that the Liouville quantum gravity metric can be defined directly from the two-dimensional Gaussian free field. On the other hand, fascinating results have been obtained for maps on surfaces of higher genus, especially when the genus tends to infinity. Still many questions remain open, concerning in particular the extension of these models to higher dimensions, or the conformal embeddings of random planar maps. The workshop will gather the most prominent specialists coming from probability theory, combinatorics and theoretical physics, who will present the recent achievements and discuss further developments.
