We study the scaling limit of uniform random quadrangulations of a connected, orientable and compact surface of a fixed topology, when the number of quadrangles tends to infinity, as well as the perimeters of the boundary faces. The limiting random metric spaces are called the (orientable) compact Brownian surfaces. This result generalises the known cases where the surface is either the sphere or the disk. We use an interpretation of a bijection by Chapuy-Marcus-Schaeffer (generalised by Bettinelli) as a way to decompose a quadrangulation along well-chosen geodesics into a finite number of elementary pieces whose scaling limits are obtained separately, by using in a crucial way the known results for the sphere and the disk, and their extensions to the non-compact settings of the Brownian plane and half-plane This paper presents the main objects and ideas of the proof, as well as some further properties of the law of the Brownian cylinder. This paper and its pictures are mostly based on joint work with Jérémie Bettinelli.
Digital Object Identifier: 10.2969/aspm/08710173