Abstract

Co-author: Curien, Nicolas (Universit Paris-Sud)

We discuss some recent results concerning the geometry of discrete and continuous random planar maps. In the continuous setting, we consider the so-called Brownian plane, which is an infinite-volume version of the Brownian map and is conjectured to be the universal scaling limit of many discrete random lattices such as the UIPT (uniform infinite planar triangulation) or the UIPQ (uniform infinite planar quadrangulation). The hull of radius r in the Brownian plane is obtained by filling in the holes in the ball of radius r centered at the distinguished point. We obtain a complete description of the process of hull volumes, as well as several explicit formulas for related distributions. In the discrete setting of the UIPT or the UIPQ , we derive similar results via a detailed study of the peeling process already inverstigated by Angel. We also apply our results to first-passage percolation on these infinite random lattices. This is a joint work with Nicolas Curien.