Compact Brownian surfaces I: Brownian disks

Abstract

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family (BD_L, 0 < L < infinity) of random metric spaces homeomorphic to the closed unit disk of R^2, the space BD_L being called the Brownian disk of perimeter L and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangulations to the Brownian map, which intuitively corresponds to the limit case where L = 0. Similar results are obtained for maps following a Boltzmann distribution, in which the perimeter is fixed but the area is random.