Compact Brownian surfaces I: Brownian disks

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family (BDL,0<L<∞) of random metric spaces homeomorphic to the closed unit disk of R², 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.