Title: Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk,
Abstract: We prove that the free Boltzmann quadrangulation with simple boundary and
fixed perimeter, equipped with its graph metric, natural area measure, and the
path which traces its boundary converges in the scaling limit to the free
Boltzmann Brownian disk. The topology of convergence is the so-called
Gromov-Hausdorff-Prokhorov-uniform (GHPU) topology, the natural analog of the
Gromov-Hausdorff topology for curve-decorated metric measure spaces. From this
we deduce that a random quadrangulation of the sphere decorated by a $2l$-step
self-avoiding loop converges in law in the GHPU topology to the random
curve-decorated metric measure space obtained by gluing together two
independent Brownian disks along their boundaries.