Compact Brownian Surfaces II

Fix an arbitrary compact orientable surface with a boundary and consider a uniform bipartite random quadrangulation of this surface with $n$ faces and boundary component lengths of order $\sqrt{n}$ or of lower order. Endow this quadrangulation with the usual graph metric renormalized by $n^{-1/4}$, mark it on each boundary component, and endow it with the counting measure on its vertex set renormalized by $n^{-1}$, as well as the counting measure on each boundary component renormalized by $n^{-1/2}$. We show that, as $n$ goes to infinity, this random marked measured metric space converges in distribution for the Gromov–Hausdorff–Prokhorov topology, toward a random limiting marked measured metric space called a Brownian surface.

This extends known convergence results of uniform random planar quadrangulations with at most one boundary component toward the Brownian sphere and toward the Brownian disk, by considering the case of quadrangulations on general compact orientable surfaces. Our approach consists in cutting a Brownian surface into elementary pieces that are naturally related to the Brownian sphere and the Brownian disk and their noncompact analogs, the Brownian plane and the Brownian half-plane, and to prove convergence results for these elementary pieces, which are of independent interest.