Compact Brownian surfaces II. Orientable surfaces

TL;DR

This paper proves that large uniform bipartite quadrangulations of orientable surfaces with boundary converge to a new class of random metric spaces called Brownian surfaces, extending known results from planar cases.

Contribution

It generalizes convergence results of random quadrangulations from planar to arbitrary orientable surfaces with boundary, introducing Brownian surfaces as limits.

Findings

Convergence of quadrangulations to Brownian surfaces in Gromov--Hausdorff--Prokhorov topology.

Extension of Brownian map and disk convergence to general orientable surfaces.

Development of cutting techniques for Brownian surfaces into elementary components.

Abstract

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 $n\to\infty$, 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.