A marked scheme representation for Voronoi volumes in the Brownian map

Source abstract

For any fixed number $k\ge3$ of independent volume-sampled sites in the unit-area Brownian map, we represent the negative-power transform of their ordinary Voronoi volumes as a finite sum of marked planar scheme integrals. These integrals arise from the critical asymptotics of generating functions for face-constrained labelled planar maps. To connect them to Voronoi volumes, we use Miermont's zero-delay correspondence to interpret the labelled maps as quadrangulations with distinguished sites and then pass to the Brownian map scaling limit.