  In a recent paper, Chapuy conjectured that, for any positive integer k, the
law for the fractions of total area covered by the k Voronoi cells defined by k
points picked uniformly at random in the Brownian map of any fixed genus is the
same law as that of a uniform k-division of the unit interval. For k=2, i.e.
with two points chosen uniformly at random, it means that the law for the ratio
of the area of one of the two Voronoi cells by the total area of the map is
uniform between 0 and 1. Here, by a direct computation of the desired law, we
show that this latter conjecture for k=2 actually holds in the case of large
planar (genus 0) quadrangulations as well as for large general planar maps
(i.e. maps whose faces have arbitrary degrees). This corroborates Chapuy's
conjecture in its simplest realizations.
