A universal law for Voronoï cell volumes in infinitely large maps

Abstract

We discuss the volume of Voronoï cells defined by two marked vertices picked randomly at a fixed given mutual distance 2s in random planar quadrangulations. We consider the regime where the mutual distance 2s is kept finite while the total volume of the quadrangulation tends to infinity. In this regime, exactly one of the Voronoï cells keeps a finite volume, which scales as s4 for large s. We analyze the universal probability distribution of this, properly rescaled, finite volume and present an explicit formula for its Laplace transform.