A universal law for Voronoï cell volumes in infinitely large maps. (8th January 2018) - Guitter, Emmanuel
Abstract: We discuss the volume of Voronoï cells defined by two marked vertices picked randomly at a fixed given mutual distance 2 s in random planar quadrangulations. We consider the regime where the mutual distance 2 s 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 s 4 for large s . We analyze the universal probability distribution of this, properly rescaled, finite volume and present an explicit formula for its Laplace transform.