A universal law for Voronoi cell volumes in infinitely large maps
Résumé
We discuss the volume of Voronoi 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 Voronoi 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.
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