无限大映射中Voronoi胞体积的一个普适律
A universal law for Voronoi cell volumes in infinitely large maps
摘要：
我们讨论了在随机平面四边形中以固定的给定相互距离2s随机选取的两个标记顶点所定义的Voronoi单元的体积。我们考虑的制度，其中相互距离2s是有限的，而四边形的总体积趋于无穷大。在这种情况下，恰好有一个Voronoi单元保持有限的体积，对于大的s，其体积的尺度为s^4。我们分析了这种普遍的概率分布，适当重新标度，有限体积，并提出了一个明确的公式，其拉普拉斯变换。
英文摘要：
We discuss the volume of Voronoi 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 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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